Theorem plyco0 25040

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 773	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝐴‘𝑘) ≠ 0)
2		ffun 6526	. . . . . . . . . . . 12 ⊢ (𝐴:ℕ0⟶ℂ → Fun 𝐴)
3	2	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → Fun 𝐴)
4		peano2nn0 12095	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
5	4	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 + 1) ∈ ℕ0)
6		eluznn0 12478	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ ℕ0)
7	6	ex 416	. . . . . . . . . . . . . 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 3893	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (ℤ≥‘(𝑁 + 1)) ⊆ ℕ0)
10		fdm 6532	. . . . . . . . . . . . 13 ⊢ (𝐴:ℕ0⟶ℂ → dom 𝐴 = ℕ0)
11	10	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → dom 𝐴 = ℕ0)
12	9, 11	sseqtrrd 3928	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴)
13		funfvima2 7025	. . . . . . . . . . 11 ⊢ ((Fun 𝐴 ∧ (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1)))))
14	3, 12, 13	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1)))))
15	14	ad2antrr 726	. . . . . . . . 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 12165	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
17	16	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → 𝑁 ∈ ℤ)
18	17	peano2zd 12250	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 + 1) ∈ ℤ)
19	18	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑁 + 1) ∈ ℤ)
20		nn0z 12165	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ)
21	20	ad2antrl 728	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ∈ ℤ)
22		eluz 12417	. . . . . . . . . 10 ⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) ↔ (𝑁 + 1) ≤ 𝑘))
23	19, 21, 22	syl2anc 587	. . . . . . . . 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 769	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})
25	24	eleq2d 2816	. . . . . . . . . 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 6708	. . . . . . . . . . 11 ⊢ (𝐴‘𝑘) ∈ V
27	26	elsn 4542	. . . . . . . . . 10 ⊢ ((𝐴‘𝑘) ∈ {0} ↔ (𝐴‘𝑘) = 0)
28	25, 27	bitrdi 290	. . . . . . . . 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 296	. . . . . . . 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 2945	. . . . . . 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 12064	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
33	32	ad2antrl 728	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ∈ ℝ)
34	18	zred 12247	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 + 1) ∈ ℝ)
35	34	ad2antrr 726	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑁 + 1) ∈ ℝ)
36	33, 35	ltnled 10944	. . . . . 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 260	. . . . 5 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 < (𝑁 + 1))
38	17	ad2antrr 726	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑁 ∈ ℤ)
39		zleltp1 12193	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑘 ≤ 𝑁 ↔ 𝑘 < (𝑁 + 1)))
40	21, 38, 39	syl2anc 587	. . . . 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 260	. . . 4 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ≤ 𝑁)
42	41	expr 460	. . 3 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
43	42	ralrimiva 3095	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
44		simpr 488	. . . . . . . 8 ⊢ ((∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))
45		eluznn0 12478	. . . . . . . 8 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑛 ∈ ℕ0)
46	5, 44, 45	syl2an 599	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈ ℕ0)
47		nn0re 12064	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
48	47	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → 𝑁 ∈ ℝ)
49	48	adantr 484	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑁 ∈ ℝ)
50	34	adantr 484	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (𝑁 + 1) ∈ ℝ)
51	46	nn0red 12116	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈ ℝ)
52	49	ltp1d 11727	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑁 < (𝑁 + 1))
53		eluzle 12416	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → (𝑁 + 1) ≤ 𝑛)
54	53	ad2antll 729	. . . . . . . . . 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 10957	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑁 < 𝑛)
56	49, 51	ltnled 10944	. . . . . . . . 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 235	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → ¬ 𝑛 ≤ 𝑁)
58		fveq2 6695	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝐴‘𝑘) = (𝐴‘𝑛))
59	58	neeq1d 2991	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((𝐴‘𝑘) ≠ 0 ↔ (𝐴‘𝑛) ≠ 0))
60		breq1 5042	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (𝑘 ≤ 𝑁 ↔ 𝑛 ≤ 𝑁))
61	59, 60	imbi12d 348	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ↔ ((𝐴‘𝑛) ≠ 0 → 𝑛 ≤ 𝑁)))
62		simprl 771	. . . . . . . . . 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 3529	. . . . . . . . 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 2950	. . . . . . . 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 6523	. . . . . . . . 9 ⊢ (𝐴:ℕ0⟶ℂ → 𝐴 Fn ℕ0)
67	66	ad2antlr 727	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝐴 Fn ℕ0)
68		fniniseg 6858	. . . . . . . 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 713	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈ (◡𝐴 “ {0}))
71	70	expr 460	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → 𝑛 ∈ (◡𝐴 “ {0})))
72	71	ssrdv 3893	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0}))
73		funimass3 6852	. . . . . 6 ⊢ ((Fun 𝐴 ∧ (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) ⊆ {0} ↔ (ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0})))
74	3, 12, 73	syl2anc 587	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) ⊆ {0} ↔ (ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0})))
75	74	adantr 484	. . . 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 260	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) ⊆ {0})
77	48	ltp1d 11727	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → 𝑁 < (𝑁 + 1))
78	48, 34	ltnled 10944	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁))
79	77, 78	mpbid 235	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ¬ (𝑁 + 1) ≤ 𝑁)
80	79	adantr 484	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ¬ (𝑁 + 1) ≤ 𝑁)
81		fveq2 6695	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑁 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑁 + 1)))
82	81	neeq1d 2991	. . . . . . . . . 10 ⊢ (𝑘 = (𝑁 + 1) → ((𝐴‘𝑘) ≠ 0 ↔ (𝐴‘(𝑁 + 1)) ≠ 0))
83		breq1 5042	. . . . . . . . . 10 ⊢ (𝑘 = (𝑁 + 1) → (𝑘 ≤ 𝑁 ↔ (𝑁 + 1) ≤ 𝑁))
84	82, 83	imbi12d 348	. . . . . . . . 9 ⊢ (𝑘 = (𝑁 + 1) → (((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ↔ ((𝐴‘(𝑁 + 1)) ≠ 0 → (𝑁 + 1) ≤ 𝑁)))
85	84	rspcva 3525	. . . . . . . 8 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ((𝐴‘(𝑁 + 1)) ≠ 0 → (𝑁 + 1) ≤ 𝑁))
86	5, 85	sylan 583	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ((𝐴‘(𝑁 + 1)) ≠ 0 → (𝑁 + 1) ≤ 𝑁))
87	86	necon1bd 2950	. . . . . 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 12418	. . . . . . . 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 7025	. . . . . . . 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 587	. . . . . . 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 484	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴‘(𝑁 + 1)) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1))))
95	88, 94	eqeltrrd 2832	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → 0 ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1))))
96	95	snssd 4708	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → {0} ⊆ (𝐴 “ (ℤ≥‘(𝑁 + 1))))
97	76, 96	eqssd 3904	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})
98	43, 97	impbida 801	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 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112   ≠ wne 2932  ∀wral 3051   ⊆ wss 3853  {csn 4527   class class class wbr 5039  ^◡ccnv 5535  dom cdm 5536   “ cima 5539  Fun wfun 6352   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7191  ℂcc 10692  ℝcr 10693  0cc0 10694  1c1 10695   + caddc 10697   < clt 10832   ≤ cle 10833  ℕ₀cn0 12055  ℤcz 12141  ℤ_≥cuz 12403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-cnex 10750  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-mulcom 10758  ax-addass 10759  ax-mulass 10760  ax-distr 10761  ax-i2m1 10762  ax-1ne0 10763  ax-1rid 10764  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767  ax-pre-lttri 10768  ax-pre-lttrn 10769  ax-pre-ltadd 10770  ax-pre-mulgt0 10771

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-er 8369  df-en 8605  df-dom 8606  df-sdom 8607  df-pnf 10834  df-mnf 10835  df-xr 10836  df-ltxr 10837  df-le 10838  df-sub 11029  df-neg 11030  df-nn 11796  df-n0 12056  df-z 12142  df-uz 12404

This theorem is referenced by:  elply2  25044  plyeq0lem  25058  coeeulem  25072  dgrlem  25077  dgrub2  25083  dgrlb  25084  coeeq2  25090  dgrle  25091  coeaddlem  25097  coemullem  25098  coe1termlem  25106  dgreq0  25113  coecj  25126  basellem2  25918