Theorem ecopovtrn 7850

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 5193	. . . . . . 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 3635	. . . . . 6 |- .~ C_ ( ( S X. S ) X. ( S X. S ) )
4	3	brel 5168	. . . . 5 |- ( A .~ B -> ( A e. ( S X. S ) /\ B e. ( S X. S ) ) )
5	4	simpld 475	. . . 4 |- ( A .~ B -> A e. ( S X. S ) )
6	3	brel 5168	. . . 4 |- ( B .~ C -> ( B e. ( S X. S ) /\ C e. ( S X. S ) ) )
7	5, 6	anim12i 590	. . 3 |- ( ( A .~ B /\ B .~ C ) -> ( A e. ( S X. S ) /\ ( B e. ( S X. S ) /\ C e. ( S X. S ) ) ) )
8		3anass 1042	. . 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 224	. 2 |- ( ( A .~ B /\ B .~ C ) -> ( A e. ( S X. S ) /\ B e. ( S X. S ) /\ C e. ( S X. S ) ) )
10		eqid 2622	. . 3 |- ( S X. S ) = ( S X. S )
11		breq1 4656	. . . . 5 |- ( <. f , g >. = A -> ( <. f , g >. .~ <. h , t >. <-> A .~ <. h , t >. ) )
12	11	anbi1d 741	. . . 4 |- ( <. f , g >. = A -> ( ( <. f , g >. .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) <-> ( A .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) ) )
13		breq1 4656	. . . 4 |- ( <. f , g >. = A -> ( <. f , g >. .~ <. s , r >. <-> A .~ <. s , r >. ) )
14	12, 13	imbi12d 334	. . 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 4657	. . . . 5 |- ( <. h , t >. = B -> ( A .~ <. h , t >. <-> A .~ B ) )
16		breq1 4656	. . . . 5 |- ( <. h , t >. = B -> ( <. h , t >. .~ <. s , r >. <-> B .~ <. s , r >. ) )
17	15, 16	anbi12d 747	. . . 4 |- ( <. h , t >. = B -> ( ( A .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) <-> ( A .~ B /\ B .~ <. s , r >. ) ) )
18	17	imbi1d 331	. . 3 |- ( <. h , t >. = B -> ( ( ( A .~ <. h , t >. /\ <. h , t >. .~ <. s , r >. ) -> A .~ <. s , r >. ) <-> ( ( A .~ B /\ B .~ <. s , r >. ) -> A .~ <. s , r >. ) ) )
19		breq2 4657	. . . . 5 |- ( <. s , r >. = C -> ( B .~ <. s , r >. <-> B .~ C ) )
20	19	anbi2d 740	. . . 4 |- ( <. s , r >. = C -> ( ( A .~ B /\ B .~ <. s , r >. ) <-> ( A .~ B /\ B .~ C ) ) )
21		breq2 4657	. . . 4 |- ( <. s , r >. = C -> ( A .~ <. s , r >. <-> A .~ C ) )
22	20, 21	imbi12d 334	. . 3 |- ( <. s , r >. = C -> ( ( ( A .~ B /\ B .~ <. s , r >. ) -> A .~ <. s , r >. ) <-> ( ( A .~ B /\ B .~ C ) -> A .~ C ) ) )
23	1	ecopoveq 7848	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( <. f , g >. .~ <. h , t >. <-> ( f .+ t ) = ( g .+ h ) ) )
24	23	3adant3 1081	. . . . . . 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 7848	. . . . . . . 8 |- ( ( ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( <. h , t >. .~ <. s , r >. <-> ( h .+ r ) = ( t .+ s ) ) )
26	25	3adant1 1079	. . . . . . 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 747	. . . . . 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 6659	. . . . . . 7 |- ( ( ( f .+ t ) = ( g .+ h ) /\ ( h .+ r ) = ( t .+ s ) ) -> ( ( f .+ t ) .+ ( h .+ r ) ) = ( ( g .+ h ) .+ ( t .+ s ) ) )
29		vex 3203	. . . . . . . 8 |- h e. _V
30		vex 3203	. . . . . . . 8 |- t e. _V
31		vex 3203	. . . . . . . 8 |- f e. _V
32		ecopopr.com	. . . . . . . 8 |- ( x .+ y ) = ( y .+ x )
33		ecopopr.ass	. . . . . . . 8 |- ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) )
34		vex 3203	. . . . . . . 8 |- r e. _V
35	29, 30, 31, 32, 33, 34	caov411 6866	. . . . . . 7 |- ( ( h .+ t ) .+ ( f .+ r ) ) = ( ( f .+ t ) .+ ( h .+ r ) )
36		vex 3203	. . . . . . . . 9 |- g e. _V
37		vex 3203	. . . . . . . . 9 |- s e. _V
38	36, 30, 29, 32, 33, 37	caov411 6866	. . . . . . . 8 |- ( ( g .+ t ) .+ ( h .+ s ) ) = ( ( h .+ t ) .+ ( g .+ s ) )
39	36, 30, 29, 32, 33, 37	caov4 6865	. . . . . . . 8 |- ( ( g .+ t ) .+ ( h .+ s ) ) = ( ( g .+ h ) .+ ( t .+ s ) )
40	38, 39	eqtr3i 2646	. . . . . . 7 |- ( ( h .+ t ) .+ ( g .+ s ) ) = ( ( g .+ h ) .+ ( t .+ s ) )
41	28, 35, 40	3eqtr4g 2681	. . . . . 6 |- ( ( ( f .+ t ) = ( g .+ h ) /\ ( h .+ r ) = ( t .+ s ) ) -> ( ( h .+ t ) .+ ( f .+ r ) ) = ( ( h .+ t ) .+ ( g .+ s ) ) )
42	27, 41	syl6bi 243	. . . . 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 ) ) ) )
43		ecopopr.cl	. . . . . . . . . . 11 |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S )
44	43	caovcl 6828	. . . . . . . . . 10 |- ( ( h e. S /\ t e. S ) -> ( h .+ t ) e. S )
45	43	caovcl 6828	. . . . . . . . . 10 |- ( ( f e. S /\ r e. S ) -> ( f .+ r ) e. S )
46		ovex 6678	. . . . . . . . . . 11 |- ( g .+ s ) e. _V
47		ecopopr.can	. . . . . . . . . . 11 |- ( ( x e. S /\ y e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) )
48	46, 47	caovcan 6838	. . . . . . . . . 10 |- ( ( ( h .+ t ) e. S /\ ( f .+ r ) e. S ) -> ( ( ( h .+ t ) .+ ( f .+ r ) ) = ( ( h .+ t ) .+ ( g .+ s ) ) -> ( f .+ r ) = ( g .+ s ) ) )
49	44, 45, 48	syl2an 494	. . . . . . . . 9 |- ( ( ( 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 1260	. . . . . . . 8 |- ( ( ( 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 1269	. . . . . . 7 |- ( ( 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 1322	. . . . . 6 |- ( ( 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 1319	. . . . 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 ) ) )
54	42, 53	syld 47	. . . 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 ) ) )
55	1	ecopoveq 7848	. . . . 5 |- ( ( ( f e. S /\ g e. S ) /\ ( s e. S /\ r e. S ) ) -> ( <. f , g >. .~ <. s , r >. <-> ( f .+ r ) = ( g .+ s ) ) )
56	55	3adant2 1080	. . . 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 ) ) )
57	54, 56	sylibrd 249	. . 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 >. ) )
58	10, 14, 18, 22, 57	3optocl 5197	. 2 |- ( ( A e. ( S X. S ) /\ B e. ( S X. S ) /\ C e. ( S X. S ) ) -> ( ( A .~ B /\ B .~ C ) -> A .~ C ) )
59	9, 58	mpcom 38	1 |- ( ( A .~ B /\ B .~ C ) -> A .~ C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   <.cop 4183   class class class wbr 4653   {copab 4712   X. cxp 5112  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  ecopover  7851  ecopoverOLD  7852