Theorem suppssfz 13950

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 8806	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐵 ↑m ℕ0) → 𝐹 Fn ℕ0)
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℕ0)
5		nn0ex 12437	. . . . . . . 8 ⊢ ℕ0 ∈ V
6	5	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ0 ∈ V)
7		suppssfz.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
8	4, 6, 7	3jca 1129	. . . . . 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 8114	. . . . 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 5090	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → (𝑆 < 𝑥 ↔ 𝑆 < 𝑛))
13		fveqeq2 6844	. . . . . . . . . . . . 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 3563	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) → (𝑆 < 𝑛 → (𝐹‘𝑛) = 𝑍))
16		simplr 769	. . . . . . . . . . . . . . . . . 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 12440	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
21		nn0re 12440	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆 ∈ ℕ0 → 𝑆 ∈ ℝ)
22	17, 21	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑆 ∈ ℝ)
23		lenlt 11218	. . . . . . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛 ≤ 𝑆)
26		elfz2nn0 13566	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (0...𝑆) ↔ (𝑛 ∈ ℕ0 ∧ 𝑆 ∈ ℕ0 ∧ 𝑛 ≤ 𝑆))
27	16, 19, 25, 26	syl3anbrc 1345	. . . . . . . . . . . . . . . . 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 2944	. . . . . . . . . . . . . . . 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 3928	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) → (𝐹 supp 𝑍) ⊆ (0...𝑆))
45	1, 44	mpdan 688	1 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (0...𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3430   ⊆ wss 3890   class class class wbr 5086   Fn wfn 6488  ‘cfv 6493  (class class class)co 7361   supp csupp 8104   ↑_m cmap 8767  ℝcr 11031  0cc0 11032   < clt 11173   ≤ cle 11174  ℕ₀cn0 12431  ...cfz 13455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-n0 12432  df-z 12519  df-uz 12783  df-fz 13456

This theorem is referenced by:  fsuppmapnn0fz  13952  fsfnn0gsumfsffz  19952