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Theorem fsuppcor 8855
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 (𝜑0𝑊)
fsuppcor.z (𝜑𝑍𝐵)
fsuppcor.f (𝜑𝐹:𝐴𝐶)
fsuppcor.g (𝜑𝐺:𝐵𝐷)
fsuppcor.s (𝜑𝐶𝐵)
fsuppcor.a (𝜑𝐴𝑈)
fsuppcor.b (𝜑𝐵𝑉)
fsuppcor.n (𝜑𝐹 finSupp 𝑍)
fsuppcor.i (𝜑 → (𝐺𝑍) = 0 )
Assertion
Ref Expression
fsuppcor (𝜑 → (𝐺𝐹) finSupp 0 )

Proof of Theorem fsuppcor
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fsuppcor.g . . . 4 (𝜑𝐺:𝐵𝐷)
21ffund 6495 . . 3 (𝜑 → Fun 𝐺)
3 fsuppcor.f . . . 4 (𝜑𝐹:𝐴𝐶)
43ffund 6495 . . 3 (𝜑 → Fun 𝐹)
5 funco 6368 . . 3 ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺𝐹))
62, 4, 5syl2anc 587 . 2 (𝜑 → Fun (𝐺𝐹))
7 fsuppcor.n . . . 4 (𝜑𝐹 finSupp 𝑍)
87fsuppimpd 8828 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9 fsuppcor.s . . . . . 6 (𝜑𝐶𝐵)
101, 9fssresd 6523 . . . . 5 (𝜑 → (𝐺𝐶):𝐶𝐷)
11 fco2 6511 . . . . 5 (((𝐺𝐶):𝐶𝐷𝐹:𝐴𝐶) → (𝐺𝐹):𝐴𝐷)
1210, 3, 11syl2anc 587 . . . 4 (𝜑 → (𝐺𝐹):𝐴𝐷)
13 eldifi 4057 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
14 fvco3 6741 . . . . . 6 ((𝐹:𝐴𝐶𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
153, 13, 14syl2an 598 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
16 ssidd 3941 . . . . . . 7 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
17 fsuppcor.a . . . . . . 7 (𝜑𝐴𝑈)
18 fsuppcor.z . . . . . . 7 (𝜑𝑍𝐵)
193, 16, 17, 18suppssr 7848 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
2019fveq2d 6653 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹𝑥)) = (𝐺𝑍))
21 fsuppcor.i . . . . . 6 (𝜑 → (𝐺𝑍) = 0 )
2221adantr 484 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺𝑍) = 0 )
2315, 20, 223eqtrd 2840 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = 0 )
2412, 23suppss 7847 . . 3 (𝜑 → ((𝐺𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍))
258, 24ssfid 8729 . 2 (𝜑 → ((𝐺𝐹) supp 0 ) ∈ Fin)
26 fsuppcor.b . . . . 5 (𝜑𝐵𝑉)
27 fex 6970 . . . . 5 ((𝐺:𝐵𝐷𝐵𝑉) → 𝐺 ∈ V)
281, 26, 27syl2anc 587 . . . 4 (𝜑𝐺 ∈ V)
29 fex 6970 . . . . 5 ((𝐹:𝐴𝐶𝐴𝑈) → 𝐹 ∈ V)
303, 17, 29syl2anc 587 . . . 4 (𝜑𝐹 ∈ V)
31 coexg 7620 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺𝐹) ∈ V)
3228, 30, 31syl2anc 587 . . 3 (𝜑 → (𝐺𝐹) ∈ V)
33 fsuppcor.0 . . 3 (𝜑0𝑊)
34 isfsupp 8825 . . 3 (((𝐺𝐹) ∈ V ∧ 0𝑊) → ((𝐺𝐹) finSupp 0 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 0 ) ∈ Fin)))
3532, 33, 34syl2anc 587 . 2 (𝜑 → ((𝐺𝐹) finSupp 0 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 0 ) ∈ Fin)))
366, 25, 35mpbir2and 712 1 (𝜑 → (𝐺𝐹) finSupp 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  Vcvv 3444  cdif 3881  wss 3884   class class class wbr 5033  cres 5525  ccom 5527  Fun wfun 6322  wf 6324  cfv 6328  (class class class)co 7139   supp csupp 7817  Fincfn 8496   finSupp cfsupp 8821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-supp 7818  df-er 8276  df-en 8497  df-fin 8500  df-fsupp 8822
This theorem is referenced by:  mapfienlem1  8856  mapfienlem2  8857  cpmadumatpolylem2  21490  selvval2lem4  39418
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