Theorem caoprdistr 4059

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 3968	. . . 4 |- (x = A -> (xG(yFz)) = (AG(yFz)))
5		opreq1 3968	. . . . 5 |- (x = A -> (xGy) = (AGy))
6		opreq1 3968	. . . . 5 |- (x = A -> (xGz) = (AGz))
7	5, 6	opreq12d 3978	. . . 4 |- (x = A -> ((xGy)F(xGz)) = ((AGy)F(AGz)))
8	4, 7	eqeq12d 1489	. . 3 |- (x = A -> ((xG(yFz)) = ((xGy)F(xGz)) <-> (AG(yFz)) = ((AGy)F(AGz))))
9		opreq1 3968	. . . . 5 |- (y = B -> (yFz) = (BFz))
10	9	opreq2d 3976	. . . 4 |- (y = B -> (AG(yFz)) = (AG(BFz)))
11		opreq2 3969	. . . . 5 |- (y = B -> (AGy) = (AGB))
12	11	opreq1d 3975	. . . 4 |- (y = B -> ((AGy)F(AGz)) = ((AGB)F(AGz)))
13	10, 12	eqeq12d 1489	. . 3 |- (y = B -> ((AG(yFz)) = ((AGy)F(AGz)) <-> (AG(BFz)) = ((AGB)F(AGz))))
14		opreq2 3969	. . . . 5 |- (z = C -> (BFz) = (BFC))
15	14	opreq2d 3976	. . . 4 |- (z = C -> (AG(BFz)) = (AG(BFC)))
16		opreq2 3969	. . . . 5 |- (z = C -> (AGz) = (AGC))
17	16	opreq2d 3976	. . . 4 |- (z = C -> ((AGB)F(AGz)) = ((AGB)F(AGC)))
18	15, 17	eqeq12d 1489	. . 3 |- (z = C -> ((AG(BFz)) = ((AGB)F(AGz)) <-> (AG(BFC)) = ((AGB)F(AGC))))
19	8, 13, 18	syl3an9b 891	. 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 1844	1 |- (AG(BFC)) = ((AGB)F(AGC))

Syntax hints:   = wceq 956   e. wcel 958  Vcvv 1811  (class class class)co 3963

This theorem is referenced by:  caoprdistrr 4067  caoprlem2 4069

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965

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