Theorem caoprdilem 3082

Description: Lemma used by real number construction.

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))
caoprdl.4	⊢ D ∈ V
caoprdl.5	⊢ H ∈ V
caoprdl.ass	⊢ ((xGy)Gz) = (xG(yGz))

Assertion

Ref	Expression
caoprdilem	⊢ (((AGC)F(BGD))GH) = ((AG(CGH))F(BG(DGH)))

Distinct variable group(s):   x,y,z,F   x,A,y,z   x,B,y,z   x,C,y,z   x,D,y,z   x,G,y,z   x,H,y,z

Proof of Theorem caoprdilem

Step	Hyp	Ref	Expression
1		oprex 3018	. . 3 ⊢ (AGC) ∈ V
2		oprex 3018	. . 3 ⊢ (BGD) ∈ V
3		caoprdl.5	. . 3 ⊢ H ∈ V
4		caoprd.com	. . 3 ⊢ (xGy) = (yGx)
5		caoprd.distr	. . 3 ⊢ (xG(yFz)) = ((xGy)F(xGz))
6	1, 2, 3, 4, 5	caoprdistrr 3081	. 2 ⊢ (((AGC)F(BGD))GH) = (((AGC)GH)F((BGD)GH))
7		caoprd.1	. . . 4 ⊢ A ∈ V
8		caoprd.3	. . . 4 ⊢ C ∈ V
9		caoprdl.ass	. . . 4 ⊢ ((xGy)Gz) = (xG(yGz))
10	7, 8, 3, 9	caoprass 3068	. . 3 ⊢ ((AGC)GH) = (AG(CGH))
11		caoprd.2	. . . 4 ⊢ B ∈ V
12		caoprdl.4	. . . 4 ⊢ D ∈ V
13	11, 12, 3, 9	caoprass 3068	. . 3 ⊢ ((BGD)GH) = (BG(DGH))
14	10, 13	opreq12i 3011	. 2 ⊢ (((AGC)GH)F((BGD)GH)) = ((AG(CGH))F(BG(DGH)))
15	6, 14	eqtr 1119	1 ⊢ (((AGC)F(BGD))GH) = ((AG(CGH))F(BG(DGH)))

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

This theorem is referenced by:  caoprlem2 3083  addasspq 3857  axmulass 4073

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp   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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