Article HOMOTHET, MML version 4.99.1005
:: HOMOTHET:th 1
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
st not LIN b2,b3,b4 & LIN b2,b3,b5 & LIN b2,b3,b6 & LIN b2,b3,b7 & LIN b2,b4,b8 & LIN b2,b4,b9 & LIN b2,b4,b10 & b4 <> b9 & b3 <> b6 & b2 <> b9 & b2 <> b6 & b3,b4 // b5,b8 & b3,b9 // b5,b10 & b6,b4 // b7,b8 & b1 is Desarguesian
holds b6,b9 // b7,b10;
:: HOMOTHET:th 2
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st for b2, b3, b4 being Element of the carrier of b1
st b2 <> b3 & b2 <> b4 & LIN b2,b3,b4
holds ex b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
b5 is dilatation(b1) & b5 . b2 = b2 & b5 . b3 = b4
holds b1 is Desarguesian;
:: HOMOTHET:th 3
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st b1 is Desarguesian
for b2, b3, b4 being Element of the carrier of b1
st b2 <> b3 & b2 <> b4 & LIN b2,b3,b4
holds ex b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
b5 is dilatation(b1) & b5 . b2 = b2 & b5 . b3 = b4;
:: HOMOTHET:th 4
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is Desarguesian
iff
for b2, b3, b4 being Element of the carrier of b1
st b2 <> b3 & b2 <> b4 & LIN b2,b3,b4
holds ex b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
b5 is dilatation(b1) & b5 . b2 = b2 & b5 . b3 = b4;
:: HOMOTHET:prednot 1 => HOMOTHET:pred 1
definition
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
let a3 be Element of bool the carrier of a1;
pred A2 is_Sc A3 means
a2 is collineation(a1) &
a3 is being_line(a1) &
(for b1 being Element of the carrier of a1
st b1 in a3
holds a2 . b1 = b1) &
(for b1 being Element of the carrier of a1 holds
b1,a2 . b1 // a3);
end;
:: HOMOTHET:dfs 1
definiens
let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
let a3 be Element of bool the carrier of a1;
To prove
a2 is_Sc a3
it is sufficient to prove
thus a2 is collineation(a1) &
a3 is being_line(a1) &
(for b1 being Element of the carrier of a1
st b1 in a3
holds a2 . b1 = b1) &
(for b1 being Element of the carrier of a1 holds
b1,a2 . b1 // a3);
:: HOMOTHET:def 1
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b2 is_Sc b3
iff
b2 is collineation(b1) &
b3 is being_line(b1) &
(for b4 being Element of the carrier of b1
st b4 in b3
holds b2 . b4 = b4) &
(for b4 being Element of the carrier of b1 holds
b4,b2 . b4 // b3);
:: HOMOTHET:th 5
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
for b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
st b4 is_Sc b3 & b4 . b2 = b2 & not b2 in b3
holds b4 = id the carrier of b1;
:: HOMOTHET:th 6
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
st b2,b3 // b4 & not b2 in b4
holds ex b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
b5 is_Sc b4 & b5 . b2 = b3
holds b1 is Moufangian;
:: HOMOTHET:th 7
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
for b10, b11 being Element of bool the carrier of b1
st b10 // b11 & b2 in b10 & b3 in b10 & b4 in b10 & b5 in b10 & b1 is Moufangian & b6 in b11 & b7 in b11 & b8 in b11 & b9 in b11 & b6 <> b7 & b3 <> b2 & b2,b6 // b4,b8 & b2,b7 // b4,b9 & b3,b6 // b5,b8
holds b3,b7 // b5,b9;
:: HOMOTHET:th 8
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
st b2,b3 // b4 & not b2 in b4 & b1 is Moufangian
holds ex b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
b5 is_Sc b4 & b5 . b2 = b3;
:: HOMOTHET:th 9
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is Moufangian
iff
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
st b2,b3 // b4 & not b2 in b4
holds ex b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
b5 is_Sc b4 & b5 . b2 = b3;