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Theorem fsuppcor 8294
 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)
2 ffun 6035 . . . 4 (𝐺:𝐵𝐷 → Fun 𝐺)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐺)
4 fsuppcor.f . . . 4 (𝜑𝐹:𝐴𝐶)
5 ffun 6035 . . . 4 (𝐹:𝐴𝐶 → Fun 𝐹)
64, 5syl 17 . . 3 (𝜑 → Fun 𝐹)
7 funco 5916 . . 3 ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺𝐹))
83, 6, 7syl2anc 692 . 2 (𝜑 → Fun (𝐺𝐹))
9 fsuppcor.n . . . 4 (𝜑𝐹 finSupp 𝑍)
109fsuppimpd 8267 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
11 fsuppcor.s . . . . . 6 (𝜑𝐶𝐵)
121, 11fssresd 6058 . . . . 5 (𝜑 → (𝐺𝐶):𝐶𝐷)
13 fco2 6046 . . . . 5 (((𝐺𝐶):𝐶𝐷𝐹:𝐴𝐶) → (𝐺𝐹):𝐴𝐷)
1412, 4, 13syl2anc 692 . . . 4 (𝜑 → (𝐺𝐹):𝐴𝐷)
15 eldifi 3724 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
16 fvco3 6262 . . . . . 6 ((𝐹:𝐴𝐶𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
174, 15, 16syl2an 494 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
18 ssid 3616 . . . . . . . 8 (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍)
1918a1i 11 . . . . . . 7 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
20 fsuppcor.a . . . . . . 7 (𝜑𝐴𝑈)
21 fsuppcor.z . . . . . . 7 (𝜑𝑍𝐵)
224, 19, 20, 21suppssr 7311 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
2322fveq2d 6182 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹𝑥)) = (𝐺𝑍))
24 fsuppcor.i . . . . . 6 (𝜑 → (𝐺𝑍) = 0 )
2524adantr 481 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺𝑍) = 0 )
2617, 23, 253eqtrd 2658 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = 0 )
2714, 26suppss 7310 . . 3 (𝜑 → ((𝐺𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍))
28 ssfi 8165 . . 3 (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐺𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍)) → ((𝐺𝐹) supp 0 ) ∈ Fin)
2910, 27, 28syl2anc 692 . 2 (𝜑 → ((𝐺𝐹) supp 0 ) ∈ Fin)
30 fsuppcor.b . . . . 5 (𝜑𝐵𝑉)
31 fex 6475 . . . . 5 ((𝐺:𝐵𝐷𝐵𝑉) → 𝐺 ∈ V)
321, 30, 31syl2anc 692 . . . 4 (𝜑𝐺 ∈ V)
33 fex 6475 . . . . 5 ((𝐹:𝐴𝐶𝐴𝑈) → 𝐹 ∈ V)
344, 20, 33syl2anc 692 . . . 4 (𝜑𝐹 ∈ V)
35 coexg 7102 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺𝐹) ∈ V)
3632, 34, 35syl2anc 692 . . 3 (𝜑 → (𝐺𝐹) ∈ V)
37 fsuppcor.0 . . 3 (𝜑0𝑊)
38 isfsupp 8264 . . 3 (((𝐺𝐹) ∈ V ∧ 0𝑊) → ((𝐺𝐹) finSupp 0 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 0 ) ∈ Fin)))
3936, 37, 38syl2anc 692 . 2 (𝜑 → ((𝐺𝐹) finSupp 0 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 0 ) ∈ Fin)))
408, 29, 39mpbir2and 956 1 (𝜑 → (𝐺𝐹) finSupp 0 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1481   ∈ wcel 1988  Vcvv 3195   ∖ cdif 3564   ⊆ wss 3567   class class class wbr 4644   ↾ cres 5106   ∘ ccom 5108  Fun wfun 5870  ⟶wf 5872  ‘cfv 5876  (class class class)co 6635   supp csupp 7280  Fincfn 7940   finSupp cfsupp 8260 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-supp 7281  df-er 7727  df-en 7941  df-fin 7944  df-fsupp 8261 This theorem is referenced by:  mapfienlem1  8295  mapfienlem2  8296  cpmadumatpolylem2  20668
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