Theorem caoprdistr 3073

Description: Convert an operation distributive law to class notation.

Hypotheses

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

Assertion

Ref	Expression
caoprdistr	⊢ (AG(BFC)) = ((AGB)F(AGC))

Distinct variable group(s):   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 ∈ V
2		caoprdistr.2	. 2 ⊢ B ∈ V
3		caoprdistr.3	. 2 ⊢ C ∈ V
4		opreq1 3006	. . . 4 ⊢ (x = A → (xG(yFz)) = (AG(yFz)))
5		opreq1 3006	. . . . 5 ⊢ (x = A → (xGy) = (AGy))
6		opreq1 3006	. . . . 5 ⊢ (x = A → (xGz) = (AGz))
7	5, 6	opreq12d 3014	. . . 4 ⊢ (x = A → ((xGy)F(xGz)) = ((AGy)F(AGz)))
8	4, 7	cleq12d 1115	. . 3 ⊢ (x = A → ((xG(yFz)) = ((xGy)F(xGz)) ↔ (AG(yFz)) = ((AGy)F(AGz))))
9		opreq1 3006	. . . . 5 ⊢ (y = B → (yFz) = (BFz))
10	9	opreq2d 3013	. . . 4 ⊢ (y = B → (AG(yFz)) = (AG(BFz)))
11		opreq2 3007	. . . . 5 ⊢ (y = B → (AGy) = (AGB))
12	11	opreq1d 3012	. . . 4 ⊢ (y = B → ((AGy)F(AGz)) = ((AGB)F(AGz)))
13	10, 12	cleq12d 1115	. . 3 ⊢ (y = B → ((AG(yFz)) = ((AGy)F(AGz)) ↔ (AG(BFz)) = ((AGB)F(AGz))))
14		opreq2 3007	. . . . 5 ⊢ (z = C → (BFz) = (BFC))
15	14	opreq2d 3013	. . . 4 ⊢ (z = C → (AG(BFz)) = (AG(BFC)))
16		opreq2 3007	. . . . 5 ⊢ (z = C → (AGz) = (AGC))
17	16	opreq2d 3013	. . . 4 ⊢ (z = C → ((AGB)F(AGz)) = ((AGB)F(AGC)))
18	15, 17	cleq12d 1115	. . 3 ⊢ (z = C → ((AG(BFz)) = ((AGB)F(AGz)) ↔ (AG(BFC)) = ((AGB)F(AGC))))
19	8, 13, 18	syl3an9b 634	. 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 1380	1 ⊢ (AG(BFC)) = ((AGB)F(AGC))

Syntax hints:   = wceq 1091   ∈ wcel 1092  Vcvv 1348  (class class class)co 3001

This theorem is referenced by:  caoprdistrr 3081  caoprlem2 3083

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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