Theorem caofass 6931

Description: Transfer an associative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)

Hypotheses

Ref	Expression
caofref.1	|- ( ph -> A e. V )
caofref.2	|- ( ph -> F : A --> S )
caofcom.3	|- ( ph -> G : A --> S )
caofass.4	|- ( ph -> H : A --> S )
caofass.5	|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x R y ) T z ) = ( x O ( y P z ) ) )

Assertion

Ref	Expression
caofass	|- ( ph -> ( ( F oF R G ) oF T H ) = ( F oF O ( G oF P H ) ) )

Distinct variable groups:   x, y, z, F   x, G, y, z   x, H, y, z   x, O, y, z   x, P, y, z   ph, x, y, z   x, R, y, z   x, S, y, z   x, T, y, z

Allowed substitution hints:   A( x, y, z)   V( x, y, z)

Proof of Theorem caofass

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofass.5	. . . . . 6 |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x R y ) T z ) = ( x O ( y P z ) ) )
2	1	ralrimivvva 2972	. . . . 5 |- ( ph -> A. x e. S A. y e. S A. z e. S ( ( x R y ) T z ) = ( x O ( y P z ) ) )
3	2	adantr 481	. . . 4 |- ( ( ph /\ w e. A ) -> A. x e. S A. y e. S A. z e. S ( ( x R y ) T z ) = ( x O ( y P z ) ) )
4		caofref.2	. . . . . 6 |- ( ph -> F : A --> S )
5	4	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
6		caofcom.3	. . . . . 6 |- ( ph -> G : A --> S )
7	6	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ w e. A ) -> ( G ` w ) e. S )
8		caofass.4	. . . . . 6 |- ( ph -> H : A --> S )
9	8	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ w e. A ) -> ( H ` w ) e. S )
10		oveq1 6657	. . . . . . . 8 |- ( x = ( F ` w ) -> ( x R y ) = ( ( F ` w ) R y ) )
11	10	oveq1d 6665	. . . . . . 7 |- ( x = ( F ` w ) -> ( ( x R y ) T z ) = ( ( ( F ` w ) R y ) T z ) )
12		oveq1 6657	. . . . . . 7 |- ( x = ( F ` w ) -> ( x O ( y P z ) ) = ( ( F ` w ) O ( y P z ) ) )
13	11, 12	eqeq12d 2637	. . . . . 6 |- ( x = ( F ` w ) -> ( ( ( x R y ) T z ) = ( x O ( y P z ) ) <-> ( ( ( F ` w ) R y ) T z ) = ( ( F ` w ) O ( y P z ) ) ) )
14		oveq2 6658	. . . . . . . 8 |- ( y = ( G ` w ) -> ( ( F ` w ) R y ) = ( ( F ` w ) R ( G ` w ) ) )
15	14	oveq1d 6665	. . . . . . 7 |- ( y = ( G ` w ) -> ( ( ( F ` w ) R y ) T z ) = ( ( ( F ` w ) R ( G ` w ) ) T z ) )
16		oveq1 6657	. . . . . . . 8 |- ( y = ( G ` w ) -> ( y P z ) = ( ( G ` w ) P z ) )
17	16	oveq2d 6666	. . . . . . 7 |- ( y = ( G ` w ) -> ( ( F ` w ) O ( y P z ) ) = ( ( F ` w ) O ( ( G ` w ) P z ) ) )
18	15, 17	eqeq12d 2637	. . . . . 6 |- ( y = ( G ` w ) -> ( ( ( ( F ` w ) R y ) T z ) = ( ( F ` w ) O ( y P z ) ) <-> ( ( ( F ` w ) R ( G ` w ) ) T z ) = ( ( F ` w ) O ( ( G ` w ) P z ) ) ) )
19		oveq2 6658	. . . . . . 7 |- ( z = ( H ` w ) -> ( ( ( F ` w ) R ( G ` w ) ) T z ) = ( ( ( F ` w ) R ( G ` w ) ) T ( H ` w ) ) )
20		oveq2 6658	. . . . . . . 8 |- ( z = ( H ` w ) -> ( ( G ` w ) P z ) = ( ( G ` w ) P ( H ` w ) ) )
21	20	oveq2d 6666	. . . . . . 7 |- ( z = ( H ` w ) -> ( ( F ` w ) O ( ( G ` w ) P z ) ) = ( ( F ` w ) O ( ( G ` w ) P ( H ` w ) ) ) )
22	19, 21	eqeq12d 2637	. . . . . 6 |- ( z = ( H ` w ) -> ( ( ( ( F ` w ) R ( G ` w ) ) T z ) = ( ( F ` w ) O ( ( G ` w ) P z ) ) <-> ( ( ( F ` w ) R ( G ` w ) ) T ( H ` w ) ) = ( ( F ` w ) O ( ( G ` w ) P ( H ` w ) ) ) ) )
23	13, 18, 22	rspc3v 3325	. . . . 5 |- ( ( ( F ` w ) e. S /\ ( G ` w ) e. S /\ ( H ` w ) e. S ) -> ( A. x e. S A. y e. S A. z e. S ( ( x R y ) T z ) = ( x O ( y P z ) ) -> ( ( ( F ` w ) R ( G ` w ) ) T ( H ` w ) ) = ( ( F ` w ) O ( ( G ` w ) P ( H ` w ) ) ) ) )
24	5, 7, 9, 23	syl3anc 1326	. . . 4 |- ( ( ph /\ w e. A ) -> ( A. x e. S A. y e. S A. z e. S ( ( x R y ) T z ) = ( x O ( y P z ) ) -> ( ( ( F ` w ) R ( G ` w ) ) T ( H ` w ) ) = ( ( F ` w ) O ( ( G ` w ) P ( H ` w ) ) ) ) )
25	3, 24	mpd 15	. . 3 |- ( ( ph /\ w e. A ) -> ( ( ( F ` w ) R ( G ` w ) ) T ( H ` w ) ) = ( ( F ` w ) O ( ( G ` w ) P ( H ` w ) ) ) )
26	25	mpteq2dva 4744	. 2 |- ( ph -> ( w e. A |-> ( ( ( F ` w ) R ( G ` w ) ) T ( H ` w ) ) ) = ( w e. A |-> ( ( F ` w ) O ( ( G ` w ) P ( H ` w ) ) ) ) )
27		caofref.1	. . 3 |- ( ph -> A e. V )
28		ovexd 6680	. . 3 |- ( ( ph /\ w e. A ) -> ( ( F ` w ) R ( G ` w ) ) e. _V )
29	4	feqmptd 6249	. . . 4 |- ( ph -> F = ( w e. A |-> ( F ` w ) ) )
30	6	feqmptd 6249	. . . 4 |- ( ph -> G = ( w e. A |-> ( G ` w ) ) )
31	27, 5, 7, 29, 30	offval2 6914	. . 3 |- ( ph -> ( F oF R G ) = ( w e. A |-> ( ( F ` w ) R ( G ` w ) ) ) )
32	8	feqmptd 6249	. . 3 |- ( ph -> H = ( w e. A |-> ( H ` w ) ) )
33	27, 28, 9, 31, 32	offval2 6914	. 2 |- ( ph -> ( ( F oF R G ) oF T H ) = ( w e. A |-> ( ( ( F ` w ) R ( G ` w ) ) T ( H ` w ) ) ) )
34		ovexd 6680	. . 3 |- ( ( ph /\ w e. A ) -> ( ( G ` w ) P ( H ` w ) ) e. _V )
35	27, 7, 9, 30, 32	offval2 6914	. . 3 |- ( ph -> ( G oF P H ) = ( w e. A |-> ( ( G ` w ) P ( H ` w ) ) ) )
36	27, 5, 34, 29, 35	offval2 6914	. 2 |- ( ph -> ( F oF O ( G oF P H ) ) = ( w e. A |-> ( ( F ` w ) O ( ( G ` w ) P ( H ` w ) ) ) ) )
37	26, 33, 36	3eqtr4d 2666	1 |- ( ph -> ( ( F oF R G ) oF T H ) = ( F oF O ( G oF P H ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897

This theorem is referenced by:  psrgrp  19398  psrlmod  19401  mndvass  20198  itg2mulc  23514  plydivlem4  24051  dchrabl  24979  lfladdass  34360  lflvsass  34368  expgrowth  38534