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Theorem ofco 6361
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 5864 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( H `  x )  e.  C )
31feqmptd 5765 . . 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 2444 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( F `  y ) )
10 eqidd 2444 . . . 4  |-  ( (
ph  /\  y  e.  B )  ->  ( G `  y )  =  ( G `  y ) )
114, 5, 6, 7, 8, 9, 10offval 6348 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( y  e.  C  |->  ( ( F `  y ) R ( G `  y ) ) ) )
12 fveq2 5712 . . . 4  |-  ( y  =  ( H `  x )  ->  ( F `  y )  =  ( F `  ( H `  x ) ) )
13 fveq2 5712 . . . 4  |-  ( y  =  ( H `  x )  ->  ( G `  y )  =  ( G `  ( H `  x ) ) )
1412, 13oveq12d 6130 . . 3  |-  ( y  =  ( H `  x )  ->  (
( F `  y
) R ( G `
 y ) )  =  ( ( F `
 ( H `  x ) ) R ( G `  ( H `  x )
) ) )
152, 3, 11, 14fmptco 5897 . 2  |-  ( ph  ->  ( ( F  oF R G )  o.  H )  =  ( x  e.  D  |->  ( ( F `  ( H `  x ) ) R ( G `
 ( H `  x ) ) ) ) )
16 inss1 3591 . . . . . 6  |-  ( A  i^i  B )  C_  A
178, 16eqsstr3i 3408 . . . . 5  |-  C  C_  A
18 fss 5588 . . . . 5  |-  ( ( H : D --> C  /\  C  C_  A )  ->  H : D --> A )
191, 17, 18sylancl 662 . . . 4  |-  ( ph  ->  H : D --> A )
20 fnfco 5598 . . . 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 3592 . . . . . 6  |-  ( A  i^i  B )  C_  B
238, 22eqsstr3i 3408 . . . . 5  |-  C  C_  B
24 fss 5588 . . . . 5  |-  ( ( H : D --> C  /\  C  C_  B )  ->  H : D --> B )
251, 23, 24sylancl 662 . . . 4  |-  ( ph  ->  H : D --> B )
26 fnfco 5598 . . . 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 3580 . . 3  |-  ( D  i^i  D )  =  D
30 ffn 5580 . . . . 5  |-  ( H : D --> C  ->  H  Fn  D )
311, 30syl 16 . . . 4  |-  ( ph  ->  H  Fn  D )
32 fvco2 5787 . . . 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 5787 . . . 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 6348 . 2  |-  ( ph  ->  ( ( F  o.  H )  oF R ( G  o.  H ) )  =  ( x  e.  D  |->  ( ( F `  ( H `  x ) ) R ( G `
 ( H `  x ) ) ) ) )
3715, 36eqtr4d 2478 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 1369    e. wcel 1756    i^i cin 3348    C_ wss 3349    e. cmpt 4371    o. ccom 4865    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112    oFcof 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341
This theorem is referenced by:  gsumzaddlem  16429  gsumzaddlemOLD  16431  coe1add  17740  pf1ind  17811  mendrng  29575
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