Theorem ecovcom 6754

Description: Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6755 instead. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Hypotheses

Ref	Expression
ecovcom.1	|- C = ( ( S X. S ) /. .~ )
ecovcom.2	|- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. D , G >. ] .~ )
ecovcom.3	|- ( ( ( z e. S /\ w e. S ) /\ ( x e. S /\ y e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = [ <. H , J >. ] .~ )
ecovcom.4	|- D = H
ecovcom.5	|- G = J

Assertion

Ref	Expression
ecovcom	|- ( ( A e. C /\ B e. C ) -> ( A .+ B ) = ( B .+ A ) )

Distinct variable groups:   x, y, z, w, A   z, B, w   x, .+ , y, z, w   x, .~ , y, z, w   x, S, y, z, w   z, C, w

Allowed substitution hints:   B( x, y)   C( x, y)   D( x, y, z, w)   G( x, y, z, w)   H( x, y, z, w)   J( x, y, z, w)

Proof of Theorem ecovcom

Step	Hyp	Ref	Expression
1		ecovcom.1	. 2 |- C = ( ( S X. S ) /. .~ )
2		oveq1 5976	. . 3 |- ( [ <. x , y >. ] .~ = A -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = ( A .+ [ <. z , w >. ] .~ ) )
3		oveq2 5977	. . 3 |- ( [ <. x , y >. ] .~ = A -> ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = ( [ <. z , w >. ] .~ .+ A ) )
4	2, 3	eqeq12d 2222	. 2 |- ( [ <. x , y >. ] .~ = A -> ( ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) <-> ( A .+ [ <. z , w >. ] .~ ) = ( [ <. z , w >. ] .~ .+ A ) ) )
5		oveq2 5977	. . 3 |- ( [ <. z , w >. ] .~ = B -> ( A .+ [ <. z , w >. ] .~ ) = ( A .+ B ) )
6		oveq1 5976	. . 3 |- ( [ <. z , w >. ] .~ = B -> ( [ <. z , w >. ] .~ .+ A ) = ( B .+ A ) )
7	5, 6	eqeq12d 2222	. 2 |- ( [ <. z , w >. ] .~ = B -> ( ( A .+ [ <. z , w >. ] .~ ) = ( [ <. z , w >. ] .~ .+ A ) <-> ( A .+ B ) = ( B .+ A ) ) )
8		ecovcom.4	. . . 4 |- D = H
9		ecovcom.5	. . . 4 |- G = J
10		opeq12 3836	. . . . 5 |- ( ( D = H /\ G = J ) -> <. D , G >. = <. H , J >. )
11	10	eceq1d 6681	. . . 4 |- ( ( D = H /\ G = J ) -> [ <. D , G >. ] .~ = [ <. H , J >. ] .~ )
12	8, 9, 11	mp2an 426	. . 3 |- [ <. D , G >. ] .~ = [ <. H , J >. ] .~
13		ecovcom.2	. . 3 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. D , G >. ] .~ )
14		ecovcom.3	. . . 4 |- ( ( ( z e. S /\ w e. S ) /\ ( x e. S /\ y e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = [ <. H , J >. ] .~ )
15	14	ancoms 268	. . 3 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = [ <. H , J >. ] .~ )
16	12, 13, 15	3eqtr4a 2266	. 2 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) )
17	1, 4, 7, 16	2ecoptocl 6735	1 |- ( ( A e. C /\ B e. C ) -> ( A .+ B ) = ( B .+ A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   <.cop 3647   X. cxp 4692  (class class class)co 5969   [cec 6643   /.cqs 6644

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2779  df-un 3179  df-in 3181  df-ss 3188  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-br 4061  df-opab 4123  df-xp 4700  df-cnv 4702  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-iota 5252  df-fv 5299  df-ov 5972  df-ec 6647  df-qs 6651

This theorem is referenced by: (None)