Theorem caoprdistrr 3081

Description: Reverse distributive law.

Hypotheses

Ref	Expression
caoprd.1	⊢ A ∈ V
caoprd.2	⊢ B ∈ V
caoprd.3	⊢ C ∈ V
caoprd.com	⊢ (xGy) = (yGx)
caoprd.distr	⊢ (xG(yFz)) = ((xGy)F(xGz))

Assertion

Ref	Expression
caoprdistrr	⊢ ((AFB)GC) = ((AGC)F(BGC))

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 caoprdistrr

Step	Hyp	Ref	Expression
1		caoprd.3	. . 3 ⊢ C ∈ V
2		caoprd.1	. . 3 ⊢ A ∈ V
3		caoprd.2	. . 3 ⊢ B ∈ V
4		caoprd.distr	. . 3 ⊢ (xG(yFz)) = ((xGy)F(xGz))
5	1, 2, 3, 4	caoprdistr 3073	. 2 ⊢ (CG(AFB)) = ((CGA)F(CGB))
6		oprex 3018	. . 3 ⊢ (AFB) ∈ V
7		caoprd.com	. . 3 ⊢ (xGy) = (yGx)
8	1, 6, 7	caoprcom 3067	. 2 ⊢ (CG(AFB)) = ((AFB)GC)
9	1, 2, 7	caoprcom 3067	. . 3 ⊢ (CGA) = (AGC)
10	1, 3, 7	caoprcom 3067	. . 3 ⊢ (CGB) = (BGC)
11	9, 10	opreq12i 3011	. 2 ⊢ ((CGA)F(CGB)) = ((AGC)F(BGC))
12	5, 8, 11	3eqtr3 1124	1 ⊢ ((AFB)GC) = ((AGC)F(BGC))

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

This theorem is referenced by:  caoprdilem 3082  addcmpblnq 3846  ltapq 3870  prlem934a 3931  mulcmpblnrlem 3976  recexsrlem 4006  mulgt0sr 4008  axrecex 4079

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-un 1076  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-eu 1009  df-mo 1010  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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