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Theorem fsuppcor 7926
Description: The composition of a function which maps the zero of the range of a finitely supported function to the zero of its range with this finitely supported function is finitely supported. (Contributed by AV, 6-Jun-2019.)
Hypotheses
Ref Expression
fsuppcor.0  |-  ( ph  ->  .0.  e.  W )
fsuppcor.z  |-  ( ph  ->  Z  e.  B )
fsuppcor.f  |-  ( ph  ->  F : A --> C )
fsuppcor.g  |-  ( ph  ->  G : B --> D )
fsuppcor.s  |-  ( ph  ->  C  C_  B )
fsuppcor.a  |-  ( ph  ->  A  e.  U )
fsuppcor.b  |-  ( ph  ->  B  e.  V )
fsuppcor.n  |-  ( ph  ->  F finSupp  Z )
fsuppcor.i  |-  ( ph  ->  ( G `  Z
)  =  .0.  )
Assertion
Ref Expression
fsuppcor  |-  ( ph  ->  ( G  o.  F
) finSupp  .0.  )

Proof of Theorem fsuppcor
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fsuppcor.g . . . 4  |-  ( ph  ->  G : B --> D )
2 ffun 5748 . . . 4  |-  ( G : B --> D  ->  Fun  G )
31, 2syl 17 . . 3  |-  ( ph  ->  Fun  G )
4 fsuppcor.f . . . 4  |-  ( ph  ->  F : A --> C )
5 ffun 5748 . . . 4  |-  ( F : A --> C  ->  Fun  F )
64, 5syl 17 . . 3  |-  ( ph  ->  Fun  F )
7 funco 5639 . . 3  |-  ( ( Fun  G  /\  Fun  F )  ->  Fun  ( G  o.  F ) )
83, 6, 7syl2anc 665 . 2  |-  ( ph  ->  Fun  ( G  o.  F ) )
9 fsuppcor.n . . . 4  |-  ( ph  ->  F finSupp  Z )
109fsuppimpd 7899 . . 3  |-  ( ph  ->  ( F supp  Z )  e.  Fin )
11 fsuppcor.s . . . . . 6  |-  ( ph  ->  C  C_  B )
121, 11fssresd 5767 . . . . 5  |-  ( ph  ->  ( G  |`  C ) : C --> D )
13 fco2 5757 . . . . 5  |-  ( ( ( G  |`  C ) : C --> D  /\  F : A --> C )  ->  ( G  o.  F ) : A --> D )
1412, 4, 13syl2anc 665 . . . 4  |-  ( ph  ->  ( G  o.  F
) : A --> D )
15 eldifi 3587 . . . . . 6  |-  ( x  e.  ( A  \ 
( F supp  Z )
)  ->  x  e.  A )
16 fvco3 5958 . . . . . 6  |-  ( ( F : A --> C  /\  x  e.  A )  ->  ( ( G  o.  F ) `  x
)  =  ( G `
 ( F `  x ) ) )
174, 15, 16syl2an 479 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( ( G  o.  F ) `  x )  =  ( G `  ( F `
 x ) ) )
18 ssid 3483 . . . . . . . 8  |-  ( F supp 
Z )  C_  ( F supp  Z )
1918a1i 11 . . . . . . 7  |-  ( ph  ->  ( F supp  Z ) 
C_  ( F supp  Z
) )
20 fsuppcor.a . . . . . . 7  |-  ( ph  ->  A  e.  U )
21 fsuppcor.z . . . . . . 7  |-  ( ph  ->  Z  e.  B )
224, 19, 20, 21suppssr 6957 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( F `  x )  =  Z )
2322fveq2d 5885 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( G `  ( F `  x ) )  =  ( G `
 Z ) )
24 fsuppcor.i . . . . . 6  |-  ( ph  ->  ( G `  Z
)  =  .0.  )
2524adantr 466 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( G `  Z )  =  .0.  )
2617, 23, 253eqtrd 2467 . . . 4  |-  ( (
ph  /\  x  e.  ( A  \  ( F supp  Z ) ) )  ->  ( ( G  o.  F ) `  x )  =  .0.  )
2714, 26suppss 6956 . . 3  |-  ( ph  ->  ( ( G  o.  F ) supp  .0.  )  C_  ( F supp  Z ) )
28 ssfi 7801 . . 3  |-  ( ( ( F supp  Z )  e.  Fin  /\  (
( G  o.  F
) supp  .0.  )  C_  ( F supp  Z )
)  ->  ( ( G  o.  F ) supp  .0.  )  e.  Fin )
2910, 27, 28syl2anc 665 . 2  |-  ( ph  ->  ( ( G  o.  F ) supp  .0.  )  e.  Fin )
30 fsuppcor.b . . . . 5  |-  ( ph  ->  B  e.  V )
31 fex 6153 . . . . 5  |-  ( ( G : B --> D  /\  B  e.  V )  ->  G  e.  _V )
321, 30, 31syl2anc 665 . . . 4  |-  ( ph  ->  G  e.  _V )
33 fex 6153 . . . . 5  |-  ( ( F : A --> C  /\  A  e.  U )  ->  F  e.  _V )
344, 20, 33syl2anc 665 . . . 4  |-  ( ph  ->  F  e.  _V )
35 coexg 6758 . . . 4  |-  ( ( G  e.  _V  /\  F  e.  _V )  ->  ( G  o.  F
)  e.  _V )
3632, 34, 35syl2anc 665 . . 3  |-  ( ph  ->  ( G  o.  F
)  e.  _V )
37 fsuppcor.0 . . 3  |-  ( ph  ->  .0.  e.  W )
38 isfsupp 7896 . . 3  |-  ( ( ( G  o.  F
)  e.  _V  /\  .0.  e.  W )  -> 
( ( G  o.  F ) finSupp  .0.  <->  ( Fun  ( G  o.  F
)  /\  ( ( G  o.  F ) supp  .0.  )  e.  Fin ) ) )
3936, 37, 38syl2anc 665 . 2  |-  ( ph  ->  ( ( G  o.  F ) finSupp  .0.  <->  ( Fun  ( G  o.  F
)  /\  ( ( G  o.  F ) supp  .0.  )  e.  Fin ) ) )
408, 29, 39mpbir2and 930 1  |-  ( ph  ->  ( G  o.  F
) finSupp  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3080    \ cdif 3433    C_ wss 3436   class class class wbr 4423    |` cres 4855    o. ccom 4857   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6305   supp csupp 6925   Fincfn 7580   finSupp cfsupp 7892
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 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
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 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-supp 6926  df-er 7374  df-en 7581  df-fin 7584  df-fsupp 7893
This theorem is referenced by:  mapfienlem1  7927  mapfienlem2  7928  cpmadumatpolylem2  19904
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