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Theorem fsuppco2 7869
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 7870 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 5691 . . . 4  |-  ( G : B --> B  ->  Fun  G )
31, 2syl 17 . . 3  |-  ( ph  ->  Fun  G )
4 fsuppco2.f . . . 4  |-  ( ph  ->  F : A --> B )
5 ffun 5691 . . . 4  |-  ( F : A --> B  ->  Fun  F )
64, 5syl 17 . . 3  |-  ( ph  ->  Fun  F )
7 funco 5582 . . 3  |-  ( ( Fun  G  /\  Fun  F )  ->  Fun  ( G  o.  F ) )
83, 6, 7syl2anc 665 . 2  |-  ( ph  ->  Fun  ( G  o.  F ) )
9 fsuppco2.n . . . 4  |-  ( ph  ->  F finSupp  Z )
109fsuppimpd 7843 . . 3  |-  ( ph  ->  ( F supp  Z )  e.  Fin )
11 fco 5699 . . . . 5  |-  ( ( G : B --> B  /\  F : A --> B )  ->  ( G  o.  F ) : A --> B )
121, 4, 11syl2anc 665 . . . 4  |-  ( ph  ->  ( G  o.  F
) : A --> B )
13 eldifi 3530 . . . . . 6  |-  ( x  e.  ( A  \ 
( F supp  Z )
)  ->  x  e.  A )
14 fvco3 5902 . . . . . 6  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( ( G  o.  F ) `  x
)  =  ( G `
 ( F `  x ) ) )
154, 13, 14syl2an 479 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( ( G  o.  F ) `  x )  =  ( G `  ( F `
 x ) ) )
16 ssid 3426 . . . . . . . 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 6901 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( F `  x )  =  Z )
2120fveq2d 5829 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( G `  ( F `  x ) )  =  ( G `
 Z ) )
22 fsuppco2.i . . . . . 6  |-  ( ph  ->  ( G `  Z
)  =  Z )
2322adantr 466 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( G `  Z )  =  Z )
2415, 21, 233eqtrd 2466 . . . 4  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( ( G  o.  F ) `  x )  =  Z )
2512, 24suppss 6900 . . 3  |-  ( ph  ->  ( ( G  o.  F ) supp  Z )  C_  ( F supp  Z ) )
26 ssfi 7745 . . 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 665 . 2  |-  ( ph  ->  ( ( G  o.  F ) supp  Z )  e.  Fin )
28 fsuppco2.b . . . . 5  |-  ( ph  ->  B  e.  V )
29 fex 6097 . . . . 5  |-  ( ( G : B --> B  /\  B  e.  V )  ->  G  e.  _V )
301, 28, 29syl2anc 665 . . . 4  |-  ( ph  ->  G  e.  _V )
31 fex 6097 . . . . 5  |-  ( ( F : A --> B  /\  A  e.  U )  ->  F  e.  _V )
324, 18, 31syl2anc 665 . . . 4  |-  ( ph  ->  F  e.  _V )
33 coexg 6702 . . . 4  |-  ( ( G  e.  _V  /\  F  e.  _V )  ->  ( G  o.  F
)  e.  _V )
3430, 32, 33syl2anc 665 . . 3  |-  ( ph  ->  ( G  o.  F
)  e.  _V )
35 isfsupp 7840 . . 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 665 . 2  |-  ( ph  ->  ( ( G  o.  F ) finSupp  Z  <->  ( Fun  ( G  o.  F
)  /\  ( ( G  o.  F ) supp  Z )  e.  Fin )
) )
378, 27, 36mpbir2and 930 1  |-  ( ph  ->  ( G  o.  F
) finSupp  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3022    \ cdif 3376    C_ wss 3379   class class class wbr 4366    o. ccom 4800   Fun wfun 5538   -->wf 5540   ` cfv 5544  (class class class)co 6249   supp csupp 6869   Fincfn 7524   finSupp cfsupp 7836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-supp 6870  df-er 7318  df-en 7525  df-fin 7528  df-fsupp 7837
This theorem is referenced by:  gsumzinv  17521  gsumsub  17524
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