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Theorem ofco 6113
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 5679 . . . 4  |-  ( ( H : D --> C  /\  x  e.  D )  ->  ( H `  x
)  e.  C )
31, 2sylan 457 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( H `  x )  e.  C )
41feqmptd 5591 . . 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 2297 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( F `  y ) )
11 eqidd 2297 . . . 4  |-  ( (
ph  /\  y  e.  B )  ->  ( G `  y )  =  ( G `  y ) )
125, 6, 7, 8, 9, 10, 11offval 6101 . . 3  |-  ( ph  ->  ( F  o F R G )  =  ( y  e.  C  |->  ( ( F `  y ) R ( G `  y ) ) ) )
13 fveq2 5541 . . . 4  |-  ( y  =  ( H `  x )  ->  ( F `  y )  =  ( F `  ( H `  x ) ) )
14 fveq2 5541 . . . 4  |-  ( y  =  ( H `  x )  ->  ( G `  y )  =  ( G `  ( H `  x ) ) )
1513, 14oveq12d 5892 . . 3  |-  ( y  =  ( H `  x )  ->  (
( F `  y
) R ( G `
 y ) )  =  ( ( F `
 ( H `  x ) ) R ( G `  ( H `  x )
) ) )
163, 4, 12, 15fmptco 5707 . 2  |-  ( ph  ->  ( ( F  o F R G )  o.  H )  =  ( x  e.  D  |->  ( ( F `  ( H `  x )
) R ( G `
 ( H `  x ) ) ) ) )
17 inss1 3402 . . . . . 6  |-  ( A  i^i  B )  C_  A
189, 17eqsstr3i 3222 . . . . 5  |-  C  C_  A
19 fss 5413 . . . . 5  |-  ( ( H : D --> C  /\  C  C_  A )  ->  H : D --> A )
201, 18, 19sylancl 643 . . . 4  |-  ( ph  ->  H : D --> A )
21 fnfco 5423 . . . 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 3403 . . . . . 6  |-  ( A  i^i  B )  C_  B
249, 23eqsstr3i 3222 . . . . 5  |-  C  C_  B
25 fss 5413 . . . . 5  |-  ( ( H : D --> C  /\  C  C_  B )  ->  H : D --> B )
261, 24, 25sylancl 643 . . . 4  |-  ( ph  ->  H : D --> B )
27 fnfco 5423 . . . 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 3391 . . 3  |-  ( D  i^i  D )  =  D
31 ffn 5405 . . . . 5  |-  ( H : D --> C  ->  H  Fn  D )
321, 31syl 15 . . . 4  |-  ( ph  ->  H  Fn  D )
33 fvco2 5610 . . . 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 5610 . . . 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 6101 . 2  |-  ( ph  ->  ( ( F  o.  H )  o F R ( G  o.  H ) )  =  ( x  e.  D  |->  ( ( F `  ( H `  x ) ) R ( G `
 ( H `  x ) ) ) ) )
3816, 37eqtr4d 2331 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 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165    e. cmpt 4093    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092
This theorem is referenced by:  gsumzaddlem  15219  coe1add  16357  pf1ind  19454  mendrng  27603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094
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