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Theorem caofdi 6342
Description: Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-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 )
caofdi.5  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x T ( y R z ) )  =  ( ( x T y ) O ( x T z ) ) )
Assertion
Ref Expression
caofdi  |-  ( ph  ->  ( F  o F T ( G  o F R H ) )  =  ( ( F  o F T G )  o F O ( F  o F T H ) ) )
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 caofdi
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caofdi.5 . . . . 5  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x T ( y R z ) )  =  ( ( x T y ) O ( x T z ) ) )
21adantlr 697 . . . 4  |-  ( ( ( ph  /\  w  e.  A )  /\  (
x  e.  K  /\  y  e.  S  /\  z  e.  S )
)  ->  ( x T ( y R z ) )  =  ( ( x T y ) O ( x T z ) ) )
3 caofdi.2 . . . . 5  |-  ( ph  ->  F : A --> K )
43ffvelrnda 5872 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  K )
5 caofdi.3 . . . . 5  |-  ( ph  ->  G : A --> S )
65ffvelrnda 5872 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
7 caofdi.4 . . . . 5  |-  ( ph  ->  H : A --> S )
87ffvelrnda 5872 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( H `  w )  e.  S )
92, 4, 6, 8caovdid 6264 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) T ( ( G `  w ) R ( H `  w ) ) )  =  ( ( ( F `  w ) T ( G `  w ) ) O ( ( F `  w ) T ( H `  w ) ) ) )
109mpteq2dva 4297 . 2  |-  ( ph  ->  ( w  e.  A  |->  ( ( F `  w ) T ( ( G `  w
) R ( H `
 w ) ) ) )  =  ( w  e.  A  |->  ( ( ( F `  w ) T ( G `  w ) ) O ( ( F `  w ) T ( H `  w ) ) ) ) )
11 caofdi.1 . . 3  |-  ( ph  ->  A  e.  V )
12 ovex 6108 . . . 4  |-  ( ( G `  w ) R ( H `  w ) )  e. 
_V
1312a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( H `
 w ) )  e.  _V )
143feqmptd 5781 . . 3  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
155feqmptd 5781 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
167feqmptd 5781 . . . 4  |-  ( ph  ->  H  =  ( w  e.  A  |->  ( H `
 w ) ) )
1711, 6, 8, 15, 16offval2 6324 . . 3  |-  ( ph  ->  ( G  o F R H )  =  ( w  e.  A  |->  ( ( G `  w ) R ( H `  w ) ) ) )
1811, 4, 13, 14, 17offval2 6324 . 2  |-  ( ph  ->  ( F  o F T ( G  o F R H ) )  =  ( w  e.  A  |->  ( ( F `
 w ) T ( ( G `  w ) R ( H `  w ) ) ) ) )
19 ovex 6108 . . . 4  |-  ( ( F `  w ) T ( G `  w ) )  e. 
_V
2019a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) T ( G `
 w ) )  e.  _V )
21 ovex 6108 . . . 4  |-  ( ( F `  w ) T ( H `  w ) )  e. 
_V
2221a1i 11 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) T ( H `
 w ) )  e.  _V )
2311, 4, 6, 14, 15offval2 6324 . . 3  |-  ( ph  ->  ( F  o F T G )  =  ( w  e.  A  |->  ( ( F `  w ) T ( G `  w ) ) ) )
2411, 4, 8, 14, 16offval2 6324 . . 3  |-  ( ph  ->  ( F  o F T H )  =  ( w  e.  A  |->  ( ( F `  w ) T ( H `  w ) ) ) )
2511, 20, 22, 23, 24offval2 6324 . 2  |-  ( ph  ->  ( ( F  o F T G )  o F O ( F  o F T H ) )  =  ( w  e.  A  |->  ( ( ( F `  w ) T ( G `  w ) ) O ( ( F `  w ) T ( H `  w ) ) ) ) )
2610, 18, 253eqtr4d 2480 1  |-  ( ph  ->  ( F  o F T ( G  o F R H ) )  =  ( ( F  o F T G )  o F O ( F  o F T H ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    e. cmpt 4268   -->wf 5452   ` cfv 5456  (class class class)co 6083    o Fcof 6305
This theorem is referenced by:  psrlmod  16467  plydivlem4  20215  plydiveu  20217  quotcan  20228  basellem9  20873  mendlmod  27480  lflvsdi2  29879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307
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