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Theorem caofcom 6109
Description: Transfer a commutative 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 )
caofcom.4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x R y )  =  ( y R x ) )
Assertion
Ref Expression
caofcom  |-  ( ph  ->  ( F  o F R G )  =  ( G  o F R F ) )
Distinct variable groups:    x, y, F    x, G, y    ph, x, y    x, R, y    x, S, y
Allowed substitution hints:    A( x, y)    V( x, y)

Proof of Theorem caofcom
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . . 6  |-  ( ph  ->  F : A --> S )
2 ffvelrn 5663 . . . . . 6  |-  ( ( F : A --> S  /\  w  e.  A )  ->  ( F `  w
)  e.  S )
31, 2sylan 457 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
4 caofcom.3 . . . . . 6  |-  ( ph  ->  G : A --> S )
5 ffvelrn 5663 . . . . . 6  |-  ( ( G : A --> S  /\  w  e.  A )  ->  ( G `  w
)  e.  S )
64, 5sylan 457 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
73, 6jca 518 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
)  e.  S  /\  ( G `  w )  e.  S ) )
8 caofcom.4 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x R y )  =  ( y R x ) )
98caovcomg 6015 . . . 4  |-  ( (
ph  /\  ( ( F `  w )  e.  S  /\  ( G `  w )  e.  S ) )  -> 
( ( F `  w ) R ( G `  w ) )  =  ( ( G `  w ) R ( F `  w ) ) )
107, 9syldan 456 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R ( G `
 w ) )  =  ( ( G `
 w ) R ( F `  w
) ) )
1110mpteq2dva 4106 . 2  |-  ( ph  ->  ( w  e.  A  |->  ( ( F `  w ) R ( G `  w ) ) )  =  ( w  e.  A  |->  ( ( G `  w
) R ( F `
 w ) ) ) )
12 caofref.1 . . 3  |-  ( ph  ->  A  e.  V )
131feqmptd 5575 . . 3  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
144feqmptd 5575 . . 3  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
1512, 3, 6, 13, 14offval2 6095 . 2  |-  ( ph  ->  ( F  o F R G )  =  ( w  e.  A  |->  ( ( F `  w ) R ( G `  w ) ) ) )
1612, 6, 3, 14, 13offval2 6095 . 2  |-  ( ph  ->  ( G  o F R F )  =  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) ) )
1711, 15, 163eqtr4d 2325 1  |-  ( ph  ->  ( F  o F R G )  =  ( G  o F R F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076
This theorem is referenced by:  plydivlem4  19676  quotcan  19689  dchrabl  20493  expgrowth  27552  lfladdcom  29262
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
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