Theorem ecopovtrng 6782

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 Jim Kingdon, 1-Sep-2019.)

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

Assertion

Ref	Expression
ecopovtrng	|- ( ( 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 ecopovtrng

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 4793	. . . . . . 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 3256	. . . . . 6 |- .~ C_ ( ( S X. S ) X. ( S X. S ) )
4	3	brel 4771	. . . . 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 4771	. . . 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 1006	. . 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 2229	. . 3 |- ( S X. S ) = ( S X. S )
11		breq1 4086	. . . . 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 4086	. . . 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 4087	. . . . 5 |- ( <. h , t >. = B -> ( A .~ <. h , t >. <-> A .~ B ) )
16		breq1 4086	. . . . 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 4087	. . . . 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 4087	. . . 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 6777	. . . . . . . 8 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) ) -> ( <. f , g >. .~ <. h , t >. <-> ( f .+ t ) = ( g .+ h ) ) )
24	23	3adant3 1041	. . . . . . 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 6777	. . . . . . . 8 |- ( ( ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( <. h , t >. .~ <. s , r >. <-> ( h .+ r ) = ( t .+ s ) ) )
26	25	3adant1 1039	. . . . . . 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 6010	. . . . . . 7 |- ( ( ( f .+ t ) = ( g .+ h ) /\ ( h .+ r ) = ( t .+ s ) ) -> ( ( f .+ t ) .+ ( h .+ r ) ) = ( ( g .+ h ) .+ ( t .+ s ) ) )
29		simp2l 1047	. . . . . . . . 9 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> h e. S )
30		simp2r 1048	. . . . . . . . 9 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> t e. S )
31		simp1l 1045	. . . . . . . . 9 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> f e. S )
32		ecopoprg.com	. . . . . . . . . 10 |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) = ( y .+ x ) )
33	32	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 ) = ( y .+ x ) )
34		ecopoprg.ass	. . . . . . . . . 10 |- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
35	34	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 /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
36		simp3r 1050	. . . . . . . . 9 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> r e. S )
37		ecopoprg.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 6191	. . . . . . . 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 1046	. . . . . . . . . 10 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> g e. S )
41		simp3l 1049	. . . . . . . . . 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 6191	. . . . . . . . 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 6190	. . . . . . . . 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 2264	. . . . . . . 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 2244	. . . . . . 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	imbitrrid 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		ecopoprg.can	. . . . . . . 8 |- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) )
49		oveq2 6009	. . . . . . . 8 |- ( y = z -> ( x .+ y ) = ( x .+ z ) )
50	48, 49	impbid1 142	. . . . . . 7 |- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x .+ y ) = ( x .+ z ) <-> y = z ) )
51	50	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 ) )
52	37	caovcl 6160	. . . . . . 7 |- ( ( h e. S /\ t e. S ) -> ( h .+ t ) e. S )
53	29, 30, 52	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 )
54	37	caovcl 6160	. . . . . . 7 |- ( ( f e. S /\ r e. S ) -> ( f .+ r ) e. S )
55	31, 36, 54	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 )
56	38, 40, 41	caovcld 6159	. . . . . 6 |- ( ( ( f e. S /\ g e. S ) /\ ( h e. S /\ t e. S ) /\ ( s e. S /\ r e. S ) ) -> ( g .+ s ) e. S )
57	51, 53, 55, 56	caovcand 6168	. . . . 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 ) ) )
58	47, 57	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 ) ) )
59	1	ecopoveq 6777	. . . . 5 |- ( ( ( f e. S /\ g e. S ) /\ ( s e. S /\ r e. S ) ) -> ( <. f , g >. .~ <. s , r >. <-> ( f .+ r ) = ( g .+ s ) ) )
60	59	3adant2 1040	. . . 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 ) ) )
61	58, 60	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 >. ) )
62	10, 14, 18, 22, 61	3optocl 4797	. 2 |- ( ( A e. ( S X. S ) /\ B e. ( S X. S ) /\ C e. ( S X. S ) ) -> ( ( A .~ B /\ B .~ C ) -> A .~ C ) )
63	9, 62	mpcom 36	1 |- ( ( A .~ B /\ B .~ C ) -> A .~ C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   <.cop 3669   class class class wbr 4083   {copab 4144   X. cxp 4717  (class class class)co 6001

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-iota 5278  df-fv 5326  df-ov 6004

This theorem is referenced by:  ecopoverg  6783