Theorem ecovcom 6876

Description: Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6877 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 6057	. . 3 |- ( [ <. x , y >. ] .~ = A -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = ( A .+ [ <. z , w >. ] .~ ) )
3		oveq2 6058	. . 3 |- ( [ <. x , y >. ] .~ = A -> ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) = ( [ <. z , w >. ] .~ .+ A ) )
4	2, 3	eqeq12d 2247	. 2 |- ( [ <. x , y >. ] .~ = A -> ( ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = ( [ <. z , w >. ] .~ .+ [ <. x , y >. ] .~ ) <-> ( A .+ [ <. z , w >. ] .~ ) = ( [ <. z , w >. ] .~ .+ A ) ) )
5		oveq2 6058	. . 3 |- ( [ <. z , w >. ] .~ = B -> ( A .+ [ <. z , w >. ] .~ ) = ( A .+ B ) )
6		oveq1 6057	. . 3 |- ( [ <. z , w >. ] .~ = B -> ( [ <. z , w >. ] .~ .+ A ) = ( B .+ A ) )
7	5, 6	eqeq12d 2247	. 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 3885	. . . . 5 |- ( ( D = H /\ G = J ) -> <. D , G >. = <. H , J >. )
11	10	eceq1d 6803	. . . 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 2291	. 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 6857	1 |- ( ( A e. C /\ B e. C ) -> ( A .+ B ) = ( B .+ A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   <.cop 3692   X. cxp 4747  (class class class)co 6050   [cec 6765   /.cqs 6766

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fv 5360  df-ov 6053  df-ec 6769  df-qs 6773

This theorem is referenced by: (None)