Theorem caov4d 6241

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

Hypotheses

Ref	Expression
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

Ref	Expression
caov4d	|- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   x, D, y, z   ph, x, y, z   x, F, y, z   x, S, y, z

Proof of Theorem caov4d

Step	Hyp	Ref	Expression
1		caovd.2	. . . 4 |- ( ph -> B e. S )
2		caovd.3	. . . 4 |- ( ph -> C e. S )
3		caovd.4	. . . 4 |- ( ph -> D e. S )
4		caovd.com	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
5		caovd.ass	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
6	1, 2, 3, 4, 5	caov12d 6238	. . 3 |- ( ph -> ( B F ( C F D ) ) = ( C F ( B F D ) ) )
7	6	oveq2d 6068	. 2 |- ( ph -> ( A F ( B F ( C F D ) ) ) = ( A F ( C F ( B F D ) ) ) )
8		caovd.1	. . 3 |- ( ph -> A e. S )
9		caovd.cl	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
10	9, 2, 3	caovcld 6210	. . 3 |- ( ph -> ( C F D ) e. S )
11	5, 8, 1, 10	caovassd 6216	. 2 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( A F ( B F ( C F D ) ) ) )
12	9, 1, 3	caovcld 6210	. . 3 |- ( ph -> ( B F D ) e. S )
13	5, 8, 2, 12	caovassd 6216	. 2 |- ( ph -> ( ( A F C ) F ( B F D ) ) = ( A F ( C F ( B F D ) ) ) )
14	7, 11, 13	3eqtr4d 2277	1 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205  (class class class)co 6052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-iota 5314  df-fv 5362  df-ov 6055

This theorem is referenced by:  caov411d  6242  caov42d  6243  ecopovtrn  6868  ecopovtrng  6871  addcmpblnq  7684  mulcmpblnq  7685  ordpipqqs  7691  distrnqg  7704  ltsonq  7715  ltanqg  7717  ltmnqg  7718  addcmpblnq0  7760  mulcmpblnq0  7761  distrnq0  7776  prarloclemlo  7811  addlocprlemeqgt  7849  addcanprleml  7931  recexprlem1ssl  7950  recexprlem1ssu  7951  mulcmpblnrlemg  8057  distrsrg  8076  ltasrg  8087  mulgt0sr  8095  prsradd  8103  axdistr  8191