MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caofdir Unicode version

Theorem caofdir 6273
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 5802 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
5 caofdi.4 . . . . 5  |-  ( ph  ->  H : A --> S )
65ffvelrnda 5802 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
7 caofdi.2 . . . . 5  |-  ( ph  ->  F : A --> K )
87ffvelrnda 5802 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  K )
92, 4, 6, 8caovdird 6197 . . 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 4229 . 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 6038 . . . 4  |-  ( ( G `  w ) R ( H `  w ) )  e. 
_V
1312a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( H `
 w ) )  e.  _V )
143feqmptd 5711 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
155feqmptd 5711 . . . 4  |-  ( ph  ->  H  =  ( w  e.  A  |->  ( H `
 w ) ) )
1611, 4, 6, 14, 15offval2 6254 . . 3  |-  ( ph  ->  ( G  o F R H )  =  ( w  e.  A  |->  ( ( G `  w ) R ( H `  w ) ) ) )
177feqmptd 5711 . . 3  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
1811, 13, 8, 16, 17offval2 6254 . 2  |-  ( ph  ->  ( ( G  o F R H )  o F T F )  =  ( w  e.  A  |->  ( ( ( G `  w ) R ( H `  w ) ) T ( F `  w
) ) ) )
19 ovex 6038 . . . 4  |-  ( ( G `  w ) T ( F `  w ) )  e. 
_V
2019a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) T ( F `
 w ) )  e.  _V )
21 ovex 6038 . . . 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 6254 . . 3  |-  ( ph  ->  ( G  o F T F )  =  ( w  e.  A  |->  ( ( G `  w ) T ( F `  w ) ) ) )
2411, 6, 8, 15, 17offval2 6254 . . 3  |-  ( ph  ->  ( H  o F T F )  =  ( w  e.  A  |->  ( ( H `  w ) T ( F `  w ) ) ) )
2511, 20, 22, 23, 24offval2 6254 . 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 2422 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 1717   _Vcvv 2892    e. cmpt 4200   -->wf 5383   ` cfv 5387  (class class class)co 6013    o Fcof 6235
This theorem is referenced by:  psrlmod  16385  mendlmod  27163  expgrowth  27214  lflvsdi1  29244
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237
  Copyright terms: Public domain W3C validator