Article TRANSLAC, MML version 4.99.1005
:: TRANSLAC:attrnot 1 => TRANSLAC:attr 1
definition
let a1 be non empty non trivial AffinSpace-like AffinStruct;
attr a1 is Fanoian means
for b1, b2, b3, b4 being Element of the carrier of a1
st b1,b2 // b3,b4 & b1,b3 // b2,b4 & b1,b4 // b2,b3
holds LIN b1,b2,b3;
end;
:: TRANSLAC:dfs 1
definiens
let a1 be non empty non trivial AffinSpace-like AffinStruct;
To prove
a1 is Fanoian
it is sufficient to prove
thus for b1, b2, b3, b4 being Element of the carrier of a1
st b1,b2 // b3,b4 & b1,b3 // b2,b4 & b1,b4 // b2,b3
holds LIN b1,b2,b3;
:: TRANSLAC:def 1
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct holds
b1 is Fanoian
iff
for b2, b3, b4, b5 being Element of the carrier of b1
st b2,b3 // b4,b5 & b2,b4 // b3,b5 & b2,b5 // b3,b4
holds LIN b2,b3,b4;
:: TRANSLAC:prednot 1 => TRANSLAC:attr 1
notation
let a1 be non empty non trivial AffinSpace-like AffinStruct;
synonym a1 satisfies_Fano for Fanoian;
end;
:: TRANSLAC:th 2
theorem
for b1 being non empty non trivial AffinSpace-like AffinStruct
st ex b2, b3, b4 being Element of the carrier of b1 st
LIN b2,b3,b4 & b2 <> b3 & b2 <> b4 & b3 <> b4
for b2, b3 being Element of the carrier of b1
st b2 <> b3
holds ex b4 being Element of the carrier of b1 st
LIN b2,b3,b4 & b2 <> b4 & b3 <> b4;
:: TRANSLAC:th 4
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st b1 is Fanoian & b2,b3 // b4,b5 & b2,b4 // b3,b5 & not LIN b2,b3,b4
holds ex b6 being Element of the carrier of b1 st
LIN b3,b4,b6 & LIN b2,b5,b6;
:: TRANSLAC:th 5
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
st b5 is translation(b1) & not LIN b2,b5 . b2,b3 & b2,b5 . b2 // b3,b4 & b2,b3 // b5 . b2,b4
holds b4 = b5 . b3;
:: TRANSLAC:th 6
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
b1 is translational
iff
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st not LIN b2,b3,b4 & not LIN b2,b3,b5 & b2,b3 // b4,b6 & b2,b3 // b5,b7 & b2,b4 // b3,b6 & b2,b5 // b3,b7
holds b4,b5 // b6,b7;
:: TRANSLAC:th 7
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2 being Element of the carrier of b1 holds
ex b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
b3 is translation(b1) & b3 . b2 = b2;
:: TRANSLAC:th 8
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st (for b6, b7, b8 being Element of the carrier of b1
st b6 <> b7 & LIN b6,b7,b8 & b8 <> b6
holds b8 = b7) &
b2,b3 // b4,b5 &
b2,b4 // b3,b5 &
not LIN b2,b3,b4
holds b2,b5 // b3,b4;
:: TRANSLAC:th 9
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3 being Element of the carrier of b1
st b1 is translational
holds ex b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
b4 is translation(b1) & b4 . b2 = b3;
:: TRANSLAC:th 10
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
st for b2, b3 being Element of the carrier of b1 holds
ex b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
b4 is translation(b1) & b4 . b2 = b3
holds b1 is translational;
:: TRANSLAC:th 11
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
st b3 is translation(b1) & b4 is translation(b1) & not LIN b2,b3 . b2,b4 . b2
holds b3 * b4 = b4 * b3;
:: TRANSLAC:th 12
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
st b1 is translational & b2 is translation(b1) & b3 is translation(b1)
holds b2 * b3 = b3 * b2;
:: TRANSLAC:th 13
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2 being Element of the carrier of b1
for b3, b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
st b3 is translation(b1) & b4 is translation(b1) & b2,b3 . b2 // b2,b4 . b2
holds b2,b3 . b2 // b2,(b3 * b4) . b2;
:: TRANSLAC:th 14
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
st b1 is Fanoian & b1 is translational & b2 is translation(b1)
holds ex b3 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
b3 is translation(b1) & b3 * b3 = b2;
:: TRANSLAC:th 15
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
st b1 is Fanoian & b2 is translation(b1) & b2 * b2 = id the carrier of b1
holds b2 = id the carrier of b1;
:: TRANSLAC:th 16
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
st b1 is translational & b1 is Fanoian & b2 is translation(b1) & b3 is translation(b1) & b4 is translation(b1) & b2 = b3 * b3 & b2 = b4 * b4
holds b3 = b4;