Theorem ecovass 6754

Description: Lemma used to transfer an associative law via an equivalence relation. In most cases ecoviass 6755 will be more useful. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Hypotheses

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

Assertion

Ref	Expression
ecovass	|- ( ( 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 ecovass

Step	Hyp	Ref	Expression
1		ecovass.1	. 2 |- D = ( ( S X. S ) /. .~ )
2		oveq1 5974	. . . 4 |- ( [ <. x , y >. ] .~ = A -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = ( A .+ [ <. z , w >. ] .~ ) )
3	2	oveq1d 5982	. . 3 |- ( [ <. x , y >. ] .~ = A -> ( ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) = ( ( A .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) )
4		oveq1 5974	. . 3 |- ( [ <. x , y >. ] .~ = A -> ( [ <. x , y >. ] .~ .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( A .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) )
5	3, 4	eqeq12d 2222	. 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 5975	. . . 4 |- ( [ <. z , w >. ] .~ = B -> ( A .+ [ <. z , w >. ] .~ ) = ( A .+ B ) )
7	6	oveq1d 5982	. . 3 |- ( [ <. z , w >. ] .~ = B -> ( ( A .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) = ( ( A .+ B ) .+ [ <. v , u >. ] .~ ) )
8		oveq1 5974	. . . 4 |- ( [ <. z , w >. ] .~ = B -> ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = ( B .+ [ <. v , u >. ] .~ ) )
9	8	oveq2d 5983	. . 3 |- ( [ <. z , w >. ] .~ = B -> ( A .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( A .+ ( B .+ [ <. v , u >. ] .~ ) ) )
10	7, 9	eqeq12d 2222	. 2 |- ( [ <. z , w >. ] .~ = B -> ( ( ( A .+ [ <. z , w >. ] .~ ) .+ [ <. v , u >. ] .~ ) = ( A .+ ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) <-> ( ( A .+ B ) .+ [ <. v , u >. ] .~ ) = ( A .+ ( B .+ [ <. v , u >. ] .~ ) ) ) )
11		oveq2 5975	. . 3 |- ( [ <. v , u >. ] .~ = C -> ( ( A .+ B ) .+ [ <. v , u >. ] .~ ) = ( ( A .+ B ) .+ C ) )
12		oveq2 5975	. . . 4 |- ( [ <. v , u >. ] .~ = C -> ( B .+ [ <. v , u >. ] .~ ) = ( B .+ C ) )
13	12	oveq2d 5983	. . 3 |- ( [ <. v , u >. ] .~ = C -> ( A .+ ( B .+ [ <. v , u >. ] .~ ) ) = ( A .+ ( B .+ C ) ) )
14	11, 13	eqeq12d 2222	. 2 |- ( [ <. v , u >. ] .~ = C -> ( ( ( A .+ B ) .+ [ <. v , u >. ] .~ ) = ( A .+ ( B .+ [ <. v , u >. ] .~ ) ) <-> ( ( A .+ B ) .+ C ) = ( A .+ ( B .+ C ) ) ) )
15		ecovass.8	. . . 4 |- J = L
16		ecovass.9	. . . 4 |- K = M
17		opeq12 3835	. . . . 5 |- ( ( J = L /\ K = M ) -> <. J , K >. = <. L , M >. )
18	17	eceq1d 6679	. . . 4 |- ( ( J = L /\ K = M ) -> [ <. J , K >. ] .~ = [ <. L , M >. ] .~ )
19	15, 16, 18	mp2an 426	. . 3 |- [ <. J , K >. ] .~ = [ <. L , M >. ] .~
20		ecovass.2	. . . . . . 7 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. z , w >. ] .~ ) = [ <. G , H >. ] .~ )
21	20	oveq1d 5982	. . . . . 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 276	. . . . 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		ecovass.6	. . . . . 6 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( G e. S /\ H e. S ) )
24		ecovass.4	. . . . . 6 |- ( ( ( G e. S /\ H e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. G , H >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. J , K >. ] .~ )
25	23, 24	sylan 283	. . . . 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 2240	. . . 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 1197	. . 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		ecovass.3	. . . . . . 7 |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. N , Q >. ] .~ )
29	28	oveq2d 5983	. . . . . 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 277	. . . . 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		ecovass.7	. . . . . 6 |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( N e. S /\ Q e. S ) )
32		ecovass.5	. . . . . 6 |- ( ( ( x e. S /\ y e. S ) /\ ( N e. S /\ Q e. S ) ) -> ( [ <. x , y >. ] .~ .+ [ <. N , Q >. ] .~ ) = [ <. L , M >. ] .~ )
33	31, 32	sylan2 286	. . . . 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 2240	. . . 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 1202	. . 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	3eqtr4a 2266	. 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 6734	1 |- ( ( A e. D /\ B e. D /\ C e. D ) -> ( ( A .+ B ) .+ C ) = ( A .+ ( B .+ C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2178   <.cop 3646   X. cxp 4691  (class class class)co 5967   [cec 6641   /.cqs 6642

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 4178  ax-pow 4234  ax-pr 4269

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 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fv 5298  df-ov 5970  df-ec 6645  df-qs 6649

This theorem is referenced by: (None)