Article EUCLMETR, MML version 4.99.1005
:: EUCLMETR:attrnot 1 => EUCLMETR:attr 1
definition
let a1 be non empty OrtAfSp-like ParOrtStr;
attr a1 is Euclidean means
for b1, b2, b3, b4 being Element of the carrier of a1
st b1,b2 _|_ b3,b4 & b2,b3 _|_ b1,b4
holds b2,b4 _|_ b1,b3;
end;
:: EUCLMETR:dfs 1
definiens
let a1 be non empty OrtAfSp-like ParOrtStr;
To prove
a1 is Euclidean
it is sufficient to prove
thus for b1, b2, b3, b4 being Element of the carrier of a1
st b1,b2 _|_ b3,b4 & b2,b3 _|_ b1,b4
holds b2,b4 _|_ b1,b3;
:: EUCLMETR:def 1
theorem
for b1 being non empty OrtAfSp-like ParOrtStr holds
b1 is Euclidean
iff
for b2, b3, b4, b5 being Element of the carrier of b1
st b2,b3 _|_ b4,b5 & b3,b4 _|_ b2,b5
holds b3,b5 _|_ b2,b4;
:: EUCLMETR:attrnot 2 => EUCLMETR:attr 2
definition
let a1 be non empty OrtAfSp-like ParOrtStr;
attr a1 is Pappian means
Af a1 is Pappian;
end;
:: EUCLMETR:dfs 2
definiens
let a1 be non empty OrtAfSp-like ParOrtStr;
To prove
a1 is Pappian
it is sufficient to prove
thus Af a1 is Pappian;
:: EUCLMETR:def 2
theorem
for b1 being non empty OrtAfSp-like ParOrtStr holds
b1 is Pappian
iff
Af b1 is Pappian;
:: EUCLMETR:attrnot 3 => EUCLMETR:attr 3
definition
let a1 be non empty OrtAfSp-like ParOrtStr;
attr a1 is Desarguesian means
Af a1 is Desarguesian;
end;
:: EUCLMETR:dfs 3
definiens
let a1 be non empty OrtAfSp-like ParOrtStr;
To prove
a1 is Desarguesian
it is sufficient to prove
thus Af a1 is Desarguesian;
:: EUCLMETR:def 3
theorem
for b1 being non empty OrtAfSp-like ParOrtStr holds
b1 is Desarguesian
iff
Af b1 is Desarguesian;
:: EUCLMETR:attrnot 4 => EUCLMETR:attr 4
definition
let a1 be non empty OrtAfSp-like ParOrtStr;
attr a1 is Fanoian means
Af a1 is Fanoian;
end;
:: EUCLMETR:dfs 4
definiens
let a1 be non empty OrtAfSp-like ParOrtStr;
To prove
a1 is Fanoian
it is sufficient to prove
thus Af a1 is Fanoian;
:: EUCLMETR:def 4
theorem
for b1 being non empty OrtAfSp-like ParOrtStr holds
b1 is Fanoian
iff
Af b1 is Fanoian;
:: EUCLMETR:attrnot 5 => EUCLMETR:attr 5
definition
let a1 be non empty OrtAfSp-like ParOrtStr;
attr a1 is Moufangian means
Af a1 is Moufangian;
end;
:: EUCLMETR:dfs 5
definiens
let a1 be non empty OrtAfSp-like ParOrtStr;
To prove
a1 is Moufangian
it is sufficient to prove
thus Af a1 is Moufangian;
:: EUCLMETR:def 5
theorem
for b1 being non empty OrtAfSp-like ParOrtStr holds
b1 is Moufangian
iff
Af b1 is Moufangian;
:: EUCLMETR:attrnot 6 => EUCLMETR:attr 6
definition
let a1 be non empty OrtAfSp-like ParOrtStr;
attr a1 is translation means
Af a1 is translational;
end;
:: EUCLMETR:dfs 6
definiens
let a1 be non empty OrtAfSp-like ParOrtStr;
To prove
a1 is translation
it is sufficient to prove
thus Af a1 is translational;
:: EUCLMETR:def 6
theorem
for b1 being non empty OrtAfSp-like ParOrtStr holds
b1 is translation
iff
Af b1 is translational;
:: EUCLMETR:attrnot 7 => EUCLMETR:attr 7
definition
let a1 be non empty OrtAfSp-like ParOrtStr;
attr a1 is Homogeneous means
for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
st b1,b2 _|_ b1,b3 & b1,b4 _|_ b1,b5 & b1,b6 _|_ b1,b7 & b2,b4 _|_ b3,b5 & b2,b6 _|_ b3,b7 & not b1,b6 // b1,b2 & not b1,b2 // b1,b4
holds b4,b6 _|_ b5,b7;
end;
:: EUCLMETR:dfs 7
definiens
let a1 be non empty OrtAfSp-like ParOrtStr;
To prove
a1 is Homogeneous
it is sufficient to prove
thus for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
st b1,b2 _|_ b1,b3 & b1,b4 _|_ b1,b5 & b1,b6 _|_ b1,b7 & b2,b4 _|_ b3,b5 & b2,b6 _|_ b3,b7 & not b1,b6 // b1,b2 & not b1,b2 // b1,b4
holds b4,b6 _|_ b5,b7;
:: EUCLMETR:def 7
theorem
for b1 being non empty OrtAfSp-like ParOrtStr holds
b1 is Homogeneous
iff
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
st b2,b3 _|_ b2,b4 & b2,b5 _|_ b2,b6 & b2,b7 _|_ b2,b8 & b3,b5 _|_ b4,b6 & b3,b7 _|_ b4,b8 & not b2,b7 // b2,b3 & not b2,b3 // b2,b5
holds b5,b7 _|_ b6,b8;
:: EUCLMETR:th 1
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4 being Element of the carrier of b1
st not LIN b2,b3,b4
holds b2 <> b3 & b3 <> b4 & b2 <> b4;
:: EUCLMETR:th 2
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Element of bool the carrier of b1
st b2,b3 _|_ b6 & b4,b5 _|_ b6
holds b2,b3 // b4,b5 & b2,b3 // b5,b4;
:: EUCLMETR:th 3
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of bool the carrier of b1
st b2 <> b3 & (b2,b3 _|_ b5 or b3,b2 _|_ b5) & b2,b3 _|_ b4
holds b5 // b4;
:: EUCLMETR:th 4
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4 being Element of the carrier of b1
st LIN b2,b3,b4
holds LIN b2,b4,b3 & LIN b3,b2,b4 & LIN b3,b4,b2 & LIN b4,b2,b3 & LIN b4,b3,b2;
:: EUCLMETR:th 5
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4 being Element of the carrier of b1
st not LIN b2,b3,b4
holds ex b5 being Element of the carrier of b1 st
b5,b2 _|_ b3,b4 & b5,b3 _|_ b2,b4;
:: EUCLMETR:th 6
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st not LIN b2,b3,b4 & b5,b2 _|_ b3,b4 & b5,b3 _|_ b2,b4 & b6,b2 _|_ b3,b4 & b6,b3 _|_ b2,b4
holds b5 = b6;
:: EUCLMETR:th 7
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4, b5 being Element of the carrier of b1
st b2,b3 _|_ b4,b5 & b3,b4 _|_ b2,b5 & LIN b2,b3,b4 & b2 <> b4 & b2 <> b3
holds b3 = b4;
:: EUCLMETR:th 8
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
b1 is Euclidean
iff
b1 is satisfying_3H;
:: EUCLMETR:th 9
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
b1 is Homogeneous
iff
b1 is satisfying_ODES;
:: EUCLMETR:th 10
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
b1 is Pappian
iff
b1 is satisfying_PAP;
:: EUCLMETR:th 11
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
b1 is Desarguesian
iff
b1 is satisfying_DES;
:: EUCLMETR:th 12
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
b1 is Moufangian
iff
b1 is satisfying_TDES;
:: EUCLMETR:th 13
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
b1 is translation
iff
b1 is satisfying_des;
:: EUCLMETR:th 14
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
st b1 is Homogeneous
holds b1 is Desarguesian;
:: EUCLMETR:th 15
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
st b1 is Euclidean & b1 is Desarguesian
holds b1 is Pappian;
:: EUCLMETR:th 16
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st not LIN b2,b3,b5 & b2 <> b4 & b2,b3 _|_ b2,b4 & b2,b5 _|_ b2,b6 & b2,b5 _|_ b2,b7 & b3,b5 _|_ b4,b6 & b3,b5 _|_ b4,b7
holds b6 = b7;
:: EUCLMETR:th 17
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4, b5 being Element of the carrier of b1
st not LIN b2,b3,b5 & b2 <> b4
holds ex b6 being Element of the carrier of b1 st
b2,b5 _|_ b2,b6 & b3,b5 _|_ b4,b6;
:: EUCLMETR:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7 being Element of REAL
st Gen b2,b3 & 0. b1 <> b4 & 0. b1 <> b5 & b4,b5 are_Ort_wrt b2,b3 & b4 = (b6 * b2) + (b7 * b3)
holds ex b8 being Element of REAL st
b8 <> 0 &
b5 = ((b8 * b7) * b2) + ((- (b8 * b6)) * b3);
:: EUCLMETR:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st Gen b2,b3 & 0. b1 <> b4 & 0. b1 <> b5 & b4,b5 are_Ort_wrt b2,b3
holds ex b6 being Element of REAL st
for b7, b8 being Element of REAL holds
(b7 * b2) + (b8 * b3),((b6 * b8) * b2) + ((- (b6 * b7)) * b3) are_Ort_wrt b2,b3 &
((b7 * b2) + (b8 * b3)) - b4,(((b6 * b8) * b2) + ((- (b6 * b7)) * b3)) - b5 are_Ort_wrt b2,b3;
:: EUCLMETR:th 21
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4 being Element of the carrier of b2
st Gen b3,b4 & b1 = AMSpace(b2,b3,b4)
holds b1 is satisfying_LIN;
:: EUCLMETR:th 22
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
st b2,b3 _|_ b2,b4 & b2,b5 _|_ b2,b6 & b2,b7 _|_ b2,b8 & b3,b5 _|_ b4,b6 & b3,b7 _|_ b4,b8 & not b2,b7 // b2,b3 & not b2,b3 // b2,b5 & b2 = b4
holds b5,b7 _|_ b6,b8;
:: EUCLMETR:th 23
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4 being Element of the carrier of b2
st Gen b3,b4 & b1 = AMSpace(b2,b3,b4)
holds b1 is Homogeneous;
:: EUCLMETR:exreg 1
registration
cluster non empty OrtAfSp-like OrtAfPl-like Euclidean Pappian Desarguesian Fanoian Moufangian translation Homogeneous ParOrtStr;
end;
:: EUCLMETR:exreg 2
registration
cluster non empty OrtAfSp-like Euclidean Pappian Desarguesian Fanoian Moufangian translation Homogeneous ParOrtStr;
end;
:: EUCLMETR:th 24
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3, b4 being Element of the carrier of b2
st Gen b3,b4 & b1 = AMSpace(b2,b3,b4)
holds b1 is non empty OrtAfPl-like Euclidean Pappian Desarguesian Fanoian Moufangian translation Homogeneous ParOrtStr;
:: EUCLMETR:funcreg 1
registration
let a1 be non empty OrtAfPl-like Pappian ParOrtStr;
cluster Af a1 -> strict Pappian;
end;
:: EUCLMETR:funcreg 2
registration
let a1 be non empty OrtAfPl-like Desarguesian ParOrtStr;
cluster Af a1 -> strict Desarguesian;
end;
:: EUCLMETR:funcreg 3
registration
let a1 be non empty OrtAfPl-like Moufangian ParOrtStr;
cluster Af a1 -> strict Moufangian;
end;
:: EUCLMETR:funcreg 4
registration
let a1 be non empty OrtAfPl-like translation ParOrtStr;
cluster Af a1 -> strict translational;
end;
:: EUCLMETR:funcreg 5
registration
let a1 be non empty OrtAfPl-like Fanoian ParOrtStr;
cluster Af a1 -> strict Fanoian;
end;
:: EUCLMETR:funcreg 6
registration
let a1 be non empty OrtAfPl-like Homogeneous ParOrtStr;
cluster Af a1 -> strict Desarguesian;
end;
:: EUCLMETR:funcreg 7
registration
let a1 be non empty OrtAfPl-like Euclidean Desarguesian ParOrtStr;
cluster Af a1 -> strict Pappian;
end;
:: EUCLMETR:funcreg 8
registration
let a1 be non empty OrtAfSp-like Pappian ParOrtStr;
cluster Af a1 -> strict Pappian;
end;
:: EUCLMETR:funcreg 9
registration
let a1 be non empty OrtAfSp-like Desarguesian ParOrtStr;
cluster Af a1 -> strict Desarguesian;
end;
:: EUCLMETR:funcreg 10
registration
let a1 be non empty OrtAfSp-like Moufangian ParOrtStr;
cluster Af a1 -> strict Moufangian;
end;
:: EUCLMETR:funcreg 11
registration
let a1 be non empty OrtAfSp-like translation ParOrtStr;
cluster Af a1 -> strict translational;
end;
:: EUCLMETR:funcreg 12
registration
let a1 be non empty OrtAfSp-like Fanoian ParOrtStr;
cluster Af a1 -> strict Fanoian;
end;