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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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