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 ) ) ) ) |