Theorem ecopover 6897

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 an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-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
ecopover	|- .~ Er ( S X. S )

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

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

Proof of Theorem ecopover

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

Step	Hyp	Ref	Expression
1		ecopopr.1	. . . . 5 |- .~ = { <. 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	1	relopabi 4900	. . . 4 |- Rel .~
3	2	a1i 9	. . 3 |- ( T. -> Rel .~ )
4		ecopopr.com	. . . . 5 |- ( x .+ y ) = ( y .+ x )
5	1, 4	ecopovsym 6895	. . . 4 |- ( f .~ g -> g .~ f )
6	5	adantl 277	. . 3 |- ( ( T. /\ f .~ g ) -> g .~ f )
7		ecopopr.cl	. . . . 5 |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S )
8		ecopopr.ass	. . . . 5 |- ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) )
9		ecopopr.can	. . . . 5 |- ( ( x e. S /\ y e. S ) -> ( ( x .+ y ) = ( x .+ z ) -> y = z ) )
10	1, 4, 7, 8, 9	ecopovtrn 6896	. . . 4 |- ( ( f .~ g /\ g .~ h ) -> f .~ h )
11	10	adantl 277	. . 3 |- ( ( T. /\ ( f .~ g /\ g .~ h ) ) -> f .~ h )
12		vex 2824	. . . . . . . . . . 11 |- g e. _V
13		vex 2824	. . . . . . . . . . 11 |- h e. _V
14	12, 13, 4	caovcom 6237	. . . . . . . . . 10 |- ( g .+ h ) = ( h .+ g )
15	1	ecopoveq 6894	. . . . . . . . . 10 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> ( <. g , h >. .~ <. g , h >. <-> ( g .+ h ) = ( h .+ g ) ) )
16	14, 15	mpbiri 168	. . . . . . . . 9 |- ( ( ( g e. S /\ h e. S ) /\ ( g e. S /\ h e. S ) ) -> <. g , h >. .~ <. g , h >. )
17	16	anidms 401	. . . . . . . 8 |- ( ( g e. S /\ h e. S ) -> <. g , h >. .~ <. g , h >. )
18	17	rgen2a 2604	. . . . . . 7 |- A. g e. S A. h e. S <. g , h >. .~ <. g , h >.
19		breq12 4130	. . . . . . . . 9 |- ( ( f = <. g , h >. /\ f = <. g , h >. ) -> ( f .~ f <-> <. g , h >. .~ <. g , h >. ) )
20	19	anidms 401	. . . . . . . 8 |- ( f = <. g , h >. -> ( f .~ f <-> <. g , h >. .~ <. g , h >. ) )
21	20	ralxp 4918	. . . . . . 7 |- ( A. f e. ( S X. S ) f .~ f <-> A. g e. S A. h e. S <. g , h >. .~ <. g , h >. )
22	18, 21	mpbir 146	. . . . . 6 |- A. f e. ( S X. S ) f .~ f
23	22	rspec 2602	. . . . 5 |- ( f e. ( S X. S ) -> f .~ f )
24	23	a1i 9	. . . 4 |- ( T. -> ( f e. ( S X. S ) -> f .~ f ) )
25		opabssxp 4844	. . . . . . 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 ) )
26	1, 25	eqsstri 3280	. . . . . 6 |- .~ C_ ( ( S X. S ) X. ( S X. S ) )
27	26	ssbri 4170	. . . . 5 |- ( f .~ f -> f ( ( S X. S ) X. ( S X. S ) ) f )
28		brxp 4800	. . . . . 6 |- ( f ( ( S X. S ) X. ( S X. S ) ) f <-> ( f e. ( S X. S ) /\ f e. ( S X. S ) ) )
29	28	simplbi 274	. . . . 5 |- ( f ( ( S X. S ) X. ( S X. S ) ) f -> f e. ( S X. S ) )
30	27, 29	syl 14	. . . 4 |- ( f .~ f -> f e. ( S X. S ) )
31	24, 30	impbid1 142	. . 3 |- ( T. -> ( f e. ( S X. S ) <-> f .~ f ) )
32	3, 6, 11, 31	iserd 6823	. 2 |- ( T. -> .~ Er ( S X. S ) )
33	32	mptru 1411	1 |- .~ Er ( S X. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   T. wtru 1403   E.wex 1545   e. wcel 2209   A.wral 2528   <.cop 3708   class class class wbr 4125   {copab 4186   X. cxp 4767   Rel wrel 4774  (class class class)co 6075   Er wer 6794

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fv 5380  df-ov 6078  df-er 6797

This theorem is referenced by: (None)