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Theorem caofdir 6561
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  oF R H )  oF T F )  =  ( ( G  oF T F )  oF O ( H  oF 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 715 . . . 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 6011 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
5 caofdi.4 . . . . 5  |-  ( ph  ->  H : A --> S )
65ffvelrnda 6011 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
7 caofdi.2 . . . . 5  |-  ( ph  ->  F : A --> K )
87ffvelrnda 6011 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  K )
92, 4, 6, 8caovdird 6476 . . 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 4483 . 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 6308 . . . 4  |-  ( ( G `  w ) R ( H `  w ) )  e. 
_V
1312a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( H `
 w ) )  e.  _V )
143feqmptd 5904 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
155feqmptd 5904 . . . 4  |-  ( ph  ->  H  =  ( w  e.  A  |->  ( H `
 w ) ) )
1611, 4, 6, 14, 15offval2 6540 . . 3  |-  ( ph  ->  ( G  oF R H )  =  ( w  e.  A  |->  ( ( G `  w ) R ( H `  w ) ) ) )
177feqmptd 5904 . . 3  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
1811, 13, 8, 16, 17offval2 6540 . 2  |-  ( ph  ->  ( ( G  oF R H )  oF T F )  =  ( w  e.  A  |->  ( ( ( G `  w
) R ( H `
 w ) ) T ( F `  w ) ) ) )
19 ovex 6308 . . . 4  |-  ( ( G `  w ) T ( F `  w ) )  e. 
_V
2019a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) T ( F `
 w ) )  e.  _V )
21 ovex 6308 . . . 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 6540 . . 3  |-  ( ph  ->  ( G  oF T F )  =  ( w  e.  A  |->  ( ( G `  w ) T ( F `  w ) ) ) )
2411, 6, 8, 15, 17offval2 6540 . . 3  |-  ( ph  ->  ( H  oF T F )  =  ( w  e.  A  |->  ( ( H `  w ) T ( F `  w ) ) ) )
2511, 20, 22, 23, 24offval2 6540 . 2  |-  ( ph  ->  ( ( G  oF T F )  oF O ( H  oF T F ) )  =  ( w  e.  A  |->  ( ( ( G `
 w ) T ( F `  w
) ) O ( ( H `  w
) T ( F `
 w ) ) ) ) )
2610, 18, 253eqtr4d 2455 1  |-  ( ph  ->  ( ( G  oF R H )  oF T F )  =  ( ( G  oF T F )  oF O ( H  oF T F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   _Vcvv 3061    |-> cmpt 4455   -->wf 5567   ` cfv 5571  (class class class)co 6280    oFcof 6521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523
This theorem is referenced by:  psrlmod  18376  lflvsdi1  32109  mendlmod  35519  expgrowth  36101
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