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Theorem caofdir 6300
Description: Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.)
Hypotheses
Ref Expression
caofdi.1  |-  ( ph  ->  A  e.  V )
caofdi.2  |-  ( ph  ->  F : A --> K )
caofdi.3  |-  ( ph  ->  G : A --> S )
caofdi.4  |-  ( ph  ->  H : A --> S )
caofdir.5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  K ) )  -> 
( ( x R y ) T z )  =  ( ( x T z ) O ( y T z ) ) )
Assertion
Ref Expression
caofdir  |-  ( ph  ->  ( ( G  o F R H )  o F T F )  =  ( ( G  o F T F )  o F O ( H  o F T F ) ) )
Distinct variable groups:    x, y,
z, A    x, F, y, z    x, G, y, z    ph, x, y, z   
x, H, y, z   
x, K, y, z   
x, O, y, z   
x, R, y, z   
x, S, y, z   
x, T, y, z
Allowed substitution hints:    V( x, y, z)

Proof of Theorem caofdir
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caofdir.5 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  K ) )  -> 
( ( x R y ) T z )  =  ( ( x T z ) O ( y T z ) ) )
21adantlr 696 . . . 4  |-  ( ( ( ph  /\  w  e.  A )  /\  (
x  e.  S  /\  y  e.  S  /\  z  e.  K )
)  ->  ( (
x R y ) T z )  =  ( ( x T z ) O ( y T z ) ) )
3 caofdi.3 . . . . 5  |-  ( ph  ->  G : A --> S )
43ffvelrnda 5829 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
5 caofdi.4 . . . . 5  |-  ( ph  ->  H : A --> S )
65ffvelrnda 5829 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
7 caofdi.2 . . . . 5  |-  ( ph  ->  F : A --> K )
87ffvelrnda 5829 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  K )
92, 4, 6, 8caovdird 6224 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( ( G `  w ) R ( H `  w ) ) T ( F `
 w ) )  =  ( ( ( G `  w ) T ( F `  w ) ) O ( ( H `  w ) T ( F `  w ) ) ) )
109mpteq2dva 4255 . 2  |-  ( ph  ->  ( w  e.  A  |->  ( ( ( G `
 w ) R ( H `  w
) ) T ( F `  w ) ) )  =  ( w  e.  A  |->  ( ( ( G `  w ) T ( F `  w ) ) O ( ( H `  w ) T ( F `  w ) ) ) ) )
11 caofdi.1 . . 3  |-  ( ph  ->  A  e.  V )
12 ovex 6065 . . . 4  |-  ( ( G `  w ) R ( H `  w ) )  e. 
_V
1312a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( H `
 w ) )  e.  _V )
143feqmptd 5738 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
155feqmptd 5738 . . . 4  |-  ( ph  ->  H  =  ( w  e.  A  |->  ( H `
 w ) ) )
1611, 4, 6, 14, 15offval2 6281 . . 3  |-  ( ph  ->  ( G  o F R H )  =  ( w  e.  A  |->  ( ( G `  w ) R ( H `  w ) ) ) )
177feqmptd 5738 . . 3  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
1811, 13, 8, 16, 17offval2 6281 . 2  |-  ( ph  ->  ( ( G  o F R H )  o F T F )  =  ( w  e.  A  |->  ( ( ( G `  w ) R ( H `  w ) ) T ( F `  w
) ) ) )
19 ovex 6065 . . . 4  |-  ( ( G `  w ) T ( F `  w ) )  e. 
_V
2019a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) T ( F `
 w ) )  e.  _V )
21 ovex 6065 . . . 4  |-  ( ( H `  w ) T ( F `  w ) )  e. 
_V
2221a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( H `  w
) T ( F `
 w ) )  e.  _V )
2311, 4, 8, 14, 17offval2 6281 . . 3  |-  ( ph  ->  ( G  o F T F )  =  ( w  e.  A  |->  ( ( G `  w ) T ( F `  w ) ) ) )
2411, 6, 8, 15, 17offval2 6281 . . 3  |-  ( ph  ->  ( H  o F T F )  =  ( w  e.  A  |->  ( ( H `  w ) T ( F `  w ) ) ) )
2511, 20, 22, 23, 24offval2 6281 . 2  |-  ( ph  ->  ( ( G  o F T F )  o F O ( H  o F T F ) )  =  ( w  e.  A  |->  ( ( ( G `  w ) T ( F `  w ) ) O ( ( H `  w ) T ( F `  w ) ) ) ) )
2610, 18, 253eqtr4d 2446 1  |-  ( ph  ->  ( ( G  o F R H )  o F T F )  =  ( ( G  o F T F )  o F O ( H  o F T F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916    e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262
This theorem is referenced by:  psrlmod  16420  mendlmod  27369  expgrowth  27420  lflvsdi1  29561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264
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