Theorem caovlem2 5644

Description: Lemma used in real number construction. (Contributed by set.mm contributors, 26-Aug-1995.)

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))
caoprdl2.6	⊢ R ∈ V
caoprdl2.com	⊢ (xFy) = (yFx)
caoprdl2.ass	⊢ ((xFy)Fz) = (xF(yFz))

Assertion

Ref	Expression
caovlem2	⊢ ((((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 caovlem2

Step	Hyp	Ref	Expression
1		ovex 5551	. . 3 ⊢ (AG(CGH)) ∈ V
2		ovex 5551	. . 3 ⊢ (BG(DGH)) ∈ V
3		ovex 5551	. . 3 ⊢ (AG(DGR)) ∈ V
4		caoprdl2.com	. . 3 ⊢ (xFy) = (yFx)
5		caoprdl2.ass	. . 3 ⊢ ((xFy)Fz) = (xF(yFz))
6		ovex 5551	. . 3 ⊢ (BG(CGR)) ∈ V
7	1, 2, 3, 4, 5, 6	caov42 5641	. 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 ∈ V
9		caoprd.2	. . . 4 ⊢ B ∈ V
10		caoprd.3	. . . 4 ⊢ C ∈ V
11		caoprd.com	. . . 4 ⊢ (xGy) = (yGx)
12		caoprd.distr	. . . 4 ⊢ (xG(yFz)) = ((xGy)F(xGz))
13		caoprdl.4	. . . 4 ⊢ D ∈ V
14		caoprdl.5	. . . 4 ⊢ H ∈ V
15		caoprdl.ass	. . . 4 ⊢ ((xGy)Gz) = (xG(yGz))
16	8, 9, 10, 11, 12, 13, 14, 15	caovdilem 5643	. . 3 ⊢ (((AGC)F(BGD))GH) = ((AG(CGH))F(BG(DGH)))
17		caoprdl2.6	. . . 4 ⊢ R ∈ V
18	8, 9, 13, 11, 12, 10, 17, 15	caovdilem 5643	. . 3 ⊢ (((AGD)F(BGC))GR) = ((AG(DGR))F(BG(CGR)))
19	16, 18	oveq12i 5535	. 2 ⊢ ((((AGC)F(BGD))GH)F(((AGD)F(BGC))GR)) = (((AG(CGH))F(BG(DGH)))F((AG(DGR))F(BG(CGR))))
20		ovex 5551	. . . 4 ⊢ (CGH) ∈ V
21		ovex 5551	. . . 4 ⊢ (DGR) ∈ V
22	8, 20, 21, 12	caovdi 5634	. . 3 ⊢ (AG((CGH)F(DGR))) = ((AG(CGH))F(AG(DGR)))
23		ovex 5551	. . . 4 ⊢ (CGR) ∈ V
24		ovex 5551	. . . 4 ⊢ (DGH) ∈ V
25	9, 23, 24, 12	caovdi 5634	. . 3 ⊢ (BG((CGR)F(DGH))) = ((BG(CGR))F(BG(DGH)))
26	22, 25	oveq12i 5535	. 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	3eqtr4i 2383	1 ⊢ ((((AGC)F(BGD))GH)F(((AGD)F(BGC))GR)) = ((AG((CGH)F(DGR)))F(BG((CGR)F(DGH))))

Syntax hints:   = wceq 1642   ∈ wcel 1710  Vcvv 2859  (class class class)co 5525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640  df-fv 4795  df-ov 5526

This theorem is referenced by: (None)