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Theorem caovdilem 5643
 Description: Lemma used by 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))
Assertion
Ref Expression
caovdilem (((AGC)F(BGD))GH) = ((AG(CGH))F(BG(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,H,y,z

Proof of Theorem caovdilem
StepHypRef Expression
1 ovex 5551 . . 3 (AGC) V
2 ovex 5551 . . 3 (BGD) V
3 caoprdl.5 . . 3 H V
4 caoprd.com . . 3 (xGy) = (yGx)
5 caoprd.distr . . 3 (xG(yFz)) = ((xGy)F(xGz))
61, 2, 3, 4, 5caovdir 5642 . 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))
107, 8, 3, 9caovass 5627 . . 3 ((AGC)GH) = (AG(CGH))
11 caoprd.2 . . . 4 B V
12 caoprdl.4 . . . 4 D V
1311, 12, 3, 9caovass 5627 . . 3 ((BGD)GH) = (BG(DGH))
1410, 13oveq12i 5535 . 2 (((AGC)GH)F((BGD)GH)) = ((AG(CGH))F(BG(DGH)))
156, 14eqtri 2373 1 (((AGC)F(BGD))GH) = ((AG(CGH))F(BG(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-1 5  ax-2 6  ax-3 7  ax-mp 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:  caovlem2  5644
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