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Theorem caovdilem 6274
Description: Lemma used by 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 ) )
Assertion
Ref Expression
caovdilem  |-  ( ( ( A G C ) F ( B G D ) ) G H )  =  ( ( A G ( C G H ) ) F ( B G ( 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

Proof of Theorem caovdilem
StepHypRef Expression
1 ovex 6098 . . 3  |-  ( A G C )  e. 
_V
2 ovex 6098 . . 3  |-  ( B G D )  e. 
_V
3 caovdl.5 . . 3  |-  H  e. 
_V
4 caovdir.com . . 3  |-  ( x G y )  =  ( y G x )
5 caovdir.distr . . 3  |-  ( x G ( y F z ) )  =  ( ( x G y ) F ( x G z ) )
61, 2, 3, 4, 5caovdir 6273 . 2  |-  ( ( ( A G C ) F ( B G D ) ) G H )  =  ( ( ( A G C ) G H ) F ( ( B G D ) G H ) )
7 caovdir.1 . . . 4  |-  A  e. 
_V
8 caovdir.3 . . . 4  |-  C  e. 
_V
9 caovdl.ass . . . 4  |-  ( ( x G y ) G z )  =  ( x G ( y G z ) )
107, 8, 3, 9caovass 6239 . . 3  |-  ( ( A G C ) G H )  =  ( A G ( C G H ) )
11 caovdir.2 . . . 4  |-  B  e. 
_V
12 caovdl.4 . . . 4  |-  D  e. 
_V
1311, 12, 3, 9caovass 6239 . . 3  |-  ( ( B G D ) G H )  =  ( B G ( D G H ) )
1410, 13oveq12i 6085 . 2  |-  ( ( ( A G C ) G H ) F ( ( B G D ) G H ) )  =  ( ( A G ( C G H ) ) F ( B G ( D G H ) ) )
156, 14eqtri 2455 1  |-  ( ( ( A G C ) F ( B G D ) ) G H )  =  ( ( A G ( C G H ) ) F ( B G ( D G H ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2948  (class class class)co 6073
This theorem is referenced by:  caovlem2  6275  axmulass  9024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076
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