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Metamath Proof Explorer

Theorem caov4d

Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995) (Revised by Mario Carneiro, 30-Dec-2014)

Ref Expression
Hypotheses caovd.1 
|- ( ph -> A e. S )
caovd.2 
|- ( ph -> B e. S )
caovd.3 
|- ( ph -> C e. S )
caovd.com 
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
caovd.ass 
|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
caovd.4 
|- ( ph -> D e. S )
caovd.cl 
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
Assertion caov4d 
|- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) ) )

Proof

Step Hyp Ref Expression
1 caovd.1 
 |-  ( ph -> A e. S )
2 caovd.2 
 |-  ( ph -> B e. S )
3 caovd.3 
 |-  ( ph -> C e. S )
4 caovd.com 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
5 caovd.ass 
 |-  ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
6 caovd.4 
 |-  ( ph -> D e. S )
7 caovd.cl 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
8 2 3 6 4 5 caov12d 
 |-  ( ph -> ( B F ( C F D ) ) = ( C F ( B F D ) ) )
9 8 oveq2d 
 |-  ( ph -> ( A F ( B F ( C F D ) ) ) = ( A F ( C F ( B F D ) ) ) )
10 7 3 6 caovcld 
 |-  ( ph -> ( C F D ) e. S )
11 5 1 2 10 caovassd 
 |-  ( ph -> ( ( A F B ) F ( C F D ) ) = ( A F ( B F ( C F D ) ) ) )
12 7 2 6 caovcld 
 |-  ( ph -> ( B F D ) e. S )
13 5 1 3 12 caovassd 
 |-  ( ph -> ( ( A F C ) F ( B F D ) ) = ( A F ( C F ( B F D ) ) ) )
14 9 11 13 3eqtr4d 
 |-  ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) ) )