Theorem ecoviass 6623

Description: Lemma used to transfer an associative law via an equivalence relation. (Contributed by Jim Kingdon, 16-Sep-2019.)

Hypotheses

Ref	Expression
ecoviass.1	|- D = ( ( S X. S ) /. .~ )
ecoviass.2	|- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. G , H >. ] .~ )
ecoviass.3	|- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. N , Q >. ] .~ )
ecoviass.4	|- ( ( ( G e. S /\ H e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. G , H >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. J , K >. ] .~ )
ecoviass.5	|- ( ( ( x e. S /\ y e. S ) /\ ( N e. S /\ Q e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. N , Q >. ] .~ ) = [ <. L , M >. ] .~ )
ecoviass.6	|- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( G e. S /\ H e. S ) )
ecoviass.7	|- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( N e. S /\ Q e. S ) )
ecoviass.8	|- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> J = L )
ecoviass.9	|- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> K = M )

Assertion

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

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

Allowed substitution hints:   B( x, y)   D( x, y)   Q( x, y, z, w, v, u)   G( x, y, z, w, v, u)   H( x, y, z, w, v, u)   J( x, y, z, w, v, u)   K( x, y, z, w, v, u)   L( x, y, z, w, v, u)   M( x, y, z, w, v, u)   N( x, y, z, w, v, u)

Proof of Theorem ecoviass

Step	Hyp	Ref	Expression
1		ecoviass.1	. 2 |- D = ( ( S X. S ) /. .~ )
2		oveq1 5860	. . . 4 |- ( [ <. x , y >. ] .~ = A -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = ( A .+ [ <. z , w >. ] .~ ) )
3	2	oveq1d 5868	. . 3 |- ( [ <. x , y >. ] .~ = A -> ( ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) = ( ( A .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) )
4		oveq1 5860	. . 3 |- ( [ <. x , y >. ] .~ = A -> ( [ <. x , y >. ] .~ .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( A .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) )
5	3, 4	eqeq12d 2185	. 2 |- ( [ <. x , y >. ] .~ = A -> ( ( ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) = ( [ <. x , y >. ] .~ .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) <-> ( ( A .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) = ( A .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) ) )
6		oveq2 5861	. . . 4 |- ( [ <. z , w >. ] .~ = B -> ( A .+ [ <. z , w >. ] .~ ) = ( A .+ B ) )
7	6	oveq1d 5868	. . 3 |- ( [ <. z , w >. ] .~ = B -> ( ( A .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) = ( ( A .+ B ) .+ [ <. v , u >. ] .~ ) )
8		oveq1 5860	. . . 4 |- ( [ <. z , w >. ] .~ = B -> ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = ( B .+ [ <. v , u >. ] .~ ) )
9	8	oveq2d 5869	. . 3 |- ( [ <. z , w >. ] .~ = B -> ( A .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( A .+ ( B .+ [ <. v , u >. ] .~ ) ) )
10	7, 9	eqeq12d 2185	. 2 |- ( [ <. z , w >. ] .~ = B -> ( ( ( A .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) = ( A .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) <-> ( ( A .+ B ) .+ [ <. v , u >. ] .~ ) = ( A .+ ( B .+ [ <. v , u >. ] .~ ) ) ) )
11		oveq2 5861	. . 3 |- ( [ <. v , u >. ] .~ = C -> ( ( A .+ B ) .+ [ <. v , u >. ] .~ ) = ( ( A .+ B ) .+ C ) )
12		oveq2 5861	. . . 4 |- ( [ <. v , u >. ] .~ = C -> ( B .+ [ <. v , u >. ] .~ ) = ( B .+ C ) )
13	12	oveq2d 5869	. . 3 |- ( [ <. v , u >. ] .~ = C -> ( A .+ ( B .+ [ <. v , u >. ] .~ ) ) = ( A .+ ( B .+ C ) ) )
14	11, 13	eqeq12d 2185	. 2 |- ( [ <. v , u >. ] .~ = C -> ( ( ( A .+ B ) .+ [ <. v , u >. ] .~ ) = ( A .+ ( B .+ [ <. v , u >. ] .~ ) ) <-> ( ( A .+ B ) .+ C ) = ( A .+ ( B .+ C ) ) ) )
15		ecoviass.8	. . . 4 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> J = L )
16		ecoviass.9	. . . 4 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> K = M )
17		opeq12 3767	. . . . 5 |- ( ( J = L /\ K = M ) -> <. J , K >. = <. L , M >. )
18	17	eceq1d 6549	. . . 4 |- ( ( J = L /\ K = M ) -> [ <. J , K >. ] .~ = [ <. L , M >. ] .~ )
19	15, 16, 18	syl2anc 409	. . 3 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> [ <. J , K >. ] .~ = [ <. L , M >. ] .~ )
20		ecoviass.2	. . . . . . 7 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. G , H >. ] .~ )
21	20	oveq1d 5868	. . . . . 6 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) = ( [ <. G , H >. ] .~ .+ [ <. v , u >. ] .~ ) )
22	21	adantr 274	. . . . 5 |- ( ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) /\ ( v e. S /\ u e. S ) ) -> ( ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) = ( [ <. G , H >. ] .~ .+ [ <. v , u >. ] .~ ) )
23		ecoviass.6	. . . . . 6 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( G e. S /\ H e. S ) )
24		ecoviass.4	. . . . . 6 |- ( ( ( G e. S /\ H e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. G , H >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. J , K >. ] .~ )
25	23, 24	sylan 281	. . . . 5 |- ( ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. G , H >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. J , K >. ] .~ )
26	22, 25	eqtrd 2203	. . . 4 |- ( ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) /\ ( v e. S /\ u e. S ) ) -> ( ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) = [ <. J , K >. ] .~ )
27	26	3impa 1189	. . 3 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) = [ <. J , K >. ] .~ )
28		ecoviass.3	. . . . . . 7 |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. N , Q >. ] .~ )
29	28	oveq2d 5869	. . . . . 6 |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. x , y >. ] .~ .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( [ <. x , y >. ] .~ .+ [ <. N , Q >. ] .~ ) )
30	29	adantl 275	. . . . 5 |- ( ( ( x e. S /\ y e. S ) /\ ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ) -> ( [ <. x , y >. ] .~ .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( [ <. x , y >. ] .~ .+ [ <. N , Q >. ] .~ ) )
31		ecoviass.7	. . . . . 6 |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( N e. S /\ Q e. S ) )
32		ecoviass.5	. . . . . 6 |- ( ( ( x e. S /\ y e. S ) /\ ( N e. S /\ Q e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. N , Q >. ] .~ ) = [ <. L , M >. ] .~ )
33	31, 32	sylan2 284	. . . . 5 |- ( ( ( x e. S /\ y e. S ) /\ ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ) -> ( [ <. x , y >. ] .~ .+ [ <. N , Q >. ] .~ ) = [ <. L , M >. ] .~ )
34	30, 33	eqtrd 2203	. . . 4 |- ( ( ( x e. S /\ y e. S ) /\ ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ) -> ( [ <. x , y >. ] .~ .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = [ <. L , M >. ] .~ )
35	34	3impb 1194	. . 3 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. x , y >. ] .~ .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = [ <. L , M >. ] .~ )
36	19, 27, 35	3eqtr4d 2213	. 2 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) = ( [ <. x , y >. ] .~ .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) )
37	1, 5, 10, 14, 36	3ecoptocl 6602	1 |- ( ( A e. D /\ B e. D /\ C e. D ) -> ( ( A .+ B ) .+ C ) = ( A .+ ( B .+ C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   <.cop 3586   X. cxp 4609  (class class class)co 5853   [cec 6511   /.cqs 6512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fv 5206  df-ov 5856  df-ec 6515  df-qs 6519

This theorem is referenced by:  addassnqg  7344  mulassnqg  7346  addasssrg  7718  mulasssrg  7720  axaddass  7834  axmulass  7835