Theorem caov4d 5948

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 5945	. . 3 |- ( ph -> ( B F ( C F D ) ) = ( C F ( B F D ) ) )
7	6	oveq2d 5783	. 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 5917	. . 3 |- ( ph -> ( C F D ) e. S )
11	5, 8, 1, 10	caovassd 5923	. 2 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( A F ( B F ( C F D ) ) ) )
12	9, 1, 3	caovcld 5917	. . 3 |- ( ph -> ( B F D ) e. S )
13	5, 8, 2, 12	caovassd 5923	. 2 |- ( ph -> ( ( A F C ) F ( B F D ) ) = ( A F ( C F ( B F D ) ) ) )
14	7, 11, 13	3eqtr4d 2180	1 |- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480  (class class class)co 5767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770

This theorem is referenced by:  caov411d  5949  caov42d  5950  ecopovtrn  6519  ecopovtrng  6522  addcmpblnq  7168  mulcmpblnq  7169  ordpipqqs  7175  distrnqg  7188  ltsonq  7199  ltanqg  7201  ltmnqg  7202  addcmpblnq0  7244  mulcmpblnq0  7245  distrnq0  7260  prarloclemlo  7295  addlocprlemeqgt  7333  addcanprleml  7415  recexprlem1ssl  7434  recexprlem1ssu  7435  mulcmpblnrlemg  7541  distrsrg  7560  ltasrg  7571  mulgt0sr  7579  prsradd  7587  axdistr  7675