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Theorem caovlem2 6285
Description: Lemma used in real number construction. (Contributed by NM, 26-Aug-1995.)
Hypotheses
Ref Expression
caovdir.1  |-  A  e. 
_V
caovdir.2  |-  B  e. 
_V
caovdir.3  |-  C  e. 
_V
caovdir.com  |-  ( x G y )  =  ( y G x )
caovdir.distr  |-  ( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) )
caovdl.4  |-  D  e. 
_V
caovdl.5  |-  H  e. 
_V
caovdl.ass  |-  ( ( x G y ) G z )  =  ( x G ( y G z ) )
caovdl2.6  |-  R  e. 
_V
caovdl2.com  |-  ( x F y )  =  ( y F x )
caovdl2.ass  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
Assertion
Ref Expression
caovlem2  |-  ( ( ( ( A G C ) F ( B G D ) ) G H ) F ( ( ( A G D ) F ( B G C ) ) G R ) )  =  ( ( A G ( ( C G H ) F ( D G R ) ) ) F ( B G ( ( C G R ) F ( D G H ) ) ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    x, D, y, z    x, F, y, z    x, G, y, z    x, H, y, z    x, R, y, z

Proof of Theorem caovlem2
StepHypRef Expression
1 ovex 6108 . . 3  |-  ( A G ( C G H ) )  e. 
_V
2 ovex 6108 . . 3  |-  ( B G ( D G H ) )  e. 
_V
3 ovex 6108 . . 3  |-  ( A G ( D G R ) )  e. 
_V
4 caovdl2.com . . 3  |-  ( x F y )  =  ( y F x )
5 caovdl2.ass . . 3  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
6 ovex 6108 . . 3  |-  ( B G ( C G R ) )  e. 
_V
71, 2, 3, 4, 5, 6caov42 6282 . 2  |-  ( ( ( A G ( C G H ) ) F ( B G ( D G H ) ) ) F ( ( A G ( D G R ) ) F ( B G ( C G R ) ) ) )  =  ( ( ( A G ( C G H ) ) F ( A G ( D G R ) ) ) F ( ( B G ( C G R ) ) F ( B G ( D G H ) ) ) )
8 caovdir.1 . . . 4  |-  A  e. 
_V
9 caovdir.2 . . . 4  |-  B  e. 
_V
10 caovdir.3 . . . 4  |-  C  e. 
_V
11 caovdir.com . . . 4  |-  ( x G y )  =  ( y G x )
12 caovdir.distr . . . 4  |-  ( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) )
13 caovdl.4 . . . 4  |-  D  e. 
_V
14 caovdl.5 . . . 4  |-  H  e. 
_V
15 caovdl.ass . . . 4  |-  ( ( x G y ) G z )  =  ( x G ( y G z ) )
168, 9, 10, 11, 12, 13, 14, 15caovdilem 6284 . . 3  |-  ( ( ( A G C ) F ( B G D ) ) G H )  =  ( ( A G ( C G H ) ) F ( B G ( D G H ) ) )
17 caovdl2.6 . . . 4  |-  R  e. 
_V
188, 9, 13, 11, 12, 10, 17, 15caovdilem 6284 . . 3  |-  ( ( ( A G D ) F ( B G C ) ) G R )  =  ( ( A G ( D G R ) ) F ( B G ( C G R ) ) )
1916, 18oveq12i 6095 . 2  |-  ( ( ( ( A G C ) F ( B G D ) ) G H ) F ( ( ( A G D ) F ( B G C ) ) G R ) )  =  ( ( ( A G ( C G H ) ) F ( B G ( D G H ) ) ) F ( ( A G ( D G R ) ) F ( B G ( C G R ) ) ) )
20 ovex 6108 . . . 4  |-  ( C G H )  e. 
_V
21 ovex 6108 . . . 4  |-  ( D G R )  e. 
_V
228, 20, 21, 12caovdi 6268 . . 3  |-  ( A G ( ( C G H ) F ( D G R ) ) )  =  ( ( A G ( C G H ) ) F ( A G ( D G R ) ) )
23 ovex 6108 . . . 4  |-  ( C G R )  e. 
_V
24 ovex 6108 . . . 4  |-  ( D G H )  e. 
_V
259, 23, 24, 12caovdi 6268 . . 3  |-  ( B G ( ( C G R ) F ( D G H ) ) )  =  ( ( B G ( C G R ) ) F ( B G ( D G H ) ) )
2622, 25oveq12i 6095 . 2  |-  ( ( A G ( ( C G H ) F ( D G R ) ) ) F ( B G ( ( C G R ) F ( D G H ) ) ) )  =  ( ( ( A G ( C G H ) ) F ( A G ( D G R ) ) ) F ( ( B G ( C G R ) ) F ( B G ( D G H ) ) ) )
277, 19, 263eqtr4i 2468 1  |-  ( ( ( ( A G C ) F ( B G D ) ) G H ) F ( ( ( A G D ) F ( B G C ) ) G R ) )  =  ( ( A G ( ( C G H ) F ( D G R ) ) ) F ( B G ( ( C G R ) F ( D G H ) ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   _Vcvv 2958  (class class class)co 6083
This theorem is referenced by:  mulasssr  8967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4340
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086
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