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Theorem caovlem2 6870
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 6678 . . 3 |- ( A G ( C G H ) ) e. _V
2 ovex 6678 . . 3 |- ( B G ( D G H ) ) e. _V
3 ovex 6678 . . 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 6678 . . 3 |- ( B G ( C G R ) ) e. _V
71, 2, 3, 4, 5, 6caov42 6867 . 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 6869 . . 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 6869 . . 3 |- ( ( ( A G D ) F ( B G C ) ) G R ) = ( ( A G ( D G R ) ) F ( B G ( C G R ) ) )
1916, 18oveq12i 6662 . 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 6678 . . . 4 |- ( C G H ) e. _V
21 ovex 6678 . . . 4 |- ( D G R ) e. _V
228, 20, 21, 12caovdi 6853 . . 3 |- ( A G ( ( C G H ) F ( D G R ) ) ) = ( ( A G ( C G H ) ) F ( A G ( D G R ) ) )
23 ovex 6678 . . . 4 |- ( C G R ) e. _V
24 ovex 6678 . . . 4 |- ( D G H ) e. _V
259, 23, 24, 12caovdi 6853 . . 3 |- ( B G ( ( C G R ) F ( D G H ) ) ) = ( ( B G ( C G R ) ) F ( B G ( D G H ) ) )
2622, 25oveq12i 6662 . 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 2654 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 setvar class
Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200  (class class class)co 6650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653
This theorem is referenced by:  mulasssr  9911
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