Theorem suppssfz 13363

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 8429	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐵 ↑m ℕ0) → 𝐹 Fn ℕ0)
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℕ0)
5		nn0ex 11904	. . . . . . . 8 ⊢ ℕ0 ∈ V
6	5	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ0 ∈ V)
7		suppssfz.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
8	4, 6, 7	3jca 1124	. . . . . 6 ⊢ (𝜑 → (𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍 ∈ 𝑉))
9	8	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) → (𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍 ∈ 𝑉))
10		elsuppfn 7838	. . . . 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 5070	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → (𝑆 < 𝑥 ↔ 𝑆 < 𝑛))
13		fveqeq2 6679	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → ((𝐹‘𝑥) = 𝑍 ↔ (𝐹‘𝑛) = 𝑍))
14	12, 13	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → ((𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍) ↔ (𝑆 < 𝑛 → (𝐹‘𝑛) = 𝑍)))
15	14	rspcva 3621	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) → (𝑆 < 𝑛 → (𝐹‘𝑛) = 𝑍))
16		simplr 767	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛 ∈ ℕ0)
17		suppssfz.s	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
18	17	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑆 ∈ ℕ0)
19	18	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑆 ∈ ℕ0)
20		nn0re 11907	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
21		nn0re 11907	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆 ∈ ℕ0 → 𝑆 ∈ ℝ)
22	17, 21	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑆 ∈ ℝ)
23		lenlt 10719	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑛 ≤ 𝑆 ↔ ¬ 𝑆 < 𝑛))
24	20, 22, 23	syl2anr 598	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑛 ≤ 𝑆 ↔ ¬ 𝑆 < 𝑛))
25	24	biimpar 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛 ≤ 𝑆)
26		elfz2nn0 12999	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (0...𝑆) ↔ (𝑛 ∈ ℕ0 ∧ 𝑆 ∈ ℕ0 ∧ 𝑛 ≤ 𝑆))
27	16, 19, 25, 26	syl3anbrc 1339	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (¬ 𝑆 < 𝑛 → ((𝐹‘𝑛) ≠ 𝑍 → 𝑛 ∈ (0...𝑆))))
30		eqneqall 3027	. . . . . . . . . . . . . . . 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 189	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 415	. . . . . . . . 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 409	. . . . . 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 409	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) → ((𝑛 ∈ ℕ0 ∧ (𝐹‘𝑛) ≠ 𝑍) → 𝑛 ∈ (0...𝑆)))
43	11, 42	sylbid 242	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) → (𝑛 ∈ (𝐹 supp 𝑍) → 𝑛 ∈ (0...𝑆)))
44	43	ssrdv 3973	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) → (𝐹 supp 𝑍) ⊆ (0...𝑆))
45	1, 44	mpdan 685	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (0...𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  Vcvv 3494   ⊆ wss 3936   class class class wbr 5066   Fn wfn 6350  ‘cfv 6355  (class class class)co 7156   supp csupp 7830   ↑_m cmap 8406  ℝcr 10536  0cc0 10537   < clt 10675   ≤ cle 10676  ℕ₀cn0 11898  ...cfz 12893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-supp 7831  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894

This theorem is referenced by:  fsuppmapnn0fz  13365  fsfnn0gsumfsffz  19103