Theorem caov4d 6247

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 6244	. . 3 |- ( ph -> ( B F ( C F D ) ) = ( C F ( B F D ) ) )
7	6	oveq2d 6074	. 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 6216	. . 3 |- ( ph -> ( C F D ) e. S )
11	5, 8, 1, 10	caovassd 6222	. 2 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( A F ( B F ( C F D ) ) ) )
12	9, 1, 3	caovcld 6216	. . 3 |- ( ph -> ( B F D ) e. S )
13	5, 8, 2, 12	caovassd 6222	. 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 6058

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 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061

This theorem is referenced by:  caov411d  6248  caov42d  6249  ecopovtrn  6879  ecopovtrng  6882  addcmpblnq  7698  mulcmpblnq  7699  ordpipqqs  7705  distrnqg  7718  ltsonq  7729  ltanqg  7731  ltmnqg  7732  addcmpblnq0  7774  mulcmpblnq0  7775  distrnq0  7790  prarloclemlo  7825  addlocprlemeqgt  7863  addcanprleml  7945  recexprlem1ssl  7964  recexprlem1ssu  7965  mulcmpblnrlemg  8071  distrsrg  8090  ltasrg  8101  mulgt0sr  8109  prsradd  8117  axdistr  8205