Theorem caoprlem2 4069

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 groups:   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 3983	. . 3 |- (AG(CGH)) e. V
2		oprex 3983	. . 3 |- (BG(DGH)) e. V
3		oprex 3983	. . 3 |- (AG(DGR)) e. V
4		caoprdl2.com	. . 3 |- (xFy) = (yFx)
5		caoprdl2.ass	. . 3 |- ((xFy)Fz) = (xF(yFz))
6		oprex 3983	. . 3 |- (BG(CGR)) e. V
7	1, 2, 3, 4, 5, 6	caopr42 4066	. 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 4068	. . 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 4068	. . 3 |- (((AGD)F(BGC))GR) = ((AG(DGR))F(BG(CGR)))
19	16, 18	opreq12i 3973	. 2 |- ((((AGC)F(BGD))GH)F(((AGD)F(BGC))GR)) = (((AG(CGH))F(BG(DGH)))F((AG(DGR))F(BG(CGR))))
20		oprex 3983	. . . 4 |- (CGH) e. V
21		oprex 3983	. . . 4 |- (DGR) e. V
22	8, 20, 21, 12	caoprdistr 4059	. . 3 |- (AG((CGH)F(DGR))) = ((AG(CGH))F(AG(DGR)))
23		oprex 3983	. . . 4 |- (CGR) e. V
24		oprex 3983	. . . 4 |- (DGH) e. V
25	9, 23, 24, 12	caoprdistr 4059	. . 3 |- (BG((CGR)F(DGH))) = ((BG(CGR))F(BG(DGH)))
26	22, 25	opreq12i 3973	. 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 1505	1 |- ((((AGC)F(BGD))GH)F(((AGD)F(BGC))GR)) = ((AG((CGH)F(DGR)))F(BG((CGR)F(DGH))))

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

This theorem is referenced by:  mulasssr 5199

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  ax-un 2866

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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