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Theorem fsuppco2 7874
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 7875 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 5739 . . . 4  |-  ( G : B --> B  ->  Fun  G )
31, 2syl 16 . . 3  |-  ( ph  ->  Fun  G )
4 fsuppco2.f . . . 4  |-  ( ph  ->  F : A --> B )
5 ffun 5739 . . . 4  |-  ( F : A --> B  ->  Fun  F )
64, 5syl 16 . . 3  |-  ( ph  ->  Fun  F )
7 funco 5632 . . 3  |-  ( ( Fun  G  /\  Fun  F )  ->  Fun  ( G  o.  F ) )
83, 6, 7syl2anc 661 . 2  |-  ( ph  ->  Fun  ( G  o.  F ) )
9 fsuppco2.n . . . 4  |-  ( ph  ->  F finSupp  Z )
109fsuppimpd 7848 . . 3  |-  ( ph  ->  ( F supp  Z )  e.  Fin )
11 fco 5747 . . . . 5  |-  ( ( G : B --> B  /\  F : A --> B )  ->  ( G  o.  F ) : A --> B )
121, 4, 11syl2anc 661 . . . 4  |-  ( ph  ->  ( G  o.  F
) : A --> B )
13 eldifi 3631 . . . . . 6  |-  ( x  e.  ( A  \ 
( F supp  Z )
)  ->  x  e.  A )
14 fvco3 5951 . . . . . 6  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( ( G  o.  F ) `  x
)  =  ( G `
 ( F `  x ) ) )
154, 13, 14syl2an 477 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( ( G  o.  F ) `  x )  =  ( G `  ( F `
 x ) ) )
16 ssid 3528 . . . . . . . 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 6943 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( F `  x )  =  Z )
2120fveq2d 5876 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( G `  ( F `  x ) )  =  ( G `
 Z ) )
22 fsuppco2.i . . . . . 6  |-  ( ph  ->  ( G `  Z
)  =  Z )
2322adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( G `  Z )  =  Z )
2415, 21, 233eqtrd 2512 . . . 4  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( ( G  o.  F ) `  x )  =  Z )
2512, 24suppss 6942 . . 3  |-  ( ph  ->  ( ( G  o.  F ) supp  Z )  C_  ( F supp  Z ) )
26 ssfi 7752 . . 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 661 . 2  |-  ( ph  ->  ( ( G  o.  F ) supp  Z )  e.  Fin )
28 fsuppco2.b . . . . 5  |-  ( ph  ->  B  e.  V )
29 fex 6144 . . . . 5  |-  ( ( G : B --> B  /\  B  e.  V )  ->  G  e.  _V )
301, 28, 29syl2anc 661 . . . 4  |-  ( ph  ->  G  e.  _V )
31 fex 6144 . . . . 5  |-  ( ( F : A --> B  /\  A  e.  U )  ->  F  e.  _V )
324, 18, 31syl2anc 661 . . . 4  |-  ( ph  ->  F  e.  _V )
33 coexg 6746 . . . 4  |-  ( ( G  e.  _V  /\  F  e.  _V )  ->  ( G  o.  F
)  e.  _V )
3430, 32, 33syl2anc 661 . . 3  |-  ( ph  ->  ( G  o.  F
)  e.  _V )
35 isfsupp 7845 . . 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 661 . 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 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    \ cdif 3478    C_ wss 3481   class class class wbr 4453    o. ccom 5009   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6295   supp csupp 6913   Fincfn 7528   finSupp cfsupp 7841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-supp 6914  df-er 7323  df-en 7529  df-fin 7532  df-fsupp 7842
This theorem is referenced by:  gsumzinv  16842  gsumsub  16847
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