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Theorem caofass 6340
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  o F R G )  o F T H )  =  ( F  o F O ( G  o F 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.
StepHypRef 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 ) ) )
21ralrimivvva 2801 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  A. z  e.  S  ( ( x R y ) T z )  =  ( x O ( y P z ) ) )
32adantr 453 . . . 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 )
54ffvelrnda 5872 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
6 caofcom.3 . . . . . 6  |-  ( ph  ->  G : A --> S )
76ffvelrnda 5872 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
8 caofass.4 . . . . . 6  |-  ( ph  ->  H : A --> S )
98ffvelrnda 5872 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
10 oveq1 6090 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  (
x R y )  =  ( ( F `
 w ) R y ) )
1110oveq1d 6098 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
( x R y ) T z )  =  ( ( ( F `  w ) R y ) T z ) )
12 oveq1 6090 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
x O ( y P z ) )  =  ( ( F `
 w ) O ( y P z ) ) )
1311, 12eqeq12d 2452 . . . . . 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 6091 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) R y )  =  ( ( F `
 w ) R ( G `  w
) ) )
1514oveq1d 6098 . . . . . . 7  |-  ( y  =  ( G `  w )  ->  (
( ( F `  w ) R y ) T z )  =  ( ( ( F `  w ) R ( G `  w ) ) T z ) )
16 oveq1 6090 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
y P z )  =  ( ( G `
 w ) P z ) )
1716oveq2d 6099 . . . . . . 7  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) O ( y P z ) )  =  ( ( F `
 w ) O ( ( G `  w ) P z ) ) )
1815, 17eqeq12d 2452 . . . . . 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 6091 . . . . . . 7  |-  ( z  =  ( H `  w )  ->  (
( ( F `  w ) R ( G `  w ) ) T z )  =  ( ( ( F `  w ) R ( G `  w ) ) T ( H `  w
) ) )
20 oveq2 6091 . . . . . . . 8  |-  ( z  =  ( H `  w )  ->  (
( G `  w
) P z )  =  ( ( G `
 w ) P ( H `  w
) ) )
2120oveq2d 6099 . . . . . . 7  |-  ( z  =  ( H `  w )  ->  (
( F `  w
) O ( ( G `  w ) P z ) )  =  ( ( F `
 w ) O ( ( G `  w ) P ( H `  w ) ) ) )
2219, 21eqeq12d 2452 . . . . . 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 ) ) ) ) )
2313, 18, 22rspc3v 3063 . . . . 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
) ) ) ) )
245, 7, 9, 23syl3anc 1185 . . . 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
) ) ) ) )
253, 24mpd 15 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( ( F `  w ) R ( G `  w ) ) T ( H `
 w ) )  =  ( ( F `
 w ) O ( ( G `  w ) P ( H `  w ) ) ) )
2625mpteq2dva 4297 . 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 ovex 6108 . . . 4  |-  ( ( F `  w ) R ( G `  w ) )  e. 
_V
2928a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R ( G `
 w ) )  e.  _V )
304feqmptd 5781 . . . 4  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
316feqmptd 5781 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
3227, 5, 7, 30, 31offval2 6324 . . 3  |-  ( ph  ->  ( F  o F R G )  =  ( w  e.  A  |->  ( ( F `  w ) R ( G `  w ) ) ) )
338feqmptd 5781 . . 3  |-  ( ph  ->  H  =  ( w  e.  A  |->  ( H `
 w ) ) )
3427, 29, 9, 32, 33offval2 6324 . 2  |-  ( ph  ->  ( ( F  o F R G )  o F T H )  =  ( w  e.  A  |->  ( ( ( F `  w ) R ( G `  w ) ) T ( H `  w
) ) ) )
35 ovex 6108 . . . 4  |-  ( ( G `  w ) P ( H `  w ) )  e. 
_V
3635a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) P ( H `
 w ) )  e.  _V )
3727, 7, 9, 31, 33offval2 6324 . . 3  |-  ( ph  ->  ( G  o F P H )  =  ( w  e.  A  |->  ( ( G `  w ) P ( H `  w ) ) ) )
3827, 5, 36, 30, 37offval2 6324 . 2  |-  ( ph  ->  ( F  o F O ( G  o F P H ) )  =  ( w  e.  A  |->  ( ( F `
 w ) O ( ( G `  w ) P ( H `  w ) ) ) ) )
3926, 34, 383eqtr4d 2480 1  |-  ( ph  ->  ( ( F  o F R G )  o F T H )  =  ( F  o F O ( G  o F P H ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    e. cmpt 4268   -->wf 5452   ` cfv 5456  (class class class)co 6083    o Fcof 6305
This theorem is referenced by:  psrgrp  16464  psrlmod  16467  itg2mulc  19641  plydivlem4  20215  dchrabl  21040  mndvass  27426  expgrowth  27531  lfladdass  29933  lflvsass  29941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307
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