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Theorem ecoprdi 3257
Description: Lemma used in proving distributive laws via equivalence classes.
Hypotheses
Ref Expression
ecoprdist.1 |- D = ((S X. S)/.R)
ecoprdist.2 |- (((z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> ([<.z, w>.]RF[<.v, u>.]R) = [<.M, N>.]R)
ecoprdist.3 |- (((x e. S /\ y e. S) /\ (M e. S /\ N e. S)) -> ([<.x, y>.]RG[<.M, N>.]R) = [<.H, J>.]R)
ecoprdist.4 |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S)) -> ([<.x, y>.]RG[<.z, w>.]R) = [<.W, X>.]R)
ecoprdist.5 |- (((x e. S /\ y e. S) /\ (v e. S /\ u e. S)) -> ([<.x, y>.]RG[<.v, u>.]R) = [<.Y, Z>.]R)
ecoprdist.6 |- (((W e. S /\ X e. S) /\ (Y e. S /\ Z e. S)) -> ([<.W, X>.]RF[<.Y, Z>.]R) = [<.K, L>.]R)
ecoprdist.7 |- (((z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> (M e. S /\ N e. S))
ecoprdist.8 |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S)) -> (W e. S /\ X e. S))
ecoprdist.9 |- (((x e. S /\ y e. S) /\ (v e. S /\ u e. S)) -> (Y e. S /\ Z e. S))
ecoprdist.10 |- H = K
ecoprdist.11 |- J = L
Assertion
Ref Expression
ecoprdi |- ((A e. D /\ B e. D /\ C e. D) -> (AG(BFC)) = ((AGB)F(AGC)))
Distinct variable group(s):   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 ecoprdist.1 . 2 |- D = ((S X. S)/.R)
2 opreq1 3006 . . 3 |- ([<.x, y>.]R = A -> ([<.x, y>.]RG([<.z, w>.]RF[<.v, u>.]R)) = (AG([<.z, w>.]RF[<.v, u>.]R)))
3 opreq1 3006 . . . 4 |- ([<.x, y>.]R = A -> ([<.x, y>.]RG[<.z, w>.]R) = (AG[<.z, w>.]R))
4 opreq1 3006 . . . 4 |- ([<.x, y>.]R = A -> ([<.x, y>.]RG[<.v, u>.]R) = (AG[<.v, u>.]R))
53, 4opreq12d 3014 . . 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, 5cleq12d 1115 . 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 3006 . . . 4 |- ([<.z, w>.]R = B -> ([<.z, w>.]RF[<.v, u>.]R) = (BF[<.v, u>.]R))
87opreq2d 3013 . . 3 |- ([<.z, w>.]R = B -> (AG([<.z, w>.]RF[<.v, u>.]R)) = (AG(BF[<.v, u>.]R)))
9 opreq2 3007 . . . 4 |- ([<.z, w>.]R = B -> (AG[<.z, w>.]R) = (AGB))
109opreq1d 3012 . . 3 |- ([<.z, w>.]R = B -> ((AG[<.z, w>.]R)F(AG[<.v, u>.]R)) = ((AGB)F(AG[<.v, u>.]R)))
118, 10cleq12d 1115 . 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 3007 . . . 4 |- ([<.v, u>.]R = C -> (BF[<.v, u>.]R) = (BFC))
1312opreq2d 3013 . . 3 |- ([<.v, u>.]R = C -> (AG(BF[<.v, u>.]R)) = (AG(BFC)))
14 opreq2 3007 . . . 4 |- ([<.v, u>.]R = C -> (AG[<.v, u>.]R) = (AGC))
1514opreq2d 3013 . . 3 |- ([<.v, u>.]R = C -> ((AGB)F(AG[<.v, u>.]R)) = ((AGB)F(AGC)))
1613, 15cleq12d 1115 . 2 |- ([<.v, u>.]R = C -> ((AG(BF[<.v, u>.]R)) = ((AGB)F(AG[<.v, u>.]R)) <-> (AG(BFC)) = ((AGB)F(AGC))))
17 ecoprdist.2 . . . . . . . 8 |- (((z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> ([<.z, w>.]RF[<.v, u>.]R) = [<.M, N>.]R)
1817opreq2d 3013 . . . . . . 7 |- (((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))
1918adantl 305 . . . . . 6 |- (((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))
20 ecoprdist.3 . . . . . . 7 |- (((x e. S /\ y e. S) /\ (M e. S /\ N e. S)) -> ([<.x, y>.]RG[<.M, N>.]R) = [<.H, J>.]R)
21 ecoprdist.7 . . . . . . 7 |- (((z e. S /\ w e. S) /\ (v e. S /\ u e. S)) -> (M e. S /\ N e. S))
2220, 21sylan2 346 . . . . . 6 |- (((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)
2319, 22eqtrd 1128 . . . . 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)) = [<.H, J>.]R)
24233impb 610 . . . 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)
25 ecoprdist.10 . . . . 5 |- H = K
26 ecoprdist.11 . . . . 5 |- J = L
27 opeq12 1878 . . . . . 6 |- ((H = K /\ J = L) -> <.H, J>. = <.K, L>.)
28 eceq2 3215 . . . . . 6 |- (<.H, J>. = <.K, L>. -> [<.H, J>.]R = [<.K, L>.]R)
2927, 28syl 12 . . . . 5 |- ((H = K /\ J = L) -> [<.H, J>.]R = [<.K, L>.]R)
3025, 26, 29mp2an 520 . . . 4 |- [<.H, J>.]R = [<.K, L>.]R
3124, 30syl6eq 1140 . . 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)) = [<.K, L>.]R)
32 ecoprdist.4 . . . . . 6 |- (((x e. S /\ y e. S) /\ (z e. S /\ w e. S)) -> ([<.x, y>.]RG[<.z, w>.]R) = [<.