MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caofass Unicode version

Theorem caofass 6111
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 2636 . . . . 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 451 . . . 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 ffvelrn 5663 . . . . . 6  |-  ( ( F : A --> S  /\  w  e.  A )  ->  ( F `  w
)  e.  S )
64, 5sylan 457 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
7 caofcom.3 . . . . . 6  |-  ( ph  ->  G : A --> S )
8 ffvelrn 5663 . . . . . 6  |-  ( ( G : A --> S  /\  w  e.  A )  ->  ( G `  w
)  e.  S )
97, 8sylan 457 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
10 caofass.4 . . . . . 6  |-  ( ph  ->  H : A --> S )
11 ffvelrn 5663 . . . . . 6  |-  ( ( H : A --> S  /\  w  e.  A )  ->  ( H `  w
)  e.  S )
1210, 11sylan 457 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
13 oveq1 5865 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  (
x R y )  =  ( ( F `
 w ) R y ) )
1413oveq1d 5873 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
( x R y ) T z )  =  ( ( ( F `  w ) R y ) T z ) )
15 oveq1 5865 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
x O ( y P z ) )  =  ( ( F `
 w ) O ( y P z ) ) )
1614, 15eqeq12d 2297 . . . . . 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 ) ) ) )
17 oveq2 5866 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) R y )  =  ( ( F `
 w ) R ( G `  w
) ) )
1817oveq1d 5873 . . . . . . 7  |-  ( y  =  ( G `  w )  ->  (
( ( F `  w ) R y ) T z )  =  ( ( ( F `  w ) R ( G `  w ) ) T z ) )
19 oveq1 5865 . . . . . . . 8  |-  ( y  =  ( G `  w )  ->  (
y P z )  =  ( ( G `
 w ) P z ) )
2019oveq2d 5874 . . . . . . 7  |-  ( y  =  ( G `  w )  ->  (
( F `  w
) O ( y P z ) )  =  ( ( F `
 w ) O ( ( G `  w ) P z ) ) )
2118, 20eqeq12d 2297 . . . . . 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 ) ) ) )
22 oveq2 5866 . . . . . . 7  |-  ( z  =  ( H `  w )  ->  (
( ( F `  w ) R ( G `  w ) ) T z )  =  ( ( ( F `  w ) R ( G `  w ) ) T ( H `  w
) ) )
23 oveq2 5866 . . . . . . . 8  |-  ( z  =  ( H `  w )  ->  (
( G `  w
) P z )  =  ( ( G `
 w ) P ( H `  w
) ) )
2423oveq2d 5874 . . . . . . 7  |-  ( z  =  ( H `  w )  ->  (
( F `  w
) O ( ( G `  w ) P z ) )  =  ( ( F `
 w ) O ( ( G `  w ) P ( H `  w ) ) ) )
2522, 24eqeq12d 2297 . . . . . 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 ) ) ) ) )
2616, 21, 25rspc3v 2893 . . . . 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
) ) ) ) )
276, 9, 12, 26syl3anc 1182 . . . 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
) ) ) ) )
283, 27mpd 14 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( ( F `  w ) R ( G `  w ) ) T ( H `
 w ) )  =  ( ( F `
 w ) O ( ( G `  w ) P ( H `  w ) ) ) )
2928mpteq2dva 4106 . 2  |-  ( ph  ->  ( w  e.  A  |->  ( ( ( F `
 w ) R ( G `  w
) ) T ( H `  w ) ) )  =  ( w  e.  A  |->  ( ( F `  w
) O ( ( G `  w ) P ( H `  w ) ) ) ) )
30 caofref.1 . . 3  |-  ( ph  ->  A  e.  V )
31 ovex 5883 . . . 4  |-  ( ( F `  w ) R ( G `  w ) )  e. 
_V
3231a1i 10 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R ( G `
 w ) )  e.  _V )
334feqmptd 5575 . . . 4  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
347feqmptd 5575 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
3530, 6, 9, 33, 34offval2 6095 . . 3  |-  ( ph  ->  ( F  o F R G )  =  ( w  e.  A  |->  ( ( F `  w ) R ( G `  w ) ) ) )
3610feqmptd 5575 . . 3  |-  ( ph  ->  H  =  ( w  e.  A  |->  ( H `
 w ) ) )
3730, 32, 12, 35, 36offval2 6095 . 2  |-  ( ph  ->  ( ( F  o F R G )  o F T H )  =  ( w  e.  A  |->  ( ( ( F `  w ) R ( G `  w ) ) T ( H `  w
) ) ) )
38 ovex 5883 . . . 4  |-  ( ( G `  w ) P ( H `  w ) )  e. 
_V
3938a1i 10 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) P ( H `
 w ) )  e.  _V )
4030, 9, 12, 34, 36offval2 6095 . . 3  |-  ( ph  ->  ( G  o F P H )  =  ( w  e.  A  |->  ( ( G `  w ) P ( H `  w ) ) ) )
4130, 6, 39, 33, 40offval2 6095 . 2  |-  ( ph  ->  ( F  o F O ( G  o F P H ) )  =  ( w  e.  A  |->  ( ( F `
 w ) O ( ( G `  w ) P ( H `  w ) ) ) ) )
4229, 37, 413eqtr4d 2325 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076
This theorem is referenced by:  psrgrp  16143  psrlmod  16146  itg2mulc  19102  plydivlem4  19676  dchrabl  20493  mndvass  27447  expgrowth  27552  lfladdass  29263  lflvsass  29271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078
  Copyright terms: Public domain W3C validator