Theorem ecovdi 7856

Description: Lemma used to transfer a distributive law via an equivalence relation. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Hypotheses

Ref	Expression
ecovdi.1	|- D = ( ( S X. S ) /. .~ )
ecovdi.2	|- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. M , N >. ] .~ )
ecovdi.3	|- ( ( ( x e. S /\ y e. S ) /\ ( M e. S /\ N e. S ) ) -> ( [ <. x , y >. ] .~ .x. [ <. M , N >. ] .~ ) = [ <. H , J >. ] .~ )
ecovdi.4	|- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) = [ <. W , X >. ] .~ )
ecovdi.5	|- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) = [ <. Y , Z >. ] .~ )
ecovdi.6	|- ( ( ( W e. S /\ X e. S ) /\ ( Y e. S /\ Z e. S ) ) -> ( [ <. W , X >. ] .~ .+ [ <. Y , Z >. ] .~ ) = [ <. K , L >. ] .~ )
ecovdi.7	|- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( M e. S /\ N e. S ) )
ecovdi.8	|- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( W e. S /\ X e. S ) )
ecovdi.9	|- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) -> ( Y e. S /\ Z e. S ) )
ecovdi.10	|- H = K
ecovdi.11	|- J = L

Assertion

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

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

Allowed substitution hints:   B( x, y)   C( x, y, z)   D( x, y)   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)   W( x, y, z, w, v, u)   X( x, y, z, w, v, u)   Y( x, y, z, w, v, u)   Z( x, y, z, w, v, u)

Proof of Theorem ecovdi

Step	Hyp	Ref	Expression
1		ecovdi.1	. 2 |- D = ( ( S X. S ) /. .~ )
2		oveq1 6657	. . 3 |- ( [ <. x , y >. ] .~ = A -> ( [ <. x , y >. ] .~ .x. ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( A .x. ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) )
3		oveq1 6657	. . . 4 |- ( [ <. x , y >. ] .~ = A -> ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) = ( A .x. [ <. z , w >. ] .~ ) )
4		oveq1 6657	. . . 4 |- ( [ <. x , y >. ] .~ = A -> ( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) = ( A .x. [ <. v , u >. ] .~ ) )
5	3, 4	oveq12d 6668	. . 3 |- ( [ <. x , y >. ] .~ = A -> ( ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) .+ ( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) ) = ( ( A .x. [ <. z , w >. ] .~ ) .+ ( A .x. [ <. v , u >. ] .~ ) ) )
6	2, 5	eqeq12d 2637	. 2 |- ( [ <. x , y >. ] .~ = A -> ( ( [ <. x , y >. ] .~ .x. ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) .+ ( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) ) <-> ( A .x. ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( ( A .x. [ <. z , w >. ] .~ ) .+ ( A .x. [ <. v , u >. ] .~ ) ) ) )
7		oveq1 6657	. . . 4 |- ( [ <. z , w >. ] .~ = B -> ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = ( B .+ [ <. v , u >. ] .~ ) )
8	7	oveq2d 6666	. . 3 |- ( [ <. z , w >. ] .~ = B -> ( A .x. ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( A .x. ( B .+ [ <. v , u >. ] .~ ) ) )
9		oveq2 6658	. . . 4 |- ( [ <. z , w >. ] .~ = B -> ( A .x. [ <. z , w >. ] .~ ) = ( A .x. B ) )
10	9	oveq1d 6665	. . 3 |- ( [ <. z , w >. ] .~ = B -> ( ( A .x. [ <. z , w >. ] .~ ) .+ ( A .x. [ <. v , u >. ] .~ ) ) = ( ( A .x. B ) .+ ( A .x. [ <. v , u >. ] .~ ) ) )
11	8, 10	eqeq12d 2637	. 2 |- ( [ <. z , w >. ] .~ = B -> ( ( A .x. ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( ( A .x. [ <. z , w >. ] .~ ) .+ ( A .x. [ <. v , u >. ] .~ ) ) <-> ( A .x. ( B .+ [ <. v , u >. ] .~ ) ) = ( ( A .x. B ) .+ ( A .x. [ <. v , u >. ] .~ ) ) ) )
12		oveq2 6658	. . . 4 |- ( [ <. v , u >. ] .~ = C -> ( B .+ [ <. v , u >. ] .~ ) = ( B .+ C ) )
13	12	oveq2d 6666	. . 3 |- ( [ <. v , u >. ] .~ = C -> ( A .x. ( B .+ [ <. v , u >. ] .~ ) ) = ( A .x. ( B .+ C ) ) )
14		oveq2 6658	. . . 4 |- ( [ <. v , u >. ] .~ = C -> ( A .x. [ <. v , u >. ] .~ ) = ( A .x. C ) )
15	14	oveq2d 6666	. . 3 |- ( [ <. v , u >. ] .~ = C -> ( ( A .x. B ) .+ ( A .x. [ <. v , u >. ] .~ ) ) = ( ( A .x. B ) .+ ( A .x. C ) ) )
16	13, 15	eqeq12d 2637	. 2 |- ( [ <. v , u >. ] .~ = C -> ( ( A .x. ( B .+ [ <. v , u >. ] .~ ) ) = ( ( A .x. B ) .+ ( A .x. [ <. v , u >. ] .~ ) ) <-> ( A .x. ( B .+ C ) ) = ( ( A .x. B ) .+ ( A .x. C ) ) ) )
17		ecovdi.10	. . . 4 |- H = K
18		ecovdi.11	. . . 4 |- J = L
19		opeq12 4404	. . . . 5 |- ( ( H = K /\ J = L ) -> <. H , J >. = <. K , L >. )
20	19	eceq1d 7783	. . . 4 |- ( ( H = K /\ J = L ) -> [ <. H , J >. ] .~ = [ <. K , L >. ] .~ )
21	17, 18, 20	mp2an 708	. . 3 |- [ <. H , J >. ] .~ = [ <. K , L >. ] .~
22		ecovdi.2	. . . . . . 7 |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. M , N >. ] .~ )
23	22	oveq2d 6666	. . . . . 6 |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. x , y >. ] .~ .x. ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( [ <. x , y >. ] .~ .x. [ <. M , N >. ] .~ ) )
24	23	adantl 482	. . . . 5 |- ( ( ( x e. S /\ y e. S ) /\ ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ) -> ( [ <. x , y >. ] .~ .x. ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( [ <. x , y >. ] .~ .x. [ <. M , N >. ] .~ ) )
25		ecovdi.7	. . . . . 6 |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( M e. S /\ N e. S ) )
26		ecovdi.3	. . . . . 6 |- ( ( ( x e. S /\ y e. S ) /\ ( M e. S /\ N e. S ) ) -> ( [ <. x , y >. ] .~ .x. [ <. M , N >. ] .~ ) = [ <. H , J >. ] .~ )
27	25, 26	sylan2 491	. . . . 5 |- ( ( ( x e. S /\ y e. S ) /\ ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ) -> ( [ <. x , y >. ] .~ .x. [ <. M , N >. ] .~ ) = [ <. H , J >. ] .~ )
28	24, 27	eqtrd 2656	. . . 4 |- ( ( ( x e. S /\ y e. S ) /\ ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) ) -> ( [ <. x , y >. ] .~ .x. ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = [ <. H , J >. ] .~ )
29	28	3impb 1260	. . 3 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. x , y >. ] .~ .x. ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = [ <. H , J >. ] .~ )
30		ecovdi.4	. . . . . 6 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) = [ <. W , X >. ] .~ )
31		ecovdi.5	. . . . . 6 |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) = [ <. Y , Z >. ] .~ )
32	30, 31	oveqan12d 6669	. . . . 5 |- ( ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) /\ ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) ) -> ( ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) .+ ( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) ) = ( [ <. W , X >. ] .~ .+ [ <. Y , Z >. ] .~ ) )
33		ecovdi.8	. . . . . 6 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( W e. S /\ X e. S ) )
34		ecovdi.9	. . . . . 6 |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) -> ( Y e. S /\ Z e. S ) )
35		ecovdi.6	. . . . . 6 |- ( ( ( W e. S /\ X e. S ) /\ ( Y e. S /\ Z e. S ) ) -> ( [ <. W , X >. ] .~ .+ [ <. Y , Z >. ] .~ ) = [ <. K , L >. ] .~ )
36	33, 34, 35	syl2an 494	. . . . 5 |- ( ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) /\ ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) ) -> ( [ <. W , X >. ] .~ .+ [ <. Y , Z >. ] .~ ) = [ <. K , L >. ] .~ )
37	32, 36	eqtrd 2656	. . . 4 |- ( ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) /\ ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) ) -> ( ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) .+ ( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) ) = [ <. K , L >. ] .~ )
38	37	3impdi 1381	. . 3 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) .+ ( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) ) = [ <. K , L >. ] .~ )
39	21, 29, 38	3eqtr4a 2682	. 2 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. x , y >. ] .~ .x. ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) .+ ( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) ) )
40	1, 6, 11, 16, 39	3ecoptocl 7839	1 |- ( ( A e. D /\ B e. D /\ C e. D ) -> ( A .x. ( B .+ C ) ) = ( ( A .x. B ) .+ ( A .x. C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   X. cxp 5112  (class class class)co 6650   [cec 7740   /.cqs 7741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-ov 6653  df-ec 7744  df-qs 7748

This theorem is referenced by:  distrsr  9912  axdistr  9979