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Theorem caofdir 6130
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 695 . . . 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 )
4 ffvelrn 5679 . . . . 5  |-  ( ( G : A --> S  /\  w  e.  A )  ->  ( G `  w
)  e.  S )
53, 4sylan 457 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
6 caofdi.4 . . . . 5  |-  ( ph  ->  H : A --> S )
7 ffvelrn 5679 . . . . 5  |-  ( ( H : A --> S  /\  w  e.  A )  ->  ( H `  w
)  e.  S )
86, 7sylan 457 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
9 caofdi.2 . . . . 5  |-  ( ph  ->  F : A --> K )
10 ffvelrn 5679 . . . . 5  |-  ( ( F : A --> K  /\  w  e.  A )  ->  ( F `  w
)  e.  K )
119, 10sylan 457 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  K )
122, 5, 8, 11caovdird 6054 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( ( G `  w ) R ( H `  w ) ) T ( F `
 w ) )  =  ( ( ( G `  w ) T ( F `  w ) ) O ( ( H `  w ) T ( F `  w ) ) ) )
1312mpteq2dva 4122 . 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 ) ) ) ) )
14 caofdi.1 . . 3  |-  ( ph  ->  A  e.  V )
15 ovex 5899 . . . 4  |-  ( ( G `  w ) R ( H `  w ) )  e. 
_V
1615a1i 10 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( H `
 w ) )  e.  _V )
173feqmptd 5591 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
186feqmptd 5591 . . . 4  |-  ( ph  ->  H  =  ( w  e.  A  |->  ( H `
 w ) ) )
1914, 5, 8, 17, 18offval2 6111 . . 3  |-  ( ph  ->  ( G  o F R H )  =  ( w  e.  A  |->  ( ( G `  w ) R ( H `  w ) ) ) )
209feqmptd 5591 . . 3  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
2114, 16, 11, 19, 20offval2 6111 . 2  |-  ( ph  ->  ( ( G  o F R H )  o F T F )  =  ( w  e.  A  |->  ( ( ( G `  w ) R ( H `  w ) ) T ( F `  w
) ) ) )
22 ovex 5899 . . . 4  |-  ( ( G `  w ) T ( F `  w ) )  e. 
_V
2322a1i 10 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) T ( F `
 w ) )  e.  _V )
24 ovex 5899 . . . 4  |-  ( ( H `  w ) T ( F `  w ) )  e. 
_V
2524a1i 10 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( H `  w
) T ( F `
 w ) )  e.  _V )
2614, 5, 11, 17, 20offval2 6111 . . 3  |-  ( ph  ->  ( G  o F T F )  =  ( w  e.  A  |->  ( ( G `  w ) T ( F `  w ) ) ) )
2714, 8, 11, 18, 20offval2 6111 . . 3  |-  ( ph  ->  ( H  o F T F )  =  ( w  e.  A  |->  ( ( H `  w ) T ( F `  w ) ) ) )
2814, 23, 25, 26, 27offval2 6111 . 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 ) ) ) ) )
2913, 21, 283eqtr4d 2338 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092
This theorem is referenced by:  psrlmod  16162  mendlmod  27604  expgrowth  27655  lflvsdi1  29890
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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