Article YELLOW11, MML version 4.99.1005
:: YELLOW11:th 1
theorem
3 = {0,1,2};
:: YELLOW11:th 2
theorem
2 \ 1 = {1};
:: YELLOW11:th 3
theorem
3 \ 1 = {1,2};
:: YELLOW11:th 4
theorem
3 \ 2 = {2};
:: YELLOW11:th 5
theorem
for b1 being reflexive antisymmetric with_suprema with_infima RelStr
for b2, b3 being Element of the carrier of b1 holds
b2 "/\" b3 = b3
iff
b2 "\/" b3 = b2;
:: YELLOW11:th 6
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 "/\" b3) "\/" (b2 "/\" b4) <= b2 "/\" (b3 "\/" b4);
:: YELLOW11:th 7
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 "\/" (b3 "/\" b4) <= (b2 "\/" b3) "/\" (b2 "\/" b4);
:: YELLOW11:th 8
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2, b3, b4 being Element of the carrier of b1
st b2 <= b4
holds b2 "\/" (b3 "/\" b4) <= (b2 "\/" b3) "/\" b4;
:: YELLOW11:funcnot 1 => YELLOW11:func 1
definition
func N_5 -> RelStr equals
InclPoset {0,3 \ 1,2,3 \ 2,3};
end;
:: YELLOW11:def 1
theorem
N_5 = InclPoset {0,3 \ 1,2,3 \ 2,3};
:: YELLOW11:funcreg 1
registration
cluster N_5 -> strict reflexive transitive antisymmetric;
end;
:: YELLOW11:funcreg 2
registration
cluster N_5 -> with_suprema with_infima;
end;
:: YELLOW11:funcnot 2 => YELLOW11:func 2
definition
func M_3 -> RelStr equals
InclPoset {0,1,2 \ 1,3 \ 2,3};
end;
:: YELLOW11:def 2
theorem
M_3 = InclPoset {0,1,2 \ 1,3 \ 2,3};
:: YELLOW11:funcreg 3
registration
cluster M_3 -> strict reflexive transitive antisymmetric;
end;
:: YELLOW11:funcreg 4
registration
cluster M_3 -> with_suprema with_infima;
end;
:: YELLOW11:th 9
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr holds
ex b2 being full meet-inheriting join-inheriting SubRelStr of b1 st
N_5,b2 are_isomorphic
iff
ex b2, b3, b4, b5, b6 being Element of the carrier of b1 st
b2 <> b3 & b2 <> b4 & b2 <> b5 & b2 <> b6 & b3 <> b4 & b3 <> b5 & b3 <> b6 & b4 <> b5 & b4 <> b6 & b5 <> b6 & b2 "/\" b3 = b2 & b2 "/\" b4 = b2 & b4 "/\" b6 = b4 & b5 "/\" b6 = b5 & b3 "/\" b4 = b2 & b3 "/\" b5 = b3 & b4 "/\" b5 = b2 & b3 "\/" b4 = b6 & b4 "\/" b5 = b6;
:: YELLOW11:th 10
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr holds
ex b2 being full meet-inheriting join-inheriting SubRelStr of b1 st
M_3,b2 are_isomorphic
iff
ex b2, b3, b4, b5, b6 being Element of the carrier of b1 st
b2 <> b3 & b2 <> b4 & b2 <> b5 & b2 <> b6 & b3 <> b4 & b3 <> b5 & b3 <> b6 & b4 <> b5 & b4 <> b6 & b5 <> b6 & b2 "/\" b3 = b2 & b2 "/\" b4 = b2 & b2 "/\" b5 = b2 & b3 "/\" b6 = b3 & b4 "/\" b6 = b4 & b5 "/\" b6 = b5 & b3 "/\" b4 = b2 & b3 "/\" b5 = b2 & b4 "/\" b5 = b2 & b3 "\/" b4 = b6 & b3 "\/" b5 = b6 & b4 "\/" b5 = b6;
:: YELLOW11:attrnot 1 => YELLOW11:attr 1
definition
let a1 be non empty RelStr;
attr a1 is modular means
for b1, b2, b3 being Element of the carrier of a1
st b1 <= b3
holds b1 "\/" (b2 "/\" b3) = (b1 "\/" b2) "/\" b3;
end;
:: YELLOW11:dfs 3
definiens
let a1 be non empty RelStr;
To prove
a1 is modular
it is sufficient to prove
thus for b1, b2, b3 being Element of the carrier of a1
st b1 <= b3
holds b1 "\/" (b2 "/\" b3) = (b1 "\/" b2) "/\" b3;
:: YELLOW11:def 3
theorem
for b1 being non empty RelStr holds
b1 is modular
iff
for b2, b3, b4 being Element of the carrier of b1
st b2 <= b4
holds b2 "\/" (b3 "/\" b4) = (b2 "\/" b3) "/\" b4;
:: YELLOW11:condreg 1
registration
cluster non empty reflexive antisymmetric with_infima distributive -> modular (RelStr);
end;
:: YELLOW11:th 11
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr holds
b1 is modular
iff
for b2 being full meet-inheriting join-inheriting SubRelStr of b1 holds
not N_5,b2 are_isomorphic;
:: YELLOW11:th 12
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
st b1 is modular
holds b1 is distributive
iff
for b2 being full meet-inheriting join-inheriting SubRelStr of b1 holds
not M_3,b2 are_isomorphic;
:: YELLOW11:funcnot 3 => YELLOW11:func 3
definition
let a1 be non empty RelStr;
let a2, a3 be Element of the carrier of a1;
func [#A2,A3#] -> Element of bool the carrier of a1 means
for b1 being Element of the carrier of a1 holds
b1 in it
iff
a2 <= b1 & b1 <= a3;
end;
:: YELLOW11:def 4
theorem
for b1 being non empty RelStr
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1 holds
b4 = [#b2,b3#]
iff
for b5 being Element of the carrier of b1 holds
b5 in b4
iff
b2 <= b5 & b5 <= b3;
:: YELLOW11:attrnot 2 => YELLOW11:attr 2
definition
let a1 be non empty RelStr;
let a2 be Element of bool the carrier of a1;
attr a2 is interval means
ex b1, b2 being Element of the carrier of a1 st
a2 = [#b1,b2#];
end;
:: YELLOW11:dfs 5
definiens
let a1 be non empty RelStr;
let a2 be Element of bool the carrier of a1;
To prove
a2 is interval
it is sufficient to prove
thus ex b1, b2 being Element of the carrier of a1 st
a2 = [#b1,b2#];
:: YELLOW11:def 5
theorem
for b1 being non empty RelStr
for b2 being Element of bool the carrier of b1 holds
b2 is interval(b1)
iff
ex b3, b4 being Element of the carrier of b1 st
b2 = [#b3,b4#];
:: YELLOW11:condreg 2
registration
let a1 be non empty reflexive transitive RelStr;
cluster non empty interval -> directed (Element of bool the carrier of a1);
end;
:: YELLOW11:condreg 3
registration
let a1 be non empty reflexive transitive RelStr;
cluster non empty interval -> filtered (Element of bool the carrier of a1);
end;
:: YELLOW11:funcreg 5
registration
let a1 be non empty RelStr;
let a2, a3 be Element of the carrier of a1;
cluster [#a2,a3#] -> interval;
end;
:: YELLOW11:th 13
theorem
for b1 being non empty reflexive transitive RelStr
for b2, b3 being Element of the carrier of b1 holds
[#b2,b3#] = (uparrow b2) /\ downarrow b3;
:: YELLOW11:funcreg 6
registration
let a1 be reflexive transitive antisymmetric with_infima RelStr;
let a2, a3 be Element of the carrier of a1;
cluster subrelstr [#a2,a3#] -> strict full meet-inheriting;
end;
:: YELLOW11:funcreg 7
registration
let a1 be reflexive transitive antisymmetric with_suprema RelStr;
let a2, a3 be Element of the carrier of a1;
cluster subrelstr [#a2,a3#] -> strict full join-inheriting;
end;
:: YELLOW11:th 14
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2, b3 being Element of the carrier of b1
st b1 is modular
holds subrelstr [#b3,b2 "\/" b3#],subrelstr [#b2 "/\" b3,b2#] are_isomorphic;
:: YELLOW11:exreg 1
registration
cluster non empty finite total reflexive transitive antisymmetric with_suprema with_infima RelStr;
end;
:: YELLOW11:condreg 4
registration
cluster finite reflexive transitive antisymmetric with_infima -> lower-bounded (RelStr);
end;
:: YELLOW11:condreg 5
registration
cluster finite reflexive transitive antisymmetric with_suprema with_infima -> complete (RelStr);
end;