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Theorem caofass 6353
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.
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 2807 . . . . 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 462 . . . 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 5840 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
6 caofcom.3 . . . . . 6  |-  ( ph  ->  G : A --> S )
76ffvelrnda 5840 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
8 caofass.4 . . . . . 6  |-  ( ph  ->  H : A --> S )
98ffvelrnda 5840 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
10 oveq1 6097 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  (
x R y )  =  ( ( F `
 w ) R y ) )
1110oveq1d 6105 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
( x R y ) T z )  =  ( ( ( F `  w ) R y ) T z ) )
12 oveq1 6097 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
x O ( y P z ) )  =  ( ( F `
 w ) O ( y P z ) ) )
1311, 12eqeq12d 2455 . . . . . 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 6098 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) R y )  =  ( ( F `
 w ) R ( G `  w
) ) )
1514oveq1d 6105 . . . . . . 7  |-  ( y  =  ( G `  w )  ->  (
( ( F `  w ) R y ) T z )  =  ( ( ( F `  w ) R ( G `  w ) ) T z ) )
16 oveq1 6097 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
y P z )  =  ( ( G `
 w ) P z ) )
1716oveq2d 6106 . . . . . . 7  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) O ( y P z ) )  =  ( ( F `
 w ) O ( ( G `  w ) P z ) ) )
1815, 17eqeq12d 2455 . . . . . 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 6098 . . . . . . 7  |-  ( z  =  ( H `  w )  ->  (
( ( F `  w ) R ( G `  w ) ) T z )  =  ( ( ( F `  w ) R ( G `  w ) ) T ( H `  w
) ) )
20 oveq2 6098 . . . . . . . 8  |-  ( z  =  ( H `  w )  ->  (
( G `  w
) P z )  =  ( ( G `
 w ) P ( H `  w
) ) )
2120oveq2d 6106 . . . . . . 7  |-  ( z  =  ( H `  w )  ->  (
( F `  w
) O ( ( G `  w ) P z ) )  =  ( ( F `
 w ) O ( ( G `  w ) P ( H `  w ) ) ) )
2219, 21eqeq12d 2455 . . . . . 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 3079 . . . . 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 1213 . . . 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 4375 . 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 6115 . . . 4  |-  ( ( F `  w ) R ( G `  w ) )  e. 
_V
2928a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R ( G `
 w ) )  e.  _V )
304feqmptd 5741 . . . 4  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
316feqmptd 5741 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
3227, 5, 7, 30, 31offval2 6335 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( w  e.  A  |->  ( ( F `  w ) R ( G `  w ) ) ) )
338feqmptd 5741 . . 3  |-  ( ph  ->  H  =  ( w  e.  A  |->  ( H `
 w ) ) )
3427, 29, 9, 32, 33offval2 6335 . 2  |-  ( ph  ->  ( ( F  oF R G )  oF T H )  =  ( w  e.  A  |->  ( ( ( F `  w
) R ( G `
 w ) ) T ( H `  w ) ) ) )
35 ovex 6115 . . . 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 6335 . . 3  |-  ( ph  ->  ( G  oF P H )  =  ( w  e.  A  |->  ( ( G `  w ) P ( H `  w ) ) ) )
3827, 5, 36, 30, 37offval2 6335 . 2  |-  ( ph  ->  ( F  oF O ( G  oF P H ) )  =  ( w  e.  A  |->  ( ( F `  w ) O ( ( G `
 w ) P ( H `  w
) ) ) ) )
3926, 34, 383eqtr4d 2483 1  |-  ( ph  ->  ( ( F  oF R G )  oF T H )  =  ( F  oF O ( G  oF P H ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    e. cmpt 4347   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319
This theorem is referenced by:  psrgrp  17459  psrlmod  17462  mndvass  18192  itg2mulc  21125  plydivlem4  21705  dchrabl  22536  expgrowth  29518  lfladdass  32406  lflvsass  32414
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