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Theorem ofco 6559
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  oF R G )  o.  H )  =  ( ( F  o.  H )  oF 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 6032 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( H `  x )  e.  C )
31feqmptd 5926 . . 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 2458 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( F `  y ) )
10 eqidd 2458 . . . 4  |-  ( (
ph  /\  y  e.  B )  ->  ( G `  y )  =  ( G `  y ) )
114, 5, 6, 7, 8, 9, 10offval 6546 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( y  e.  C  |->  ( ( F `  y ) R ( G `  y ) ) ) )
12 fveq2 5872 . . . 4  |-  ( y  =  ( H `  x )  ->  ( F `  y )  =  ( F `  ( H `  x ) ) )
13 fveq2 5872 . . . 4  |-  ( y  =  ( H `  x )  ->  ( G `  y )  =  ( G `  ( H `  x ) ) )
1412, 13oveq12d 6314 . . 3  |-  ( y  =  ( H `  x )  ->  (
( F `  y
) R ( G `
 y ) )  =  ( ( F `
 ( H `  x ) ) R ( G `  ( H `  x )
) ) )
152, 3, 11, 14fmptco 6065 . 2  |-  ( ph  ->  ( ( F  oF R G )  o.  H )  =  ( x  e.  D  |->  ( ( F `  ( H `  x ) ) R ( G `
 ( H `  x ) ) ) ) )
16 inss1 3714 . . . . . 6  |-  ( A  i^i  B )  C_  A
178, 16eqsstr3i 3530 . . . . 5  |-  C  C_  A
18 fss 5745 . . . . 5  |-  ( ( H : D --> C  /\  C  C_  A )  ->  H : D --> A )
191, 17, 18sylancl 662 . . . 4  |-  ( ph  ->  H : D --> A )
20 fnfco 5756 . . . 4  |-  ( ( F  Fn  A  /\  H : D --> A )  ->  ( F  o.  H )  Fn  D
)
214, 19, 20syl2anc 661 . . 3  |-  ( ph  ->  ( F  o.  H
)  Fn  D )
22 inss2 3715 . . . . . 6  |-  ( A  i^i  B )  C_  B
238, 22eqsstr3i 3530 . . . . 5  |-  C  C_  B
24 fss 5745 . . . . 5  |-  ( ( H : D --> C  /\  C  C_  B )  ->  H : D --> B )
251, 23, 24sylancl 662 . . . 4  |-  ( ph  ->  H : D --> B )
26 fnfco 5756 . . . 4  |-  ( ( G  Fn  B  /\  H : D --> B )  ->  ( G  o.  H )  Fn  D
)
275, 25, 26syl2anc 661 . . 3  |-  ( ph  ->  ( G  o.  H
)  Fn  D )
28 ofco.6 . . 3  |-  ( ph  ->  D  e.  X )
29 inidm 3703 . . 3  |-  ( D  i^i  D )  =  D
30 ffn 5737 . . . . 5  |-  ( H : D --> C  ->  H  Fn  D )
311, 30syl 16 . . . 4  |-  ( ph  ->  H  Fn  D )
32 fvco2 5948 . . . 4  |-  ( ( H  Fn  D  /\  x  e.  D )  ->  ( ( F  o.  H ) `  x
)  =  ( F `
 ( H `  x ) ) )
3331, 32sylan 471 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  (
( F  o.  H
) `  x )  =  ( F `  ( H `  x ) ) )
34 fvco2 5948 . . . 4  |-  ( ( H  Fn  D  /\  x  e.  D )  ->  ( ( G  o.  H ) `  x
)  =  ( G `
 ( H `  x ) ) )
3531, 34sylan 471 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  (
( G  o.  H
) `  x )  =  ( G `  ( H `  x ) ) )
3621, 27, 28, 28, 29, 33, 35offval 6546 . 2  |-  ( ph  ->  ( ( F  o.  H )  oF R ( G  o.  H ) )  =  ( x  e.  D  |->  ( ( F `  ( H `  x ) ) R ( G `
 ( H `  x ) ) ) ) )
3715, 36eqtr4d 2501 1  |-  ( ph  ->  ( ( F  oF R G )  o.  H )  =  ( ( F  o.  H )  oF R ( G  o.  H ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    i^i cin 3470    C_ wss 3471    |-> cmpt 4515    o. ccom 5012    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    oFcof 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539
This theorem is referenced by:  gsumzaddlem  17060  gsumzaddlemOLD  17062  coe1add  18431  pf1ind  18517  mendring  31303
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