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Theorem fsuppco2 7854
Description: The composition of a function which maps the zero to zero with a finitely supported function is finitely supported. This is not only a special case of fsuppcor 7855 because it does not require that the "zero" is an element of the range of the finitely supported function. (Contributed by AV, 6-Jun-2019.)
Hypotheses
Ref Expression
fsuppco2.z  |-  ( ph  ->  Z  e.  W )
fsuppco2.f  |-  ( ph  ->  F : A --> B )
fsuppco2.g  |-  ( ph  ->  G : B --> B )
fsuppco2.a  |-  ( ph  ->  A  e.  U )
fsuppco2.b  |-  ( ph  ->  B  e.  V )
fsuppco2.n  |-  ( ph  ->  F finSupp  Z )
fsuppco2.i  |-  ( ph  ->  ( G `  Z
)  =  Z )
Assertion
Ref Expression
fsuppco2  |-  ( ph  ->  ( G  o.  F
) finSupp  Z )

Proof of Theorem fsuppco2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fsuppco2.g . . . 4  |-  ( ph  ->  G : B --> B )
2 ffun 5715 . . . 4  |-  ( G : B --> B  ->  Fun  G )
31, 2syl 16 . . 3  |-  ( ph  ->  Fun  G )
4 fsuppco2.f . . . 4  |-  ( ph  ->  F : A --> B )
5 ffun 5715 . . . 4  |-  ( F : A --> B  ->  Fun  F )
64, 5syl 16 . . 3  |-  ( ph  ->  Fun  F )
7 funco 5608 . . 3  |-  ( ( Fun  G  /\  Fun  F )  ->  Fun  ( G  o.  F ) )
83, 6, 7syl2anc 659 . 2  |-  ( ph  ->  Fun  ( G  o.  F ) )
9 fsuppco2.n . . . 4  |-  ( ph  ->  F finSupp  Z )
109fsuppimpd 7828 . . 3  |-  ( ph  ->  ( F supp  Z )  e.  Fin )
11 fco 5723 . . . . 5  |-  ( ( G : B --> B  /\  F : A --> B )  ->  ( G  o.  F ) : A --> B )
121, 4, 11syl2anc 659 . . . 4  |-  ( ph  ->  ( G  o.  F
) : A --> B )
13 eldifi 3612 . . . . . 6  |-  ( x  e.  ( A  \ 
( F supp  Z )
)  ->  x  e.  A )
14 fvco3 5925 . . . . . 6  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( ( G  o.  F ) `  x
)  =  ( G `
 ( F `  x ) ) )
154, 13, 14syl2an 475 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( ( G  o.  F ) `  x )  =  ( G `  ( F `
 x ) ) )
16 ssid 3508 . . . . . . . 8  |-  ( F supp 
Z )  C_  ( F supp  Z )
1716a1i 11 . . . . . . 7  |-  ( ph  ->  ( F supp  Z ) 
C_  ( F supp  Z
) )
18 fsuppco2.a . . . . . . 7  |-  ( ph  ->  A  e.  U )
19 fsuppco2.z . . . . . . 7  |-  ( ph  ->  Z  e.  W )
204, 17, 18, 19suppssr 6923 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( F `  x )  =  Z )
2120fveq2d 5852 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( G `  ( F `  x ) )  =  ( G `
 Z ) )
22 fsuppco2.i . . . . . 6  |-  ( ph  ->  ( G `  Z
)  =  Z )
2322adantr 463 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( G `  Z )  =  Z )
2415, 21, 233eqtrd 2499 . . . 4  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( ( G  o.  F ) `  x )  =  Z )
2512, 24suppss 6922 . . 3  |-  ( ph  ->  ( ( G  o.  F ) supp  Z )  C_  ( F supp  Z ) )
26 ssfi 7733 . . 3  |-  ( ( ( F supp  Z )  e.  Fin  /\  (
( G  o.  F
) supp  Z )  C_  ( F supp  Z ) )  -> 
( ( G  o.  F ) supp  Z )  e.  Fin )
2710, 25, 26syl2anc 659 . 2  |-  ( ph  ->  ( ( G  o.  F ) supp  Z )  e.  Fin )
28 fsuppco2.b . . . . 5  |-  ( ph  ->  B  e.  V )
29 fex 6120 . . . . 5  |-  ( ( G : B --> B  /\  B  e.  V )  ->  G  e.  _V )
301, 28, 29syl2anc 659 . . . 4  |-  ( ph  ->  G  e.  _V )
31 fex 6120 . . . . 5  |-  ( ( F : A --> B  /\  A  e.  U )  ->  F  e.  _V )
324, 18, 31syl2anc 659 . . . 4  |-  ( ph  ->  F  e.  _V )
33 coexg 6724 . . . 4  |-  ( ( G  e.  _V  /\  F  e.  _V )  ->  ( G  o.  F
)  e.  _V )
3430, 32, 33syl2anc 659 . . 3  |-  ( ph  ->  ( G  o.  F
)  e.  _V )
35 isfsupp 7825 . . 3  |-  ( ( ( G  o.  F
)  e.  _V  /\  Z  e.  W )  ->  ( ( G  o.  F ) finSupp  Z  <->  ( Fun  ( G  o.  F
)  /\  ( ( G  o.  F ) supp  Z )  e.  Fin )
) )
3634, 19, 35syl2anc 659 . 2  |-  ( ph  ->  ( ( G  o.  F ) finSupp  Z  <->  ( Fun  ( G  o.  F
)  /\  ( ( G  o.  F ) supp  Z )  e.  Fin )
) )
378, 27, 36mpbir2and 920 1  |-  ( ph  ->  ( G  o.  F
) finSupp  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    \ cdif 3458    C_ wss 3461   class class class wbr 4439    o. ccom 4992   Fun wfun 5564   -->wf 5566   ` cfv 5570  (class class class)co 6270   supp csupp 6891   Fincfn 7509   finSupp cfsupp 7821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-supp 6892  df-er 7303  df-en 7510  df-fin 7513  df-fsupp 7822
This theorem is referenced by:  gsumzinv  17167  gsumsub  17171
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