Theorem caoprlem2 3083

Description: Lemma used in real number construction.

Hypotheses

Ref	Expression
caoprd.1	|- A e. V
caoprd.2	|- B e. V
caoprd.3	|- C e. V
caoprd.com	|- (xGy) = (yGx)
caoprd.distr	|- (xG(yFz)) = ((xGy)F(xGz))
caoprdl.4	|- D e. V
caoprdl.5	|- H e. V
caoprdl.ass	|- ((xGy)Gz) = (xG(yGz))
caoprdl2.6	|- R e. V
caoprdl2.com	|- (xFy) = (yFx)
caoprdl2.ass	|- ((xFy)Fz) = (xF(yFz))

Assertion

Ref	Expression
caoprlem2	|- ((((AGC)F(BGD))GH)F(((AGD)F(BGC))GR)) = ((AG((CGH)F(DGR)))F(BG((CGR)F(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,R,y,z   x,H,y,z

Proof of Theorem caoprlem2

Step	Hyp	Ref	Expression
1		oprex 3018	. . 3 |- (AG(CGH)) e. V
2		oprex 3018	. . 3 |- (BG(DGH)) e. V
3		oprex 3018	. . 3 |- (AG(DGR)) e. V
4		caoprdl2.com	. . 3 |- (xFy) = (yFx)
5		caoprdl2.ass	. . 3 |- ((xFy)Fz) = (xF(yFz))
6		oprex 3018	. . 3 |- (BG(CGR)) e. V
7	1, 2, 3, 4, 5, 6	caopr42 3080	. 2 |- (((AG(CGH))F(BG(DGH)))F((AG(DGR))F(BG(CGR)))) = (((AG(CGH))F(AG(DGR)))F((BG(CGR))F(BG(DGH))))
8		caoprd.1	. . . 4 |- A e. V
9		caoprd.2	. . . 4 |- B e. V
10		caoprd.3	. . . 4 |- C e. V
11		caoprd.com	. . . 4 |- (xGy) = (yGx)
12		caoprd.distr	. . . 4 |- (xG(yFz)) = ((xGy)F(xGz))
13		caoprdl.4	. . . 4 |- D e. V
14		caoprdl.5	. . . 4 |- H e. V
15		caoprdl.ass	. . . 4 |- ((xGy)Gz) = (xG(yGz))
16	8, 9, 10, 11, 12, 13, 14, 15	caoprdilem 3082	. . 3 |- (((AGC)F(BGD))GH) = ((AG(CGH))F(BG(DGH)))
17		caoprdl2.6	. . . 4 |- R e. V
18	8, 9, 13, 11, 12, 10, 17, 15	caoprdilem 3082	. . 3 |- (((AGD)F(BGC))GR) = ((AG(DGR))F(BG(CGR)))
19	16, 18	opreq12i 3011	. 2 |- ((((AGC)F(BGD))GH)F(((AGD)F(BGC))GR)) = (((AG(CGH))F(BG(DGH)))F((AG(DGR))F(BG(CGR))))
20		oprex 3018	. . . 4 |- (CGH) e. V
21		oprex 3018	. . . 4 |- (DGR) e. V
22	8, 20, 21, 12	caoprdistr 3073	. . 3 |- (AG((CGH)F(DGR))) = ((AG(CGH))F(AG(DGR)))
23		oprex 3018	. . . 4 |- (CGR) e. V
24		oprex 3018	. . . 4 |- (DGH) e. V
25	9, 23, 24, 12	caoprdistr 3073	. . 3 |- (BG((CGR)F(DGH))) = ((BG(CGR))F(BG(DGH)))
26	22, 25	opreq12i 3011	. 2 |- ((AG((CGH)F(DGR)))F(BG((CGR)F(DGH)))) = (((AG(CGH))F(AG(DGR)))F((BG(CGR))F(BG(DGH))))
27	7, 19, 26	3eqtr4 1126	1 |- ((((AGC)F(BGD))GH)F(((AGD)F(BGC))GR)) = ((AG((CGH)F(DGR)))F(BG((CGR)F(DGH))))

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

This theorem is referenced by:  mulasssr 3993

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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