Theorem caov4d 5821

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 5818	. . 3 |- ( ph -> ( B F ( C F D ) ) = ( C F ( B F D ) ) )
7	6	oveq2d 5660	. 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 5790	. . 3 |- ( ph -> ( C F D ) e. S )
11	5, 8, 1, 10	caovassd 5796	. 2 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( A F ( B F ( C F D ) ) ) )
12	9, 1, 3	caovcld 5790	. . 3 |- ( ph -> ( B F D ) e. S )
13	5, 8, 2, 12	caovassd 5796	. 2 |- ( ph -> ( ( A F C ) F ( B F D ) ) = ( A F ( C F ( B F D ) ) ) )
14	7, 11, 13	3eqtr4d 2130	1 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 924   = wceq 1289   e. wcel 1438  (class class class)co 5644

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-iota 4975  df-fv 5018  df-ov 5647

This theorem is referenced by:  caov411d  5822  caov42d  5823  ecopovtrn  6379  ecopovtrng  6382  addcmpblnq  6916  mulcmpblnq  6917  ordpipqqs  6923  distrnqg  6936  ltsonq  6947  ltanqg  6949  ltmnqg  6950  addcmpblnq0  6992  mulcmpblnq0  6993  distrnq0  7008  prarloclemlo  7043  addlocprlemeqgt  7081  addcanprleml  7163  recexprlem1ssl  7182  recexprlem1ssu  7183  mulcmpblnrlemg  7276  distrsrg  7295  ltasrg  7306  mulgt0sr  7313  prsradd  7321  axdistr  7399