Step |
Hyp |
Ref |
Expression |
1 |
|
ecopopr.1 |
|- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) } |
2 |
|
ecopopr.com |
|- ( x .+ y ) = ( y .+ x ) |
3 |
|
ecopopr.cl |
|- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S ) |
4 |
|
ecopopr.ass |
|- ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) |
5 |
|
ecopopr.can |
|- ( ( x e. S /\ y e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) ) |
6 |
|
opabssxp |
|- { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .+ u ) = ( w .+ v ) ) ) } C_ ( ( S X. S ) X. ( S X. S ) ) |
7 |
1 6
|
eqsstri |
|- .~ C_ ( ( S X. S ) X. ( S X. S ) ) |
8 |
7
|
brel |
|- ( A .~ B -> ( A e. ( S X. S ) /\ B e. ( S X. S ) ) ) |
9 |
8
|
simpld |
|- ( A .~ B -> A e. ( S X. S ) ) |
10 |
7
|
brel |
|- ( B .~ C -> ( B e. ( S X. S ) /\ C e. ( S X. S ) ) ) |
11 |
9 10
|
anim12i |
|- ( ( A .~ B /\ B .~ C ) -> ( A e. ( S X. S ) /\ ( B e. ( S X. S ) /\ C e. ( S X. S ) ) ) ) |
12 |
|
3anass |
|- ( ( A e. ( S X. S ) /\ B e. ( S X. S ) /\ C e. ( S X. S ) ) <-> ( A e. ( S X. S ) /\ ( B e. ( S X. S ) /\ C e. ( S X. S ) ) ) ) |
13 |
11 12
|
sylibr |
|- ( ( A .~ B /\ B .~ C ) -> ( A e. ( S X. S ) /\ B e. ( S X. S ) /\ C e. ( S X. S ) ) ) |
14 |
|
eqid |
|- ( S X. S ) = ( S X. S ) |
15 |
|
breq1 |
|- ( <. f , g >. = A -> ( <. f , g >. .~ <. h , t >. <-> A .~ <. h , t >. ) ) |
16 |
15
|
anbi1d |
|- ( <. f , g >. = A -> ( ( <. f , g >. .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) <-> ( A .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) ) ) |
17 |
|
breq1 |
|- ( <. f , g >. = A -> ( <. f , g >. .~ <. s , r >. <-> A .~ <. s , r >. ) ) |
18 |
16 17
|
imbi12d |
|- ( <. f , g >. = A -> ( ( ( <. f , g >. .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) -> <. f , g >. .~ <. s , r >. ) <-> ( ( A .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) -> A .~ <. s , r >. ) ) ) |
19 |
|
breq2 |
|- ( <. h , t >. = B -> ( A .~ <. h , t >. <-> A .~ B ) ) |
20 |
|
breq1 |
|- ( <. h , t >. = B -> ( <. h , t >. .~ <. s , r >. <-> B .~ <. s , r >. ) ) |
21 |
19 20
|
anbi12d |
|- ( <. h , t >. = B -> ( ( A .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) <-> ( A .~ B /\ B .~ <. s , r >. ) ) ) |
22 |
21
|
imbi1d |
|- ( <. h , t >. = B -> ( ( ( A .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) -> A .~ <. s , r >. ) <-> ( ( A .~ B /\ B .~ <. s , r >. ) -> A .~ <. s , r >. ) ) ) |
23 |
|
breq2 |
|- ( <. s , r >. = C -> ( B .~ <. s , r >. <-> B .~ C ) ) |
24 |
23
|
anbi2d |
|- ( <. s , r >. = C -> ( ( A .~ B /\ B .~ <. s , r >. ) <-> ( A .~ B /\ B .~ C ) ) ) |
25 |
|
breq2 |
|- ( <. s , r >. = C -> ( A .~ <. s , r >. <-> A .~ C ) ) |
26 |
24 25
|
imbi12d |
|- ( <. s , r >. = C -> ( ( ( A .~ B /\ B .~ <. s , r >. ) -> A .~ <. s , r >. ) <-> ( ( A .~ B /\ B .~ C ) -> A .~ C ) ) ) |
27 |
1
|
ecopoveq |
|- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( <. f , g >. .~ <. h , t >. <-> ( f .+ t ) = ( g .+ h ) ) ) |
28 |
27
|
3adant3 |
|- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( <. f , g >. .~ <. h , t >. <-> ( f .+ t ) = ( g .+ h ) ) ) |
29 |
1
|
ecopoveq |
|- ( ( ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( <. h , t >. .~ <. s , r >. <-> ( h .+ r ) = ( t .+ s ) ) ) |
30 |
29
|
3adant1 |
|- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( <. h , t >. .~ <. s , r >. <-> ( h .+ r ) = ( t .+ s ) ) ) |
31 |
28 30
|
anbi12d |
|- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( ( <. f , g >. .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) <-> ( ( f .+ t ) = ( g .+ h ) /\ ( h .+ r ) = ( t .+ s ) ) ) ) |
32 |
|
oveq12 |
|- ( ( ( f .+ t ) = ( g .+ h ) /\ ( h .+ r ) = ( t .+ s ) ) -> ( ( f .+ t ) .+ ( h .+ r ) ) = ( ( g .+ h ) .+ ( t .+ s ) ) ) |
33 |
|
vex |
|- h e. _V |
34 |
|
vex |
|- t e. _V |
35 |
|
vex |
|- f e. _V |
36 |
|
vex |
|- r e. _V |
37 |
33 34 35 2 4 36
|
caov411 |
|- ( ( h .+ t ) .+ ( f .+ r ) ) = ( ( f .+ t ) .+ ( h .+ r ) ) |
38 |
|
vex |
|- g e. _V |
39 |
|
vex |
|- s e. _V |
40 |
38 34 33 2 4 39
|
caov411 |
|- ( ( g .+ t ) .+ ( h .+ s ) ) = ( ( h .+ t ) .+ ( g .+ s ) ) |
41 |
38 34 33 2 4 39
|
caov4 |
|- ( ( g .+ t ) .+ ( h .+ s ) ) = ( ( g .+ h ) .+ ( t .+ s ) ) |
42 |
40 41
|
eqtr3i |
|- ( ( h .+ t ) .+ ( g .+ s ) ) = ( ( g .+ h ) .+ ( t .+ s ) ) |
43 |
32 37 42
|
3eqtr4g |
|- ( ( ( f .+ t ) = ( g .+ h ) /\ ( h .+ r ) = ( t .+ s ) ) -> ( ( h .+ t ) .+ ( f .+ r ) ) = ( ( h .+ t ) .+ ( g .+ s ) ) ) |
44 |
31 43
|
syl6bi |
|- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( ( <. f , g >. .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) -> ( ( h .+ t ) .+ ( f .+ r ) ) = ( ( h .+ t ) .+ ( g .+ s ) ) ) ) |
45 |
3
|
caovcl |
|- ( ( h e. S /\ t e. S ) -> ( h .+ t ) e. S ) |
46 |
3
|
caovcl |
|- ( ( f e. S /\ r e. S ) -> ( f .+ r ) e. S ) |
47 |
|
ovex |
|- ( g .+ s ) e. _V |
48 |
47 5
|
caovcan |
|- ( ( ( h .+ t ) e. S /\ ( f .+ r ) e. S ) -> ( ( ( h .+ t ) .+ ( f .+ r ) ) = ( ( h .+ t ) .+ ( g .+ s ) ) -> ( f .+ r ) = ( g .+ s ) ) ) |
49 |
45 46 48
|
syl2an |
|- ( ( ( h e. S /\ t e. S ) /\ ( f e. S /\ r e. S ) ) -> ( ( ( h .+ t ) .+ ( f .+ r ) ) = ( ( h .+ t ) .+ ( g .+ s ) ) -> ( f .+ r ) = ( g .+ s ) ) ) |
50 |
49
|
3impb |
|- ( ( ( h e. S /\ t e. S ) /\ f e. S /\ r e. S ) -> ( ( ( h .+ t ) .+ ( f .+ r ) ) = ( ( h .+ t ) .+ ( g .+ s ) ) -> ( f .+ r ) = ( g .+ s ) ) ) |
51 |
50
|
3com12 |
|- ( ( f e. S /\ ( h e. S /\ t e. S ) /\ r e. S ) -> ( ( ( h .+ t ) .+ ( f .+ r ) ) = ( ( h .+ t ) .+ ( g .+ s ) ) -> ( f .+ r ) = ( g .+ s ) ) ) |
52 |
51
|
3adant3l |
|- ( ( f e. S /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( ( ( h .+ t ) .+ ( f .+ r ) ) = ( ( h .+ t ) .+ ( g .+ s ) ) -> ( f .+ r ) = ( g .+ s ) ) ) |
53 |
52
|
3adant1r |
|- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( ( ( h .+ t ) .+ ( f .+ r ) ) = ( ( h .+ t ) .+ ( g .+ s ) ) -> ( f .+ r ) = ( g .+ s ) ) ) |
54 |
44 53
|
syld |
|- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( ( <. f , g >. .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) -> ( f .+ r ) = ( g .+ s ) ) ) |
55 |
1
|
ecopoveq |
|- ( ( ( f e. S /\ g e. S ) /\ ( s e. S /\ r e. S ) ) -> ( <. f , g >. .~ <. s , r >. <-> ( f .+ r ) = ( g .+ s ) ) ) |
56 |
55
|
3adant2 |
|- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( <. f , g >. .~ <. s , r >. <-> ( f .+ r ) = ( g .+ s ) ) ) |
57 |
54 56
|
sylibrd |
|- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( ( <. f , g >. .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) -> <. f , g >. .~ <. s , r >. ) ) |
58 |
14 18 22 26 57
|
3optocl |
|- ( ( A e. ( S X. S ) /\ B e. ( S X. S ) /\ C e. ( S X. S ) ) -> ( ( A .~ B /\ B .~ C ) -> A .~ C ) ) |
59 |
13 58
|
mpcom |
|- ( ( A .~ B /\ B .~ C ) -> A .~ C ) |