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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
StepHypRef 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
71, 2, 3, 4, 5, 6caov42 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))
168, 9, 10, 11, 12, 13, 14, 15caovdilem 5643 . . 3 (((AGC)F(BGD))GH) = ((AG(CGH))F(BG(DGH)))
17 caoprdl2.6 . . . 4 R V
188, 9, 13, 11, 12, 10, 17, 15caovdilem 5643 . . 3 (((AGD)F(BGC))GR) = ((AG(DGR))F(BG(CGR)))
1916, 18oveq12i 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
228, 20, 21, 12caovdi 5634 . . 3 (AG((CGH)F(DGR))) = ((AG(CGH))F(AG(DGR)))
23 ovex 5551 . . . 4 (CGR) V
24 ovex 5551 . . . 4 (DGH) V
259, 23, 24, 12caovdi 5634 . . 3 (BG((CGR)F(DGH))) = ((BG(CGR))F(BG(DGH)))
2622, 25oveq12i 5535 . 2 ((AG((CGH)F(DGR)))F(BG((CGR)F(DGH)))) = (((AG(CGH))F(AG(DGR)))F((BG(CGR))F(BG(DGH))))
277, 19, 263eqtr4i 2383 1 ((((AGC)F(BGD))GH)F(((AGD)F(BGC))GR)) = ((AG((CGH)F(DGR)))F(BG((CGR)F(DGH))))
 Colors of variables: wff setvar class 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)
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