Theorem suppssfz 13901

Description: Condition for a function over the nonnegative integers to have a support contained in a finite set of sequential integers. (Contributed by AV, 9-Oct-2019.)

Hypotheses

Ref	Expression
suppssfz.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)
suppssfz.f	⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m ℕ0))
suppssfz.s	⊢ (𝜑 → 𝑆 ∈ ℕ0)
suppssfz.b	⊢ (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍))

Assertion

Ref	Expression
suppssfz	⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (0...𝑆))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑆   𝑥,𝑍

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem suppssfz

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppssfz.b	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍))
2		suppssfz.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m ℕ0))
3		elmapfn 8792	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐵 ↑m ℕ0) → 𝐹 Fn ℕ0)
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℕ0)
5		nn0ex 12390	. . . . . . . 8 ⊢ ℕ0 ∈ V
6	5	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ0 ∈ V)
7		suppssfz.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
8	4, 6, 7	3jca 1128	. . . . . 6 ⊢ (𝜑 → (𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍 ∈ 𝑉))
9	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) → (𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍 ∈ 𝑉))
10		elsuppfn 8103	. . . . 5 ⊢ ((𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍 ∈ 𝑉) → (𝑛 ∈ (𝐹 supp 𝑍) ↔ (𝑛 ∈ ℕ0 ∧ (𝐹‘𝑛) ≠ 𝑍)))
11	9, 10	syl 17	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) → (𝑛 ∈ (𝐹 supp 𝑍) ↔ (𝑛 ∈ ℕ0 ∧ (𝐹‘𝑛) ≠ 𝑍)))
12		breq2 5096	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → (𝑆 < 𝑥 ↔ 𝑆 < 𝑛))
13		fveqeq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → ((𝐹‘𝑥) = 𝑍 ↔ (𝐹‘𝑛) = 𝑍))
14	12, 13	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → ((𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍) ↔ (𝑆 < 𝑛 → (𝐹‘𝑛) = 𝑍)))
15	14	rspcva 3575	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) → (𝑆 < 𝑛 → (𝐹‘𝑛) = 𝑍))
16		simplr 768	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛 ∈ ℕ0)
17		suppssfz.s	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
18	17	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑆 ∈ ℕ0)
19	18	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑆 ∈ ℕ0)
20		nn0re 12393	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
21		nn0re 12393	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆 ∈ ℕ0 → 𝑆 ∈ ℝ)
22	17, 21	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑆 ∈ ℝ)
23		lenlt 11194	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑛 ≤ 𝑆 ↔ ¬ 𝑆 < 𝑛))
24	20, 22, 23	syl2anr 597	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑛 ≤ 𝑆 ↔ ¬ 𝑆 < 𝑛))
25	24	biimpar 477	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛 ≤ 𝑆)
26		elfz2nn0 13521	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (0...𝑆) ↔ (𝑛 ∈ ℕ0 ∧ 𝑆 ∈ ℕ0 ∧ 𝑛 ≤ 𝑆))
27	16, 19, 25, 26	syl3anbrc 1344	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛 ∈ (0...𝑆))
28	27	a1d 25	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → ((𝐹‘𝑛) ≠ 𝑍 → 𝑛 ∈ (0...𝑆)))
29	28	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (¬ 𝑆 < 𝑛 → ((𝐹‘𝑛) ≠ 𝑍 → 𝑛 ∈ (0...𝑆))))
30		eqneqall 2936	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑛) = 𝑍 → ((𝐹‘𝑛) ≠ 𝑍 → 𝑛 ∈ (0...𝑆)))
31	30	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝐹‘𝑛) = 𝑍 → ((𝐹‘𝑛) ≠ 𝑍 → 𝑛 ∈ (0...𝑆))))
32	29, 31	jad 187	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑆 < 𝑛 → (𝐹‘𝑛) = 𝑍) → ((𝐹‘𝑛) ≠ 𝑍 → 𝑛 ∈ (0...𝑆))))
33	32	com23 86	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝐹‘𝑛) ≠ 𝑍 → ((𝑆 < 𝑛 → (𝐹‘𝑛) = 𝑍) → 𝑛 ∈ (0...𝑆))))
34	33	ex 412	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ ℕ0 → ((𝐹‘𝑛) ≠ 𝑍 → ((𝑆 < 𝑛 → (𝐹‘𝑛) = 𝑍) → 𝑛 ∈ (0...𝑆)))))
35	34	com14 96	. . . . . . . . . . 11 ⊢ ((𝑆 < 𝑛 → (𝐹‘𝑛) = 𝑍) → (𝑛 ∈ ℕ0 → ((𝐹‘𝑛) ≠ 𝑍 → (𝜑 → 𝑛 ∈ (0...𝑆)))))
36	15, 35	syl 17	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) → (𝑛 ∈ ℕ0 → ((𝐹‘𝑛) ≠ 𝑍 → (𝜑 → 𝑛 ∈ (0...𝑆)))))
37	36	ex 412	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍) → (𝑛 ∈ ℕ0 → ((𝐹‘𝑛) ≠ 𝑍 → (𝜑 → 𝑛 ∈ (0...𝑆))))))
38	37	pm2.43a 54	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍) → ((𝐹‘𝑛) ≠ 𝑍 → (𝜑 → 𝑛 ∈ (0...𝑆)))))
39	38	com23 86	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → ((𝐹‘𝑛) ≠ 𝑍 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍) → (𝜑 → 𝑛 ∈ (0...𝑆)))))
40	39	imp 406	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0 ∧ (𝐹‘𝑛) ≠ 𝑍) → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍) → (𝜑 → 𝑛 ∈ (0...𝑆))))
41	40	com13 88	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍) → ((𝑛 ∈ ℕ0 ∧ (𝐹‘𝑛) ≠ 𝑍) → 𝑛 ∈ (0...𝑆))))
42	41	imp 406	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) → ((𝑛 ∈ ℕ0 ∧ (𝐹‘𝑛) ≠ 𝑍) → 𝑛 ∈ (0...𝑆)))
43	11, 42	sylbid 240	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) → (𝑛 ∈ (𝐹 supp 𝑍) → 𝑛 ∈ (0...𝑆)))
44	43	ssrdv 3941	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) → (𝐹 supp 𝑍) ⊆ (0...𝑆))
45	1, 44	mpdan 687	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (0...𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3436   ⊆ wss 3903   class class class wbr 5092   Fn wfn 6477  ‘cfv 6482  (class class class)co 7349   supp csupp 8093   ↑_m cmap 8753  ℝcr 11008  0cc0 11009   < clt 11149   ≤ cle 11150  ℕ₀cn0 12384  ...cfz 13410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411

This theorem is referenced by:  fsuppmapnn0fz  13903  fsfnn0gsumfsffz  19862