Theorem caovdilem 5644

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

Step	Hyp	Ref	Expression
1		ovex 5552	. . 3 ⊢ (AGC) ∈ V
2		ovex 5552	. . 3 ⊢ (BGD) ∈ V
3		caoprdl.5	. . 3 ⊢ H ∈ V
4		caoprd.com	. . 3 ⊢ (xGy) = (yGx)
5		caoprd.distr	. . 3 ⊢ (xG(yFz)) = ((xGy)F(xGz))
6	1, 2, 3, 4, 5	caovdir 5643	. 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))
10	7, 8, 3, 9	caovass 5628	. . 3 ⊢ ((AGC)GH) = (AG(CGH))
11		caoprd.2	. . . 4 ⊢ B ∈ V
12		caoprdl.4	. . . 4 ⊢ D ∈ V
13	11, 12, 3, 9	caovass 5628	. . 3 ⊢ ((BGD)GH) = (BG(DGH))
14	10, 13	oveq12i 5536	. 2 ⊢ (((AGC)GH)F((BGD)GH)) = ((AG(CGH))F(BG(DGH)))
15	6, 14	eqtri 2373	1 ⊢ (((AGC)F(BGD))GH) = ((AG(CGH))F(BG(DGH)))

Syntax hints:   = wceq 1642   ∈ wcel 1710  Vcvv 2860  (class class class)co 5526

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-fv 4796  df-ov 5527

This theorem is referenced by:  caovlem2  5645