Theorem caov4d 6131

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 6128	. . 3 |- ( ph -> ( B F ( C F D ) ) = ( C F ( B F D ) ) )
7	6	oveq2d 5960	. 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 6100	. . 3 |- ( ph -> ( C F D ) e. S )
11	5, 8, 1, 10	caovassd 6106	. 2 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( A F ( B F ( C F D ) ) ) )
12	9, 1, 3	caovcld 6100	. . 3 |- ( ph -> ( B F D ) e. S )
13	5, 8, 2, 12	caovassd 6106	. 2 |- ( ph -> ( ( A F C ) F ( B F D ) ) = ( A F ( C F ( B F D ) ) ) )
14	7, 11, 13	3eqtr4d 2248	1 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2176  (class class class)co 5944

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  caov411d  6132  caov42d  6133  ecopovtrn  6719  ecopovtrng  6722  addcmpblnq  7480  mulcmpblnq  7481  ordpipqqs  7487  distrnqg  7500  ltsonq  7511  ltanqg  7513  ltmnqg  7514  addcmpblnq0  7556  mulcmpblnq0  7557  distrnq0  7572  prarloclemlo  7607  addlocprlemeqgt  7645  addcanprleml  7727  recexprlem1ssl  7746  recexprlem1ssu  7747  mulcmpblnrlemg  7853  distrsrg  7872  ltasrg  7883  mulgt0sr  7891  prsradd  7899  axdistr  7987