Theorem suppssfz 14012

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 8879	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐵 ↑m ℕ0) → 𝐹 Fn ℕ0)
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℕ0)
5		nn0ex 12507	. . . . . . . 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 8169	. . . . 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 5123	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → (𝑆 < 𝑥 ↔ 𝑆 < 𝑛))
13		fveqeq2 6885	. . . . . . . . . . . . 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 3599	. . . . . . . . . . 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 12510	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
21		nn0re 12510	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆 ∈ ℕ0 → 𝑆 ∈ ℝ)
22	17, 21	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑆 ∈ ℝ)
23		lenlt 11313	. . . . . . . . . . . . . . . . . . . 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 13635	. . . . . . . . . . . . . . . . . 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 2943	. . . . . . . . . . . . . . . 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 3964	. 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 2108   ≠ wne 2932  ∀wral 3051  Vcvv 3459   ⊆ wss 3926   class class class wbr 5119   Fn wfn 6526  ‘cfv 6531  (class class class)co 7405   supp csupp 8159   ↑_m cmap 8840  ℝcr 11128  0cc0 11129   < clt 11269   ≤ cle 11270  ℕ₀cn0 12501  ...cfz 13524

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-n0 12502  df-z 12589  df-uz 12853  df-fz 13525

This theorem is referenced by:  fsuppmapnn0fz  14014  fsfnn0gsumfsffz  19964