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Theorem caovdilem 6222
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 6046 . . 3  |-  ( A G C )  e. 
_V
2 ovex 6046 . . 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 6221 . 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 6187 . . 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 6187 . . 3  |-  ( ( B G D ) G H )  =  ( B G ( D G H ) )
1410, 13oveq12i 6033 . 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 2408 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 1649    e. wcel 1717   _Vcvv 2900  (class class class)co 6021
This theorem is referenced by:  caovlem2  6223  axmulass  8966
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-nul 4280
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-ov 6024
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