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Theorem ecoprdi 5380
Description: Lemma used to transfer a distributive law via an equivalence relation.
Hypotheses
Ref Expression
ecoprdi.1 |- D = ((S X. S)/.R)
ecoprdi.2 |- (((z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> ([<.z, w>.]RF[<.v, u>.]R) = [<.M, N>.]R)
ecoprdi.3 |- (((x e. S /\ y e. S) /\ (M e. S /\ N e. S)) -> ([<.x, y>.]RG[<.M, N>.]R) = [<.H, J>.]R)
ecoprdi.4 |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S)) -> ([<.x, y>.]RG[<.z, w>.]R) = [<.W, X>.]R)
ecoprdi.5 |- (((x e. S /\ y e. S) /\ (v e. S /\ u e. S)) -> ([<.x, y>.]RG[<.v, u>.]R) = [<.Y, Z>.]R)
ecoprdi.6 |- (((W e. S /\ X e. S) /\ (Y e. S /\ Z e. S)) -> ([<.W, X>.]RF[<.Y, Z>.]R) = [<.K, L>.]R)
ecoprdi.7 |- (((z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> (M e. S /\ N e. S))
ecoprdi.8 |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S)) -> (W e. S /\ X e. S))
ecoprdi.9 |- (((x e. S /\ y e. S) /\ (v e. S /\ u e. S)) -> (Y e. S /\ Z e. S))
ecoprdi.10 |- H = K
ecoprdi.11 |- J = L
Assertion
Ref Expression
ecoprdi |- ((A e. D /\ B e. D /\ C e. D) -> (AG(BFC)) = ((AGB)F(AGC)))
Distinct variable groups:   x,y,z,w,v,u,A   x,B,y,z,w,v,u   x,C,y,z,w,v,u   x,F,y,z,w,v,u   x,R,y,z,w,v,u   x,S,y,z,w,v,u   x,G,y,z,w,v,u   z,D,w,v,u
Proof of Theorem ecoprdi
StepHypRef Expression
1 ecoprdi.1 . 2 |- D = ((S X. S)/.R)
2 opreq1 4889 . . 3 |- ([<.x, y>.]R = A -> ([<.x, y>.]RG([<.z, w>.]RF[<.v, u>.]R)) = (AG([<.z, w>.]RF[<.v, u>.]R)))
3 opreq1 4889 . . . 4 |- ([<.x, y>.]R = A -> ([<.x, y>.]RG[<.z, w>.]R) = (AG[<.z, w>.]R))
4 opreq1 4889 . . . 4 |- ([<.x, y>.]R = A -> ([<.x, y>.]RG[<.v, u>.]R) = (AG[<.v, u>.]R))
53, 4opreq12d 4900 . . 3 |- ([<.x, y>.]R = A -> (([<.x, y>.]RG[<.z, w>.]R)F([<.x, y>.]RG[<.v, u>.]R)) = ((AG[<.z, w>.]R)F(AG[<.v, u>.]R)))
62, 5eqeq12d 1899 . 2 |- ([<.x, y>.]R = A -> (([<.x, y>.]RG([<.z, w>.]RF[<.v, u>.]R)) = (([<.x, y>.]RG[<.z, w>.]R)F([<.x, y>.]RG[<.v, u>.]R)) <-> (AG([<.z, w>.]RF[<.v, u>.]R)) = ((AG[<.z, w>.]R)F(AG[<.v, u>.]R))))
7 opreq1 4889 . . . 4 |- ([<.z, w>.]R = B -> ([<.z, w>.]RF[<.v, u>.]R) = (BF[<.v, u>.]R))
87opreq2d 4898 . . 3 |- ([<.z, w>.]R = B -> (AG([<.z, w>.]RF[<.v, u>.]R)) = (AG(BF[<.v, u>.]R)))
9 opreq2 4890 . . . 4 |- ([<.z, w>.]R = B -> (AG[<.z, w>.]R) = (AGB))
109opreq1d 4897 . . 3 |- ([<.z, w>.]R = B -> ((AG[<.z, w>.]R)F(AG[<.v, u>.]R)) = ((AGB)F(AG[<.v, u>.]R)))
118, 10eqeq12d 1899 . 2 |- ([<.z, w>.]R = B -> ((AG([<.z, w>.]RF[<.v, u>.]R)) = ((AG[<.z, w>.]R)F(AG[<.v, u>.]R)) <-> (AG(BF[<.v, u>.]R)) = ((AGB)F(AG[<.v, u>.]R))))
12 opreq2 4890 . . . 4 |- ([<.v, u>.]R = C -> (BF[<.v, u>.]R) = (BFC))
1312opreq2d 4898 . . 3 |- ([<.v, u>.]R = C -> (AG(BF[<.v, u>.]R)) = (AG(BFC)))
14 opreq2 4890 . . . 4 |- ([<.v, u>.]R = C -> (AG[<.v, u>.]R) = (AGC))
1514opreq2d 4898 . . 3 |- ([<.v, u>.]R = C -> ((AGB)F(AG[<.v, u>.]R)) = ((AGB)F(AGC)))
1613, 15eqeq12d 1899 . 2 |- ([<.v, u>.]R = C -> ((AG(BF[<.v, u>.]R)) = ((AGB)F(AG[<.v, u>.]R)) <-> (AG(BFC)) = ((AGB)F(AGC))))
17 ecoprdi.10 . . . 4 |- H = K
18 ecoprdi.11 . . . 4 |- J = L
19 opeq12 3160 . . . . 5 |- ((H = K /\ J = L) -> <.H, J>. = <.K, L>.)
20 eceq2 5336 . . . . 5 |- (<.H, J>. = <.K, L>. -> [<.H, J>.]R = [<.K, L>.]R)
2119, 20syl 12 . . . 4 |- ((H = K /\ J = L) -> [<.H, J>.]R = [<.K, L>.]R)
2217, 18, 21mp2an 761 . . 3 |- [<.H, J>.]R = [<.K, L>.]R
23 ecoprdi.2 . . . . . . 7 |- (((z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> ([<.z, w>.]RF[<.v, u>.]R) = [<.M, N>.]R)
2423opreq2d 4898 . . . . . 6 |- (((z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> ([<.x, y>.]RG([<.z, w>.]RF[<.v, u>.]R)) = ([<.x, y>.]RG[<.M, N>.]R))
2524adantl 424 . . . . 5 |- (((x e. S /\ y e. S) /\ ((z e. S /\ w e. S) /\ (v e. S /\ u e. S))) -> ([<.x, y>.]RG([<.z, w>.]RF[<.v, u>.]R)) = ([<.x, y>.]RG[<.M, N>.]R))
26 ecoprdi.3 . . . . . 6 |- (((x e. S /\ y e. S) /\ (M e. S /\ N e. S)) -> ([<.x, y>.]RG[<.M, N>.]R) = [<.H, J>.]R)
27 ecoprdi.7 . . . . . 6 |- (((z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> (M e. S /\ N e. S))
2826, 27sylan2 500 . . . . 5 |- (((x e. S /\ y e. S) /\ ((z e. S /\ w e. S) /\ (v e. S /\ u e. S))) -> ([<.x, y>.]RG[<.M, N>.]R) = [<.H, J>.]R)
2925, 28eqtrd 1925 . . . 4 |- (((x e. S /\ y e. S) /\ ((z e. S /\ w e. S) /\ (v e. S /\ u e. S))) -> ([<.x, y>.]RG([<.z, w>.]RF[<.v, u>.]R)) = [<.H, J>.]R)
30293impb 1063 . . 3 |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> ([<.x, y>.]RG([<.z, w>.]RF[<.v, u>.]R)) = [<.H, J>.]R)
31 ecoprdi.4 . . . . . 6 |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S)) -> ([<.x, y>.]RG[<.z, w>.]R) = [<.W, X>.]R)
32 ecoprdi.5 . . . . . 6 |- (((x e. S /\ y e. S) /\ (v e. S /\ u e. S)) -> ([<.x, y>.]RG[<.v, u>.]R) = [<.Y, Z>.]R)
3331, 32opreqan12d 4902 . . . . 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>.]RG[<.z, w>.]R)F([<.x, y>.]RG[<.v, u>.]R)) = ([<.W, X>.]RF[<.Y, Z>.]R))
34 ecoprdi.6 . . . . . 6 |- (((W e. S /\ X e. S) /\ (Y e. S /\ Z e. S)) -> ([<.W, X>.]RF[<.Y, Z>.]R) = [<.K, L>.]R)
35 ecoprdi.8 . . . . . 6 |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S)) -> (W e. S /\ X e. S))
36 ecoprdi.9 . . . . . 6 |- (((x e. S /\ y e. S) /\ (v e. S /\ u e. S)) -> (Y e. S /\ Z e. S))
3734, 35, 36syl2an 503 . . . . 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>.]RF[<.Y, Z>.]R) = [<.K, L>.]R)
3833, 37eqtrd 1925 . . . 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>.]RG[<.z, w>.]R)F([<.x, y>.]RG[<.v, u>.]R)) = [<.K, L>.]R)
39383impdi 1152 . . 3 |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> (([<.x, y>.]RG[<.z, w>.]R)F([<.x, y>.]RG[<.v, u>.]R)) = [<.K, L>.]R)
4022, 30, 393eqtr4a 1954 . 2 |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> ([<.x, y>.]RG([<.z, w>.]RF[<.v, u>.]R)) = (([<.x, y>.]RG[<.z, w>.]R)F([<.x, y>.]RG[<.v, u>.]R)))
411, 6, 11, 16, 403ecoptocl 5364 1 |- ((A e. D /\ B e. D /\ C e. D) -> (AG(BFC)) = ((AGB)F(AGC)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  <.cop 3046   X. cxp 3984  (class class class)co 4884  [cec 5316  /.cqs 5317
This theorem is referenced by:  distrpq 6219  distrsr 6352  axdistr 6432
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524
This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-ec 5320  df-qs 5323
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