Theorem ecovidi 6625

Description: Lemma used to transfer a distributive law via an equivalence relation. (Contributed by Jim Kingdon, 17-Sep-2019.)

Hypotheses

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

Assertion

Ref	Expression
ecovidi	|- ( ( 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 ecovidi

Step	Hyp	Ref	Expression
1		ecovidi.1	. 2 |- D = ( ( S X. S ) /. .~ )
2		oveq1 5860	. . 3 |- ( [ <. x , y >. ] .~ = A -> ( [ <. x , y >. ] .~ .x. ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( A .x. ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) )
3		oveq1 5860	. . . 4 |- ( [ <. x , y >. ] .~ = A -> ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) = ( A .x. [ <. z , w >. ] .~ ) )
4		oveq1 5860	. . . 4 |- ( [ <. x , y >. ] .~ = A -> ( [ <. x , y >. ] .~ .x. [ <. v , u >. ] .~ ) = ( A .x. [ <. v , u >. ] .~ ) )
5	3, 4	oveq12d 5871	. . 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 2185	. 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 5860	. . . 4 |- ( [ <. z , w >. ] .~ = B -> ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = ( B .+ [ <. v , u >. ] .~ ) )
8	7	oveq2d 5869	. . 3 |- ( [ <. z , w >. ] .~ = B -> ( A .x. ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) ) = ( A .x. ( B .+ [ <. v , u >. ] .~ ) ) )
9		oveq2 5861	. . . 4 |- ( [ <. z , w >. ] .~ = B -> ( A .x. [ <. z , w >. ] .~ ) = ( A .x. B ) )
10	9	oveq1d 5868	. . 3 |- ( [ <. z , w >. ] .~ = B -> ( ( A .x. [ <. z , w >. ] .~ ) .+ ( A .x. [ <. v , u >. ] .~ ) ) = ( ( A .x. B ) .+ ( A .x. [ <. v , u >. ] .~ ) ) )
11	8, 10	eqeq12d 2185	. 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 5861	. . . 4 |- ( [ <. v , u >. ] .~ = C -> ( B .+ [ <. v , u >. ] .~ ) = ( B .+ C ) )
13	12	oveq2d 5869	. . 3 |- ( [ <. v , u >. ] .~ = C -> ( A .x. ( B .+ [ <. v , u >. ] .~ ) ) = ( A .x. ( B .+ C ) ) )
14		oveq2 5861	. . . 4 |- ( [ <. v , u >. ] .~ = C -> ( A .x. [ <. v , u >. ] .~ ) = ( A .x. C ) )
15	14	oveq2d 5869	. . 3 |- ( [ <. v , u >. ] .~ = C -> ( ( A .x. B ) .+ ( A .x. [ <. v , u >. ] .~ ) ) = ( ( A .x. B ) .+ ( A .x. C ) ) )
16	13, 15	eqeq12d 2185	. 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		ecovidi.10	. . . 4 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> H = K )
18		ecovidi.11	. . . 4 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> J = L )
19		opeq12 3767	. . . . 5 |- ( ( H = K /\ J = L ) -> <. H , J >. = <. K , L >. )
20	19	eceq1d 6549	. . . 4 |- ( ( H = K /\ J = L ) -> [ <. H , J >. ] .~ = [ <. K , L >. ] .~ )
21	17, 18, 20	syl2anc 409	. . 3 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> [ <. H , J >. ] .~ = [ <. K , L >. ] .~ )
22		ecovidi.2	. . . . . . 7 |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( [ <. z , w >. ] .~ .+ [ <. v , u >. ] .~ ) = [ <. M , N >. ] .~ )
23	22	oveq2d 5869	. . . . . 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 275	. . . . 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		ecovidi.7	. . . . . 6 |- ( ( ( z e. S /\ w e. S ) /\ ( v e. S /\ u e. S ) ) -> ( M e. S /\ N e. S ) )
26		ecovidi.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 284	. . . . 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 2203	. . . 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 1194	. . 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		ecovidi.4	. . . . . 6 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( [ <. x , y >. ] .~ .x. [ <. z , w >. ] .~ ) = [ <. W , X >. ] .~ )
31		ecovidi.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 5872	. . . . 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		ecovidi.8	. . . . . 6 |- ( ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) -> ( W e. S /\ X e. S ) )
34		ecovidi.9	. . . . . 6 |- ( ( ( x e. S /\ y e. S ) /\ ( v e. S /\ u e. S ) ) -> ( Y e. S /\ Z e. S ) )
35		ecovidi.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 287	. . . . 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 2203	. . . 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 1288	. . 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	3eqtr4d 2213	. 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 6602	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 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:  distrnqg  7349  distrsrg  7721  axdistr  7836