Metamath Proof Explorer


Theorem caovlem2

Description: Lemma used in real number construction. (Contributed by NM, 26-Aug-1995)

Ref Expression
Hypotheses 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 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 ) ) ) )

Proof

Step Hyp Ref Expression
1 caovdir.1
 |-  A e. _V
2 caovdir.2
 |-  B e. _V
3 caovdir.3
 |-  C e. _V
4 caovdir.com
 |-  ( x G y ) = ( y G x )
5 caovdir.distr
 |-  ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) )
6 caovdl.4
 |-  D e. _V
7 caovdl.5
 |-  H e. _V
8 caovdl.ass
 |-  ( ( x G y ) G z ) = ( x G ( y G z ) )
9 caovdl2.6
 |-  R e. _V
10 caovdl2.com
 |-  ( x F y ) = ( y F x )
11 caovdl2.ass
 |-  ( ( x F y ) F z ) = ( x F ( y F z ) )
12 ovex
 |-  ( A G ( C G H ) ) e. _V
13 ovex
 |-  ( B G ( D G H ) ) e. _V
14 ovex
 |-  ( A G ( D G R ) ) e. _V
15 ovex
 |-  ( B G ( C G R ) ) e. _V
16 12 13 14 10 11 15 caov42
 |-  ( ( ( 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 ) ) ) )
17 1 2 3 4 5 6 7 8 caovdilem
 |-  ( ( ( A G C ) F ( B G D ) ) G H ) = ( ( A G ( C G H ) ) F ( B G ( D G H ) ) )
18 1 2 6 4 5 3 9 8 caovdilem
 |-  ( ( ( A G D ) F ( B G C ) ) G R ) = ( ( A G ( D G R ) ) F ( B G ( C G R ) ) )
19 17 18 oveq12i
 |-  ( ( ( ( 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
 |-  ( C G H ) e. _V
21 ovex
 |-  ( D G R ) e. _V
22 1 20 21 5 caovdi
 |-  ( A G ( ( C G H ) F ( D G R ) ) ) = ( ( A G ( C G H ) ) F ( A G ( D G R ) ) )
23 ovex
 |-  ( C G R ) e. _V
24 ovex
 |-  ( D G H ) e. _V
25 2 23 24 5 caovdi
 |-  ( B G ( ( C G R ) F ( D G H ) ) ) = ( ( B G ( C G R ) ) F ( B G ( D G H ) ) )
26 22 25 oveq12i
 |-  ( ( 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 ) ) ) )
27 16 19 26 3eqtr4i
 |-  ( ( ( ( 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 ) ) ) )