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Theorem caofcom 6125
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 5679 . . . . . 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 5679 . . . . . 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 6031 . . . 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 4122 . 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 5591 . . 3  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
144feqmptd 5591 . . 3  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
1512, 3, 6, 13, 14offval2 6111 . 2  |-  ( ph  ->  ( F  o F R G )  =  ( w  e.  A  |->  ( ( F `  w ) R ( G `  w ) ) ) )
1612, 6, 3, 14, 13offval2 6111 . 2  |-  ( ph  ->  ( G  o F R F )  =  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) ) )
1711, 15, 163eqtr4d 2338 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 1632    e. wcel 1696    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092
This theorem is referenced by:  plydivlem4  19692  quotcan  19705  dchrabl  20509  expgrowth  27655  lfladdcom  29884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094
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