Theorem plyco0 25353

Description: Two ways to say that a function on the nonnegative integers has finite support. (Contributed by Mario Carneiro, 22-Jul-2014.)

Assertion

Ref	Expression
plyco0	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))

Distinct variable groups:   𝐴,𝑘   𝑘,𝑁

Proof of Theorem plyco0

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprr 770	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝐴‘𝑘) ≠ 0)
2		ffun 6603	. . . . . . . . . . . 12 ⊢ (𝐴:ℕ0⟶ℂ → Fun 𝐴)
3	2	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → Fun 𝐴)
4		peano2nn0 12273	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
5	4	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 + 1) ∈ ℕ0)
6		eluznn0 12657	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ ℕ0)
7	6	ex 413	. . . . . . . . . . . . . 14 ⊢ ((𝑁 + 1) ∈ ℕ0 → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → 𝑘 ∈ ℕ0))
8	5, 7	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → 𝑘 ∈ ℕ0))
9	8	ssrdv 3927	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (ℤ≥‘(𝑁 + 1)) ⊆ ℕ0)
10		fdm 6609	. . . . . . . . . . . . 13 ⊢ (𝐴:ℕ0⟶ℂ → dom 𝐴 = ℕ0)
11	10	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → dom 𝐴 = ℕ0)
12	9, 11	sseqtrrd 3962	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴)
13		funfvima2 7107	. . . . . . . . . . 11 ⊢ ((Fun 𝐴 ∧ (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1)))))
14	3, 12, 13	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1)))))
15	14	ad2antrr 723	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1)))))
16		nn0z 12343	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
17	16	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → 𝑁 ∈ ℤ)
18	17	peano2zd 12429	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 + 1) ∈ ℤ)
19	18	ad2antrr 723	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑁 + 1) ∈ ℤ)
20		nn0z 12343	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ)
21	20	ad2antrl 725	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ∈ ℤ)
22		eluz 12596	. . . . . . . . . 10 ⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) ↔ (𝑁 + 1) ≤ 𝑘))
23	19, 21, 22	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) ↔ (𝑁 + 1) ≤ 𝑘))
24		simplr 766	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})
25	24	eleq2d 2824	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → ((𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1))) ↔ (𝐴‘𝑘) ∈ {0}))
26		fvex 6787	. . . . . . . . . . 11 ⊢ (𝐴‘𝑘) ∈ V
27	26	elsn 4576	. . . . . . . . . 10 ⊢ ((𝐴‘𝑘) ∈ {0} ↔ (𝐴‘𝑘) = 0)
28	25, 27	bitrdi 287	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → ((𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1))) ↔ (𝐴‘𝑘) = 0))
29	15, 23, 28	3imtr3d 293	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → ((𝑁 + 1) ≤ 𝑘 → (𝐴‘𝑘) = 0))
30	29	necon3ad 2956	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → ((𝐴‘𝑘) ≠ 0 → ¬ (𝑁 + 1) ≤ 𝑘))
31	1, 30	mpd 15	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → ¬ (𝑁 + 1) ≤ 𝑘)
32		nn0re 12242	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
33	32	ad2antrl 725	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ∈ ℝ)
34	18	zred 12426	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 + 1) ∈ ℝ)
35	34	ad2antrr 723	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑁 + 1) ∈ ℝ)
36	33, 35	ltnled 11122	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑘 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑘))
37	31, 36	mpbird 256	. . . . 5 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 < (𝑁 + 1))
38	17	ad2antrr 723	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑁 ∈ ℤ)
39		zleltp1 12371	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑘 ≤ 𝑁 ↔ 𝑘 < (𝑁 + 1)))
40	21, 38, 39	syl2anc 584	. . . . 5 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑘 ≤ 𝑁 ↔ 𝑘 < (𝑁 + 1)))
41	37, 40	mpbird 256	. . . 4 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ≤ 𝑁)
42	41	expr 457	. . 3 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
43	42	ralrimiva 3103	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
44		simpr 485	. . . . . . . 8 ⊢ ((∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))
45		eluznn0 12657	. . . . . . . 8 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑛 ∈ ℕ0)
46	5, 44, 45	syl2an 596	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈ ℕ0)
47		nn0re 12242	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
48	47	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → 𝑁 ∈ ℝ)
49	48	adantr 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑁 ∈ ℝ)
50	34	adantr 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (𝑁 + 1) ∈ ℝ)
51	46	nn0red 12294	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈ ℝ)
52	49	ltp1d 11905	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑁 < (𝑁 + 1))
53		eluzle 12595	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → (𝑁 + 1) ≤ 𝑛)
54	53	ad2antll 726	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (𝑁 + 1) ≤ 𝑛)
55	49, 50, 51, 52, 54	ltletrd 11135	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑁 < 𝑛)
56	49, 51	ltnled 11122	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (𝑁 < 𝑛 ↔ ¬ 𝑛 ≤ 𝑁))
57	55, 56	mpbid 231	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → ¬ 𝑛 ≤ 𝑁)
58		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝐴‘𝑘) = (𝐴‘𝑛))
59	58	neeq1d 3003	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((𝐴‘𝑘) ≠ 0 ↔ (𝐴‘𝑛) ≠ 0))
60		breq1 5077	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (𝑘 ≤ 𝑁 ↔ 𝑛 ≤ 𝑁))
61	59, 60	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ↔ ((𝐴‘𝑛) ≠ 0 → 𝑛 ≤ 𝑁)))
62		simprl 768	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
63	61, 62, 46	rspcdva 3562	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → ((𝐴‘𝑛) ≠ 0 → 𝑛 ≤ 𝑁))
64	63	necon1bd 2961	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (¬ 𝑛 ≤ 𝑁 → (𝐴‘𝑛) = 0))
65	57, 64	mpd 15	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (𝐴‘𝑛) = 0)
66		ffn 6600	. . . . . . . . 9 ⊢ (𝐴:ℕ0⟶ℂ → 𝐴 Fn ℕ0)
67	66	ad2antlr 724	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝐴 Fn ℕ0)
68		fniniseg 6937	. . . . . . . 8 ⊢ (𝐴 Fn ℕ0 → (𝑛 ∈ (◡𝐴 “ {0}) ↔ (𝑛 ∈ ℕ0 ∧ (𝐴‘𝑛) = 0)))
69	67, 68	syl 17	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (𝑛 ∈ (◡𝐴 “ {0}) ↔ (𝑛 ∈ ℕ0 ∧ (𝐴‘𝑛) = 0)))
70	46, 65, 69	mpbir2and 710	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈ (◡𝐴 “ {0}))
71	70	expr 457	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → 𝑛 ∈ (◡𝐴 “ {0})))
72	71	ssrdv 3927	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0}))
73		funimass3 6931	. . . . . 6 ⊢ ((Fun 𝐴 ∧ (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) ⊆ {0} ↔ (ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0})))
74	3, 12, 73	syl2anc 584	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) ⊆ {0} ↔ (ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0})))
75	74	adantr 481	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) ⊆ {0} ↔ (ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0})))
76	72, 75	mpbird 256	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) ⊆ {0})
77	48	ltp1d 11905	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → 𝑁 < (𝑁 + 1))
78	48, 34	ltnled 11122	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁))
79	77, 78	mpbid 231	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ¬ (𝑁 + 1) ≤ 𝑁)
80	79	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ¬ (𝑁 + 1) ≤ 𝑁)
81		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑁 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑁 + 1)))
82	81	neeq1d 3003	. . . . . . . . . 10 ⊢ (𝑘 = (𝑁 + 1) → ((𝐴‘𝑘) ≠ 0 ↔ (𝐴‘(𝑁 + 1)) ≠ 0))
83		breq1 5077	. . . . . . . . . 10 ⊢ (𝑘 = (𝑁 + 1) → (𝑘 ≤ 𝑁 ↔ (𝑁 + 1) ≤ 𝑁))
84	82, 83	imbi12d 345	. . . . . . . . 9 ⊢ (𝑘 = (𝑁 + 1) → (((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ↔ ((𝐴‘(𝑁 + 1)) ≠ 0 → (𝑁 + 1) ≤ 𝑁)))
85	84	rspcva 3559	. . . . . . . 8 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ((𝐴‘(𝑁 + 1)) ≠ 0 → (𝑁 + 1) ≤ 𝑁))
86	5, 85	sylan 580	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ((𝐴‘(𝑁 + 1)) ≠ 0 → (𝑁 + 1) ≤ 𝑁))
87	86	necon1bd 2961	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (¬ (𝑁 + 1) ≤ 𝑁 → (𝐴‘(𝑁 + 1)) = 0))
88	80, 87	mpd 15	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴‘(𝑁 + 1)) = 0)
89		uzid 12597	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℤ → (𝑁 + 1) ∈ (ℤ≥‘(𝑁 + 1)))
90	18, 89	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 + 1) ∈ (ℤ≥‘(𝑁 + 1)))
91		funfvima2 7107	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴) → ((𝑁 + 1) ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘(𝑁 + 1)) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1)))))
92	3, 12, 91	syl2anc 584	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝑁 + 1) ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘(𝑁 + 1)) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1)))))
93	90, 92	mpd 15	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝐴‘(𝑁 + 1)) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1))))
94	93	adantr 481	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴‘(𝑁 + 1)) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1))))
95	88, 94	eqeltrrd 2840	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → 0 ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1))))
96	95	snssd 4742	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → {0} ⊆ (𝐴 “ (ℤ≥‘(𝑁 + 1))))
97	76, 96	eqssd 3938	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})
98	43, 97	impbida 798	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064   ⊆ wss 3887  {csn 4561   class class class wbr 5074  ^◡ccnv 5588  dom cdm 5589   “ cima 5592  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  ℝcr 10870  0cc0 10871  1c1 10872   + caddc 10874   < clt 11009   ≤ cle 11010  ℕ₀cn0 12233  ℤcz 12319  ℤ_≥cuz 12582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583

This theorem is referenced by:  elply2  25357  plyeq0lem  25371  coeeulem  25385  dgrlem  25390  dgrub2  25396  dgrlb  25397  coeeq2  25403  dgrle  25404  coeaddlem  25410  coemullem  25411  coe1termlem  25419  dgreq0  25426  coecj  25439  basellem2  26231