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Theorem ecoprdi 4305
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 3953 . . 3 |- ([<.x, y>.]R = A -> ([<.x, y>.]RG([<.z, w>.]RF[<.v, u>.]R)) = (AG([<.z, w>.]RF[<.v, u>.]R)))
3 opreq1 3953 . . . 4 |- ([<.x, y>.]R = A -> ([<.x, y>.]RG[<.z, w>.]R) = (AG[<.z, w>.]R))
4 opreq1 3953 . . . 4 |- ([<.x, y>.]R = A -> ([<.x, y>.]RG[<.v, u>.]R) = (AG[<.v, u>.]R))
53, 4opreq12d 3963 . . 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 1481 . 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 3953 . . . 4 |- ([<.z, w>.]R = B -> ([<.z, w>.]RF[<.v, u>.]R) = (BF[<.v, u>.]R))
87opreq2d 3961 . . 3 |- ([<.z, w>.]R = B -> (AG([<.z, w>.]RF[<.v, u>.]R)) = (AG(BF[<.v, u>.]R)))
9 opreq2 3954 . . . 4 |- ([<.z, w>.]R = B -> (AG[<.z, w>.]R) = (AGB))
109opreq1d 3960 . . 3 |- ([<.z, w>.]R = B -> ((AG[<.z, w>.]R)F(AG[<.v, u>.]R)) = ((AGB)F(AG[<.v, u>.]R)))
118, 10eqeq12d 1481 . 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 3954 . . . 4 |- ([<.v, u>.]R = C -> (BF[<.v, u>.]R) = (BFC))
1312opreq2d 3961 . . 3 |- ([<.v, u>.]R = C -> (AG(BF[<.v, u>.]R)) = (AG(BFC)))
14 opreq2 3954 . . . 4 |- ([<.v, u>.]R = C -> (AG[<.v, u>.]R) = (AGC))
1514opreq2d 3961 . . 3 |- ([<.v, u>.]R = C -> ((AGB)F(AG[<.v, u>.]R)) = ((AGB)F(AGC)))
1613, 15eqeq12d 1481 . 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 2480 . . . . 5 |- ((H = K /\ J = L) -> <.H, J>. = <.K, L>.)
20 eceq2 4262 . . . . 5 |- (<.H, J>. = <.K, L>. -> [<.H, J>.]R = [<.K, L>.]R)
2119, 20syl 10 . . . 4 |- ((H = K /\ J = L) -> [<.H, J>.]R = [<.K, L>.]R)
2217, 18, 21mp2an 695 . . 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 3961 . . . . . 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 388 . . . . 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 451 . . . . 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 1499 . . . 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 827 . . 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