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Theorem ofco 6097
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 )
2 ffvelrn 5663 . . . 4  |-  ( ( H : D --> C  /\  x  e.  D )  ->  ( H `  x
)  e.  C )
31, 2sylan 457 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( H `  x )  e.  C )
41feqmptd 5575 . . 3  |-  ( ph  ->  H  =  ( x  e.  D  |->  ( H `
 x ) ) )
5 ofco.1 . . . 4  |-  ( ph  ->  F  Fn  A )
6 ofco.2 . . . 4  |-  ( ph  ->  G  Fn  B )
7 ofco.4 . . . 4  |-  ( ph  ->  A  e.  V )
8 ofco.5 . . . 4  |-  ( ph  ->  B  e.  W )
9 ofco.7 . . . 4  |-  ( A  i^i  B )  =  C
10 eqidd 2284 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( F `  y ) )
11 eqidd 2284 . . . 4  |-  ( (
ph  /\  y  e.  B )  ->  ( G `  y )  =  ( G `  y ) )
125, 6, 7, 8, 9, 10, 11offval 6085 . . 3  |-  ( ph  ->  ( F  o F R G )  =  ( y  e.  C  |->  ( ( F `  y ) R ( G `  y ) ) ) )
13 fveq2 5525 . . . 4  |-  ( y  =  ( H `  x )  ->  ( F `  y )  =  ( F `  ( H `  x ) ) )
14 fveq2 5525 . . . 4  |-  ( y  =  ( H `  x )  ->  ( G `  y )  =  ( G `  ( H `  x ) ) )
1513, 14oveq12d 5876 . . 3  |-  ( y  =  ( H `  x )  ->  (
( F `  y
) R ( G `
 y ) )  =  ( ( F `
 ( H `  x ) ) R ( G `  ( H `  x )
) ) )
163, 4, 12, 15fmptco 5691 . 2  |-  ( ph  ->  ( ( F  o F R G )  o.  H )  =  ( x  e.  D  |->  ( ( F `  ( H `  x )
) R ( G `
 ( H `  x ) ) ) ) )
17 inss1 3389 . . . . . 6  |-  ( A  i^i  B )  C_  A
189, 17eqsstr3i 3209 . . . . 5  |-  C  C_  A
19 fss 5397 . . . . 5  |-  ( ( H : D --> C  /\  C  C_  A )  ->  H : D --> A )
201, 18, 19sylancl 643 . . . 4  |-  ( ph  ->  H : D --> A )
21 fnfco 5407 . . . 4  |-  ( ( F  Fn  A  /\  H : D --> A )  ->  ( F  o.  H )  Fn  D
)
225, 20, 21syl2anc 642 . . 3  |-  ( ph  ->  ( F  o.  H
)  Fn  D )
23 inss2 3390 . . . . . 6  |-  ( A  i^i  B )  C_  B
249, 23eqsstr3i 3209 . . . . 5  |-  C  C_  B
25 fss 5397 . . . . 5  |-  ( ( H : D --> C  /\  C  C_  B )  ->  H : D --> B )
261, 24, 25sylancl 643 . . . 4  |-  ( ph  ->  H : D --> B )
27 fnfco 5407 . . . 4  |-  ( ( G  Fn  B  /\  H : D --> B )  ->  ( G  o.  H )  Fn  D
)
286, 26, 27syl2anc 642 . . 3  |-  ( ph  ->  ( G  o.  H
)  Fn  D )
29 ofco.6 . . 3  |-  ( ph  ->  D  e.  X )
30 inidm 3378 . . 3  |-  ( D  i^i  D )  =  D
31 ffn 5389 . . . . 5  |-  ( H : D --> C  ->  H  Fn  D )
321, 31syl 15 . . . 4  |-  ( ph  ->  H  Fn  D )
33 fvco2 5594 . . . 4  |-  ( ( H  Fn  D  /\  x  e.  D )  ->  ( ( F  o.  H ) `  x
)  =  ( F `
 ( H `  x ) ) )
3432, 33sylan 457 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  (
( F  o.  H
) `  x )  =  ( F `  ( H `  x ) ) )
35 fvco2 5594 . . . 4  |-  ( ( H  Fn  D  /\  x  e.  D )  ->  ( ( G  o.  H ) `  x
)  =  ( G `
 ( H `  x ) ) )
3632, 35sylan 457 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  (
( G  o.  H
) `  x )  =  ( G `  ( H `  x ) ) )
3722, 28, 29, 29, 30, 34, 36offval 6085 . 2  |-  ( ph  ->  ( ( F  o.  H )  o F R ( G  o.  H ) )  =  ( x  e.  D  |->  ( ( F `  ( H `  x ) ) R ( G `
 ( H `  x ) ) ) ) )
3816, 37eqtr4d 2318 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 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152    e. cmpt 4077    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076
This theorem is referenced by:  gsumzaddlem  15203  coe1add  16341  pf1ind  19438  mendrng  27500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078
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