Theorem ecopovtrn 6626

Description: Assuming that operation F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation .~, specified by the first hypothesis, is transitive. (Contributed by NM, 11-Feb-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypotheses

Ref	Expression
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 ) ) ) }
ecopopr.com	|- ( x .+ y ) = ( y .+ x )
ecopopr.cl	|- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S )
ecopopr.ass	|- ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) )
ecopopr.can	|- ( ( x e. S /\ y e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) )

Assertion

Ref	Expression
ecopovtrn	|- ( ( A .~ B /\ B .~ C ) -> A .~ C )

Distinct variable groups:   x, y, z, w, v, u, .+   x, S, y, z, w, v, u

Allowed substitution hints:   A( x, y, z, w, v, u)   B( x, y, z, w, v, u)   C( x, y, z, w, v, u)   .~ ( x, y, z, w, v, u)

Proof of Theorem ecopovtrn

Dummy variables f g h t s r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ecopopr.1	. . . . . . 7 |- .~ = { <. 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		opabssxp 4697	. . . . . . 7 |- { <. 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 ) )
3	1, 2	eqsstri 3187	. . . . . 6 |- .~ C_ ( ( S X. S ) X. ( S X. S ) )
4	3	brel 4675	. . . . 5 |- ( A .~ B -> ( A e. ( S X. S ) /\ B e. ( S X. S ) ) )
5	4	simpld 112	. . . 4 |- ( A .~ B -> A e. ( S X. S ) )
6	3	brel 4675	. . . 4 |- ( B .~ C -> ( B e. ( S X. S ) /\ C e. ( S X. S ) ) )
7	5, 6	anim12i 338	. . 3 |- ( ( A .~ B /\ B .~ C ) -> ( A e. ( S X. S ) /\ ( B e. ( S X. S ) /\ C e. ( S X. S ) ) ) )
8		3anass 982	. . 3 |- ( ( 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 ) ) ) )
9	7, 8	sylibr 134	. 2 |- ( ( A .~ B /\ B .~ C ) -> ( A e. ( S X. S ) /\ B e. ( S X. S ) /\ C e. ( S X. S ) ) )
10		eqid 2177	. . 3 |- ( S X. S ) = ( S X. S )
11		breq1 4003	. . . . 5 |- ( <. f , g >. = A -> ( <. f , g >. .~ <. h , t >. <-> A .~ <. h , t >. ) )
12	11	anbi1d 465	. . . 4 |- ( <. f , g >. = A -> ( ( <. f , g >. .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) <-> ( A .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) ) )
13		breq1 4003	. . . 4 |- ( <. f , g >. = A -> ( <. f , g >. .~ <. s , r >. <-> A .~ <. s , r >. ) )
14	12, 13	imbi12d 234	. . 3 |- ( <. 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 >. ) ) )
15		breq2 4004	. . . . 5 |- ( <. h , t >. = B -> ( A .~ <. h , t >. <-> A .~ B ) )
16		breq1 4003	. . . . 5 |- ( <. h , t >. = B -> ( <. h , t >. .~ <. s , r >. <-> B .~ <. s , r >. ) )
17	15, 16	anbi12d 473	. . . 4 |- ( <. h , t >. = B -> ( ( A .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) <-> ( A .~ B /\ B .~ <. s , r >. ) ) )
18	17	imbi1d 231	. . 3 |- ( <. h , t >. = B -> ( ( ( A .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) -> A .~ <. s , r >. ) <-> ( ( A .~ B /\ B .~ <. s , r >. ) -> A .~ <. s , r >. ) ) )
19		breq2 4004	. . . . 5 |- ( <. s , r >. = C -> ( B .~ <. s , r >. <-> B .~ C ) )
20	19	anbi2d 464	. . . 4 |- ( <. s , r >. = C -> ( ( A .~ B /\ B .~ <. s , r >. ) <-> ( A .~ B /\ B .~ C ) ) )
21		breq2 4004	. . . 4 |- ( <. s , r >. = C -> ( A .~ <. s , r >. <-> A .~ C ) )
22	20, 21	imbi12d 234	. . 3 |- ( <. s , r >. = C -> ( ( ( A .~ B /\ B .~ <. s , r >. ) -> A .~ <. s , r >. ) <-> ( ( A .~ B /\ B .~ C ) -> A .~ C ) ) )
23	1	ecopoveq 6624	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( <. f , g >. .~ <. h , t >. <-> ( f .+ t ) = ( g .+ h ) ) )
24	23	3adant3 1017	. . . . . . 7 |- ( ( ( 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 ) ) )
25	1	ecopoveq 6624	. . . . . . . 8 |- ( ( ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( <. h , t >. .~ <. s , r >. <-> ( h .+ r ) = ( t .+ s ) ) )
26	25	3adant1 1015	. . . . . . 7 |- ( ( ( 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 ) ) )
27	24, 26	anbi12d 473	. . . . . 6 |- ( ( ( 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 ) ) ) )
28		oveq12 5878	. . . . . . 7 |- ( ( ( f .+ t ) = ( g .+ h ) /\ ( h .+ r ) = ( t .+ s ) ) -> ( ( f .+ t ) .+ ( h .+ r ) ) = ( ( g .+ h ) .+ ( t .+ s ) ) )
29		simp2l 1023	. . . . . . . . 9 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> h e. S )
30		simp2r 1024	. . . . . . . . 9 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> t e. S )
31		simp1l 1021	. . . . . . . . 9 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> f e. S )
32		ecopopr.com	. . . . . . . . . 10 |- ( x .+ y ) = ( y .+ x )
33	32	a1i 9	. . . . . . . . 9 |- ( ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )
34		ecopopr.ass	. . . . . . . . . 10 |- ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) )
35	34	a1i 9	. . . . . . . . 9 |- ( ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
36		simp3r 1026	. . . . . . . . 9 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> r e. S )
37		ecopopr.cl	. . . . . . . . . 10 |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S )
38	37	adantl 277	. . . . . . . . 9 |- ( ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
39	29, 30, 31, 33, 35, 36, 38	caov411d 6054	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( ( h .+ t ) .+ ( f .+ r ) ) = ( ( f .+ t ) .+ ( h .+ r ) ) )
40		simp1r 1022	. . . . . . . . . 10 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> g e. S )
41		simp3l 1025	. . . . . . . . . 10 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> s e. S )
42	40, 30, 29, 33, 35, 41, 38	caov411d 6054	. . . . . . . . 9 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( ( g .+ t ) .+ ( h .+ s ) ) = ( ( h .+ t ) .+ ( g .+ s ) ) )
43	40, 30, 29, 33, 35, 41, 38	caov4d 6053	. . . . . . . . 9 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( ( g .+ t ) .+ ( h .+ s ) ) = ( ( g .+ h ) .+ ( t .+ s ) ) )
44	42, 43	eqtr3d 2212	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( ( h .+ t ) .+ ( g .+ s ) ) = ( ( g .+ h ) .+ ( t .+ s ) ) )
45	39, 44	eqeq12d 2192	. . . . . . 7 |- ( ( ( 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 .+ t ) .+ ( h .+ r ) ) = ( ( g .+ h ) .+ ( t .+ s ) ) ) )
46	28, 45	syl5ibr 156	. . . . . 6 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( ( ( f .+ t ) = ( g .+ h ) /\ ( h .+ r ) = ( t .+ s ) ) -> ( ( h .+ t ) .+ ( f .+ r ) ) = ( ( h .+ t ) .+ ( g .+ s ) ) ) )
47	27, 46	sylbid 150	. . . . 5 |- ( ( ( 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 ) ) ) )
48		ecopopr.can	. . . . . . . . 9 |- ( ( x e. S /\ y e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) )
49	48	3adant3 1017	. . . . . . . 8 |- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) )
50		oveq2 5877	. . . . . . . 8 |- ( y = z -> ( x .+ y ) = ( x .+ z ) )
51	49, 50	impbid1 142	. . . . . . 7 |- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x .+ y ) = ( x .+ z ) <-> y = z ) )
52	51	adantl 277	. . . . . 6 |- ( ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) = ( x .+ z ) <-> y = z ) )
53	37	caovcl 6023	. . . . . . 7 |- ( ( h e. S /\ t e. S ) -> ( h .+ t ) e. S )
54	29, 30, 53	syl2anc 411	. . . . . 6 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( h .+ t ) e. S )
55	37	caovcl 6023	. . . . . . 7 |- ( ( f e. S /\ r e. S ) -> ( f .+ r ) e. S )
56	31, 36, 55	syl2anc 411	. . . . . 6 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( f .+ r ) e. S )
57	38, 40, 41	caovcld 6022	. . . . . 6 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( g .+ s ) e. S )
58	52, 54, 56, 57	caovcand 6031	. . . . 5 |- ( ( ( 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 ) ) )
59	47, 58	sylibd 149	. . . 4 |- ( ( ( 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 ) ) )
60	1	ecopoveq 6624	. . . . 5 |- ( ( ( f e. S /\ g e. S ) /\ ( s e. S /\ r e. S ) ) -> ( <. f , g >. .~ <. s , r >. <-> ( f .+ r ) = ( g .+ s ) ) )
61	60	3adant2 1016	. . . 4 |- ( ( ( 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 ) ) )
62	59, 61	sylibrd 169	. . 3 |- ( ( ( 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 >. ) )
63	10, 14, 18, 22, 62	3optocl 4701	. 2 |- ( ( A e. ( S X. S ) /\ B e. ( S X. S ) /\ C e. ( S X. S ) ) -> ( ( A .~ B /\ B .~ C ) -> A .~ C ) )
64	9, 63	mpcom 36	1 |- ( ( A .~ B /\ B .~ C ) -> A .~ C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   E.wex 1492   e. wcel 2148   <.cop 3594   class class class wbr 4000   {copab 4060   X. cxp 4621  (class class class)co 5869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4629  df-iota 5174  df-fv 5220  df-ov 5872

This theorem is referenced by:  ecopover  6627