Theorem caoprdistr 4051

Description: Convert an operation distributive law to class notation.

Hypotheses

Ref	Expression
caoprdistr.1	|- A e. V
caoprdistr.2	|- B e. V
caoprdistr.3	|- C e. V
caoprdistr.4	|- (xG(yFz)) = ((xGy)F(xGz))

Assertion

Ref	Expression
caoprdistr	|- (AG(BFC)) = ((AGB)F(AGC))

Distinct variable groups:   x,y,z,F   x,A,y,z   x,B,y,z   x,C,y,z   x,G,y,z

Proof of Theorem caoprdistr

Step	Hyp	Ref	Expression
1		caoprdistr.1	. 2 |- A e. V
2		caoprdistr.2	. 2 |- B e. V
3		caoprdistr.3	. 2 |- C e. V
4		opreq1 3959	. . . 4 |- (x = A -> (xG(yFz)) = (AG(yFz)))
5		opreq1 3959	. . . . 5 |- (x = A -> (xGy) = (AGy))
6		opreq1 3959	. . . . 5 |- (x = A -> (xGz) = (AGz))
7	5, 6	opreq12d 3969	. . . 4 |- (x = A -> ((xGy)F(xGz)) = ((AGy)F(AGz)))
8	4, 7	eqeq12d 1486	. . 3 |- (x = A -> ((xG(yFz)) = ((xGy)F(xGz)) <-> (AG(yFz)) = ((AGy)F(AGz))))
9		opreq1 3959	. . . . 5 |- (y = B -> (yFz) = (BFz))
10	9	opreq2d 3967	. . . 4 |- (y = B -> (AG(yFz)) = (AG(BFz)))
11		opreq2 3960	. . . . 5 |- (y = B -> (AGy) = (AGB))
12	11	opreq1d 3966	. . . 4 |- (y = B -> ((AGy)F(AGz)) = ((AGB)F(AGz)))
13	10, 12	eqeq12d 1486	. . 3 |- (y = B -> ((AG(yFz)) = ((AGy)F(AGz)) <-> (AG(BFz)) = ((AGB)F(AGz))))
14		opreq2 3960	. . . . 5 |- (z = C -> (BFz) = (BFC))
15	14	opreq2d 3967	. . . 4 |- (z = C -> (AG(BFz)) = (AG(BFC)))
16		opreq2 3960	. . . . 5 |- (z = C -> (AGz) = (AGC))
17	16	opreq2d 3967	. . . 4 |- (z = C -> ((AGB)F(AGz)) = ((AGB)F(AGC)))
18	15, 17	eqeq12d 1486	. . 3 |- (z = C -> ((AG(BFz)) = ((AGB)F(AGz)) <-> (AG(BFC)) = ((AGB)F(AGC))))
19	8, 13, 18	syl3an9b 889	. 2 |- ((x = A /\ y = B /\ z = C) -> ((xG(yFz)) = ((xGy)F(xGz)) <-> (AG(BFC)) = ((AGB)F(AGC))))
20		caoprdistr.4	. 2 |- (xG(yFz)) = ((xGy)F(xGz))
21	1, 2, 3, 19, 20	vtocl3 1840	1 |- (AG(BFC)) = ((AGB)F(AGC))

Syntax hints:   = wceq 954   e. wcel 956  Vcvv 1807  (class class class)co 3954

This theorem is referenced by:  caoprdistrr 4059  caoprlem2 4061

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193  df-opr 3956

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