Article ROBBINS2, MML version 4.99.1005
:: ROBBINS2:attrnot 1 => ROBBINS2:attr 1
definition
let a1 be non empty ComplLattStr;
attr a1 is satisfying_DN_1 means
for b1, b2, b3, b4 being Element of the carrier of a1 holds
(((b1 "\/" b2) ` "\/" b3) ` "\/" ((b1 "\/" ((b3 ` "\/" ((b3 "\/" b4) `)) `)) `)) ` = b3;
end;
:: ROBBINS2:dfs 1
definiens
let a1 be non empty ComplLattStr;
To prove
a1 is satisfying_DN_1
it is sufficient to prove
thus for b1, b2, b3, b4 being Element of the carrier of a1 holds
(((b1 "\/" b2) ` "\/" b3) ` "\/" ((b1 "\/" ((b3 ` "\/" ((b3 "\/" b4) `)) `)) `)) ` = b3;
:: ROBBINS2:def 1
theorem
for b1 being non empty ComplLattStr holds
b1 is satisfying_DN_1
iff
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(((b2 "\/" b3) ` "\/" b4) ` "\/" ((b2 "\/" ((b4 ` "\/" ((b4 "\/" b5) `)) `)) `)) ` = b4;
:: ROBBINS2:funcreg 1
registration
cluster TrivComplLat -> strict satisfying_DN_1;
end;
:: ROBBINS2:funcreg 2
registration
cluster TrivOrtLat -> strict satisfying_DN_1;
end;
:: ROBBINS2:exreg 1
registration
cluster non empty join-commutative join-associative satisfying_DN_1 ComplLattStr;
end;
:: ROBBINS2:th 1
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
((b2 "\/" b3) ` "\/" ((((b4 "\/" b5) ` "\/" b2) ` "\/" ((b3 ` "\/" ((b3 "\/" b6) `)) `)) `)) ` = b3;
:: ROBBINS2:th 2
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
((b2 "\/" b3) ` "\/" (((b4 "\/" b2) ` "\/" ((b3 ` "\/" ((b3 "\/" b5) `)) `)) `)) ` = b3;
:: ROBBINS2:th 3
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2 being Element of the carrier of b1 holds
((b2 "\/" (b2 `)) ` "\/" b2) ` = b2 `;
:: ROBBINS2:th 4
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
((b2 "\/" b3) ` "\/" (((b4 "\/" b2) ` "\/" ((((b3 "\/" (b3 `)) ` "\/" b3) ` "\/" ((b3 "\/" b5) `)) `)) `)) ` = b3;
:: ROBBINS2:th 5
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 "\/" b3) ` "\/" (((b4 "\/" b2) ` "\/" b3) `)) ` = b3;
:: ROBBINS2:th 6
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
((b2 "\/" b3) ` "\/" ((b2 ` "\/" b3) `)) ` = b3;
:: ROBBINS2:th 7
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
(((b2 "\/" b3) ` "\/" b2) ` "\/" ((b2 "\/" b3) `)) ` = b2;
:: ROBBINS2:th 8
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
(b2 "\/" (((b2 "\/" b3) ` "\/" b2) `)) ` = (b2 "\/" b3) `;
:: ROBBINS2:th 9
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
(((b2 "\/" b3) ` "\/" b4) ` "\/" ((b2 "\/" b4) `)) ` = b4;
:: ROBBINS2:th 10
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 "\/" (((b3 "\/" b4) ` "\/" ((b3 "\/" b2) `)) `)) ` = (b3 "\/" b2) `;
:: ROBBINS2:th 11
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
((((b2 "\/" b3) ` "\/" b4) ` "\/" ((b2 ` "\/" b3) `)) ` "\/" b3) ` = (b2 ` "\/" b3) `;
:: ROBBINS2:th 12
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 "\/" (((b3 "\/" b4) ` "\/" ((b4 "\/" b2) `)) `)) ` = (b4 "\/" b2) `;
:: ROBBINS2:th 13
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
((b2 "\/" b3) ` "\/" (((b4 "\/" b2) ` "\/" ((b3 ` "\/" ((b5 "\/" b3) `)) `)) `)) ` = b3;
:: ROBBINS2:th 14
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
(b2 "\/" b3) ` = (b3 "\/" b2) `;
:: ROBBINS2:th 15
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
(((b2 "\/" b3) ` "\/" ((b3 "\/" b4) `)) ` "\/" b4) ` = (b3 "\/" b4) `;
:: ROBBINS2:th 16
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 "\/" (((b2 "\/" b3) ` "\/" b4) `)) ` "\/" b4) ` = ((b2 "\/" b3) ` "\/" b4) `;
:: ROBBINS2:th 17
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
(((b2 "\/" b3) ` "\/" b2) ` "\/" b3) ` = (b3 "\/" b3) `;
:: ROBBINS2:th 18
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
(b2 ` "\/" ((b3 "\/" b2) `)) ` = b2;
:: ROBBINS2:th 19
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
((b2 "\/" b3) ` "\/" (b3 `)) ` = b3;
:: ROBBINS2:th 20
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
(b2 "\/" ((b3 "\/" (b2 `)) `)) ` = b2 `;
:: ROBBINS2:th 21
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2 being Element of the carrier of b1 holds
(b2 "\/" b2) ` = b2 `;
:: ROBBINS2:th 22
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
(((b2 "\/" b3) ` "\/" b2) ` "\/" b3) ` = b3 `;
:: ROBBINS2:th 23
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2 being Element of the carrier of b1 holds
b2 ` ` = b2;
:: ROBBINS2:th 24
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
((b2 "\/" b3) ` "\/" b2) ` "\/" b3 = b3 ` `;
:: ROBBINS2:th 25
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
(b2 "\/" b3) ` ` = b3 "\/" b2;
:: ROBBINS2:th 26
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 "\/" (((b3 "\/" b4) ` "\/" ((b3 "\/" b2) `)) `) = (b3 "\/" b2) ` `;
:: ROBBINS2:th 27
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
b2 "\/" b3 = b3 "\/" b2;
:: ROBBINS2:condreg 1
registration
cluster non empty satisfying_DN_1 -> join-commutative (ComplLattStr);
end;
:: ROBBINS2:th 28
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
((b2 + b3) ` + b2) ` + b3 = b3;
:: ROBBINS2:th 29
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
((b2 + b3) ` + b3) ` + b2 = b2;
:: ROBBINS2:th 30
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
b2 + (((b3 + b2) ` + b3) `) = b2;
:: ROBBINS2:th 31
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
(b2 + (b3 `)) ` + ((b3 ` + b3) `) = (b2 + (b3 `)) `;
:: ROBBINS2:th 32
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
(b2 + b3) ` + ((b3 + (b3 `)) `) = (b2 + b3) `;
:: ROBBINS2:th 33
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
(b2 + b3) ` + ((b3 ` + b3) `) = (b2 + b3) `;
:: ROBBINS2:th 34
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
((b2 + (b3 `)) ` ` + b3) ` = (b3 ` + b3) `;
:: ROBBINS2:th 35
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
((b2 + (b3 `)) + b3) ` = (b3 ` + b3) `;
:: ROBBINS2:th 36
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
((((b2 + (b3 `)) + b4) ` + b3) ` + ((b3 ` + b3) `)) ` = b3;
:: ROBBINS2:th 37
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 + (((b3 + b4) ` + ((b3 + b2) `)) `) = b3 + b2;
:: ROBBINS2:th 38
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 + ((b3 + (((b4 + b3) ` + b2) `)) `) = (b4 + b3) ` + b2;
:: ROBBINS2:th 39
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 + (((b3 + b2) ` + ((b3 + b4) `)) `) = b3 + b2;
:: ROBBINS2:th 40
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 + b3) ` + (((b2 + b3) ` + ((b2 + b4) `)) `)) ` + b3 = b3;
:: ROBBINS2:th 41
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
(((b2 + (b3 `)) + b4) ` + b3) ` ` = b3;
:: ROBBINS2:th 42
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 + (((b3 + (b2 `)) + b4) `) = b2;
:: ROBBINS2:th 43
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 ` + (((b3 + b2) + b4) `) = b2 `;
:: ROBBINS2:th 44
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
(b2 + b3) ` + b2 = b2 + (b3 `);
:: ROBBINS2:th 45
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3 being Element of the carrier of b1 holds
(b2 + ((b2 + (b3 `)) `)) ` = (b2 + b3) `;
:: ROBBINS2:th 46
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 + b3) ` + (b2 + b4)) ` + b3 = b3;
:: ROBBINS2:th 47
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
(((b2 + b3) ` + b4) ` + ((b2 ` + b3) `)) ` + b3 = (b2 ` + b3) ` `;
:: ROBBINS2:th 48
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
(((b2 + b3) ` + b4) ` + ((b2 ` + b3) `)) ` + b3 = b2 ` + b3;
:: ROBBINS2:th 49
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 ` + (((b3 + b2) ` ` + (b3 + b4)) `)) ` + (b3 + b4) = (b3 + b2) ` ` + (b3 + b4);
:: ROBBINS2:th 50
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 ` + (((b3 + b2) + (b3 + b4)) `)) ` + (b3 + b4) = (b3 + b2) ` ` + (b3 + b4);
:: ROBBINS2:th 51
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 ` + (((b3 + b2) + (b3 + b4)) `)) ` + (b3 + b4) = (b3 + b2) + (b3 + b4);
:: ROBBINS2:th 52
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 ` ` + (b3 + b4) = (b3 + b2) + (b3 + b4);
:: ROBBINS2:th 53
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 + b3) + (b2 + b4) = b3 + (b2 + b4);
:: ROBBINS2:th 54
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 + b3) + (b2 + b4) = b4 + (b2 + b3);
:: ROBBINS2:th 55
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 + (b3 + b4) = b4 + (b3 + b2);
:: ROBBINS2:th 56
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 + (b3 + b4) = b3 + (b4 + b2);
:: ROBBINS2:th 57
theorem
for b1 being non empty satisfying_DN_1 ComplLattStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 + b3) + b4 = b2 + (b3 + b4);
:: ROBBINS2:condreg 2
registration
cluster non empty satisfying_DN_1 -> join-associative (ComplLattStr);
end;
:: ROBBINS2:condreg 3
registration
cluster non empty satisfying_DN_1 -> Robbins (ComplLattStr);
end;
:: ROBBINS2:th 58
theorem
for b1 being non empty ComplLattStr
for b2, b3 being Element of the carrier of b1
st b1 is join-commutative & b1 is join-associative & b1 is Huntington
holds (b3 "\/" b2) *' (b3 "\/" (b2 `)) = b3;
:: ROBBINS2:th 59
theorem
for b1 being non empty ComplLattStr
st b1 is join-commutative & b1 is join-associative & b1 is Robbins
holds b1 is satisfying_DN_1;
:: ROBBINS2:condreg 4
registration
cluster non empty join-commutative join-associative Robbins -> satisfying_DN_1 (ComplLattStr);
end;
:: ROBBINS2:exreg 2
registration
cluster non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like de_Morgan satisfying_DN_1 OrthoLattStr;
end;
:: ROBBINS2:condreg 5
registration
cluster non empty Lattice-like de_Morgan satisfying_DN_1 -> Boolean (OrthoLattStr);
end;
:: ROBBINS2:condreg 6
registration
cluster non empty Lattice-like Boolean well-complemented -> satisfying_DN_1 (OrthoLattStr);
end;
:: ROBBINS2:attrnot 2 => ROBBINS2:attr 2
definition
let a1 be non empty ComplLattStr;
attr a1 is satisfying_MD_1 means
for b1, b2 being Element of the carrier of a1 holds
(b1 ` "\/" b2) ` "\/" b1 = b1;
end;
:: ROBBINS2:dfs 2
definiens
let a1 be non empty ComplLattStr;
To prove
a1 is satisfying_MD_1
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
(b1 ` "\/" b2) ` "\/" b1 = b1;
:: ROBBINS2:def 2
theorem
for b1 being non empty ComplLattStr holds
b1 is satisfying_MD_1
iff
for b2, b3 being Element of the carrier of b1 holds
(b2 ` "\/" b3) ` "\/" b2 = b2;
:: ROBBINS2:attrnot 3 => ROBBINS2:attr 3
definition
let a1 be non empty ComplLattStr;
attr a1 is satisfying_MD_2 means
for b1, b2, b3 being Element of the carrier of a1 holds
(b1 ` "\/" b2) ` "\/" (b3 "\/" b2) = b2 "\/" (b3 "\/" b1);
end;
:: ROBBINS2:dfs 3
definiens
let a1 be non empty ComplLattStr;
To prove
a1 is satisfying_MD_2
it is sufficient to prove
thus for b1, b2, b3 being Element of the carrier of a1 holds
(b1 ` "\/" b2) ` "\/" (b3 "\/" b2) = b2 "\/" (b3 "\/" b1);
:: ROBBINS2:def 3
theorem
for b1 being non empty ComplLattStr holds
b1 is satisfying_MD_2
iff
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 ` "\/" b3) ` "\/" (b4 "\/" b3) = b3 "\/" (b4 "\/" b2);
:: ROBBINS2:condreg 7
registration
cluster non empty satisfying_MD_1 satisfying_MD_2 -> join-commutative join-associative Huntington (ComplLattStr);
end;
:: ROBBINS2:condreg 8
registration
cluster non empty join-commutative join-associative Huntington -> satisfying_MD_1 satisfying_MD_2 (ComplLattStr);
end;
:: ROBBINS2:exreg 3
registration
cluster non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like de_Morgan satisfying_DN_1 satisfying_MD_1 satisfying_MD_2 OrthoLattStr;
end;
:: ROBBINS2:condreg 9
registration
cluster non empty Lattice-like de_Morgan satisfying_MD_1 satisfying_MD_2 -> Boolean (OrthoLattStr);
end;
:: ROBBINS2:condreg 10
registration
cluster non empty Lattice-like Boolean well-complemented -> satisfying_MD_1 satisfying_MD_2 (OrthoLattStr);
end;