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Theorem ofco 6263
Description: The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
Hypotheses
Ref Expression
ofco.1  |-  ( ph  ->  F  Fn  A )
ofco.2  |-  ( ph  ->  G  Fn  B )
ofco.3  |-  ( ph  ->  H : D --> C )
ofco.4  |-  ( ph  ->  A  e.  V )
ofco.5  |-  ( ph  ->  B  e.  W )
ofco.6  |-  ( ph  ->  D  e.  X )
ofco.7  |-  ( A  i^i  B )  =  C
Assertion
Ref Expression
ofco  |-  ( ph  ->  ( ( F  o F R G )  o.  H )  =  ( ( F  o.  H
)  o F R ( G  o.  H
) ) )

Proof of Theorem ofco
Dummy variables  y  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ofco.3 . . . 4  |-  ( ph  ->  H : D --> C )
21ffvelrnda 5809 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( H `  x )  e.  C )
31feqmptd 5718 . . 3  |-  ( ph  ->  H  =  ( x  e.  D  |->  ( H `
 x ) ) )
4 ofco.1 . . . 4  |-  ( ph  ->  F  Fn  A )
5 ofco.2 . . . 4  |-  ( ph  ->  G  Fn  B )
6 ofco.4 . . . 4  |-  ( ph  ->  A  e.  V )
7 ofco.5 . . . 4  |-  ( ph  ->  B  e.  W )
8 ofco.7 . . . 4  |-  ( A  i^i  B )  =  C
9 eqidd 2388 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( F `  y ) )
10 eqidd 2388 . . . 4  |-  ( (
ph  /\  y  e.  B )  ->  ( G `  y )  =  ( G `  y ) )
114, 5, 6, 7, 8, 9, 10offval 6251 . . 3  |-  ( ph  ->  ( F  o F R G )  =  ( y  e.  C  |->  ( ( F `  y ) R ( G `  y ) ) ) )
12 fveq2 5668 . . . 4  |-  ( y  =  ( H `  x )  ->  ( F `  y )  =  ( F `  ( H `  x ) ) )
13 fveq2 5668 . . . 4  |-  ( y  =  ( H `  x )  ->  ( G `  y )  =  ( G `  ( H `  x ) ) )
1412, 13oveq12d 6038 . . 3  |-  ( y  =  ( H `  x )  ->  (
( F `  y
) R ( G `
 y ) )  =  ( ( F `
 ( H `  x ) ) R ( G `  ( H `  x )
) ) )
152, 3, 11, 14fmptco 5840 . 2  |-  ( ph  ->  ( ( F  o F R G )  o.  H )  =  ( x  e.  D  |->  ( ( F `  ( H `  x )
) R ( G `
 ( H `  x ) ) ) ) )
16 inss1 3504 . . . . . 6  |-  ( A  i^i  B )  C_  A
178, 16eqsstr3i 3322 . . . . 5  |-  C  C_  A
18 fss 5539 . . . . 5  |-  ( ( H : D --> C  /\  C  C_  A )  ->  H : D --> A )
191, 17, 18sylancl 644 . . . 4  |-  ( ph  ->  H : D --> A )
20 fnfco 5549 . . . 4  |-  ( ( F  Fn  A  /\  H : D --> A )  ->  ( F  o.  H )  Fn  D
)
214, 19, 20syl2anc 643 . . 3  |-  ( ph  ->  ( F  o.  H
)  Fn  D )
22 inss2 3505 . . . . . 6  |-  ( A  i^i  B )  C_  B
238, 22eqsstr3i 3322 . . . . 5  |-  C  C_  B
24 fss 5539 . . . . 5  |-  ( ( H : D --> C  /\  C  C_  B )  ->  H : D --> B )
251, 23, 24sylancl 644 . . . 4  |-  ( ph  ->  H : D --> B )
26 fnfco 5549 . . . 4  |-  ( ( G  Fn  B  /\  H : D --> B )  ->  ( G  o.  H )  Fn  D
)
275, 25, 26syl2anc 643 . . 3  |-  ( ph  ->  ( G  o.  H
)  Fn  D )
28 ofco.6 . . 3  |-  ( ph  ->  D  e.  X )
29 inidm 3493 . . 3  |-  ( D  i^i  D )  =  D
30 ffn 5531 . . . . 5  |-  ( H : D --> C  ->  H  Fn  D )
311, 30syl 16 . . . 4  |-  ( ph  ->  H  Fn  D )
32 fvco2 5737 . . . 4  |-  ( ( H  Fn  D  /\  x  e.  D )  ->  ( ( F  o.  H ) `  x
)  =  ( F `
 ( H `  x ) ) )
3331, 32sylan 458 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  (
( F  o.  H
) `  x )  =  ( F `  ( H `  x ) ) )
34 fvco2 5737 . . . 4  |-  ( ( H  Fn  D  /\  x  e.  D )  ->  ( ( G  o.  H ) `  x
)  =  ( G `
 ( H `  x ) ) )
3531, 34sylan 458 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  (
( G  o.  H
) `  x )  =  ( G `  ( H `  x ) ) )
3621, 27, 28, 28, 29, 33, 35offval 6251 . 2  |-  ( ph  ->  ( ( F  o.  H )  o F R ( G  o.  H ) )  =  ( x  e.  D  |->  ( ( F `  ( H `  x ) ) R ( G `
 ( H `  x ) ) ) ) )
3715, 36eqtr4d 2422 1  |-  ( ph  ->  ( ( F  o F R G )  o.  H )  =  ( ( F  o.  H
)  o F R ( G  o.  H
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    i^i cin 3262    C_ wss 3263    e. cmpt 4207    o. ccom 4822    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020    o Fcof 6242
This theorem is referenced by:  gsumzaddlem  15453  coe1add  16584  pf1ind  19842  mendrng  27169
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244
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