Theorem plyco0 26157

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 6671	. . . . . . . . . . . 12 ⊢ (𝐴:ℕ0⟶ℂ → Fun 𝐴)
3	2	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → Fun 𝐴)
4		peano2nn0 12477	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
5	4	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 + 1) ∈ ℕ0)
6		eluznn0 12867	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ ℕ0)
7	6	ex 412	. . . . . . . . . . . . . 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 6677	. . . . . . . . . . . . 13 ⊢ (𝐴:ℕ0⟶ℂ → dom 𝐴 = ℕ0)
11	10	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → dom 𝐴 = ℕ0)
12	9, 11	sseqtrrd 3959	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴)
13		funfvima2 7186	. . . . . . . . . . 11 ⊢ ((Fun 𝐴 ∧ (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1)))))
14	3, 12, 13	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1)))))
15	14	ad2antrr 727	. . . . . . . . 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 12548	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
17	16	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → 𝑁 ∈ ℤ)
18	17	peano2zd 12636	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 + 1) ∈ ℤ)
19	18	ad2antrr 727	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑁 + 1) ∈ ℤ)
20		nn0z 12548	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ)
21	20	ad2antrl 729	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ∈ ℤ)
22		eluz 12802	. . . . . . . . . 10 ⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) ↔ (𝑁 + 1) ≤ 𝑘))
23	19, 21, 22	syl2anc 585	. . . . . . . . 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 2822	. . . . . . . . . 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 6853	. . . . . . . . . . 11 ⊢ (𝐴‘𝑘) ∈ V
27	26	elsn 4582	. . . . . . . . . 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 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 12446	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
33	32	ad2antrl 729	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ∈ ℝ)
34	18	zred 12633	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 + 1) ∈ ℝ)
35	34	ad2antrr 727	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑁 + 1) ∈ ℝ)
36	33, 35	ltnled 11293	. . . . . 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 257	. . . . 5 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 < (𝑁 + 1))
38	17	ad2antrr 727	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑁 ∈ ℤ)
39		zleltp1 12578	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑘 ≤ 𝑁 ↔ 𝑘 < (𝑁 + 1)))
40	21, 38, 39	syl2anc 585	. . . . 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 257	. . . 4 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ≤ 𝑁)
42	41	expr 456	. . 3 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
43	42	ralrimiva 3129	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
44		simpr 484	. . . . . . . 8 ⊢ ((∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))
45		eluznn0 12867	. . . . . . . 8 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑛 ∈ ℕ0)
46	5, 44, 45	syl2an 597	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈ ℕ0)
47		nn0re 12446	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
48	47	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → 𝑁 ∈ ℝ)
49	48	adantr 480	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑁 ∈ ℝ)
50	34	adantr 480	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (𝑁 + 1) ∈ ℝ)
51	46	nn0red 12499	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈ ℝ)
52	49	ltp1d 12086	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑁 < (𝑁 + 1))
53		eluzle 12801	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → (𝑁 + 1) ≤ 𝑛)
54	53	ad2antll 730	. . . . . . . . . 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 11306	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑁 < 𝑛)
56	49, 51	ltnled 11293	. . . . . . . . 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 232	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → ¬ 𝑛 ≤ 𝑁)
58		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝐴‘𝑘) = (𝐴‘𝑛))
59	58	neeq1d 2991	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((𝐴‘𝑘) ≠ 0 ↔ (𝐴‘𝑛) ≠ 0))
60		breq1 5088	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (𝑘 ≤ 𝑁 ↔ 𝑛 ≤ 𝑁))
61	59, 60	imbi12d 344	. . . . . . . . . 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 3565	. . . . . . . . 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 6668	. . . . . . . . 9 ⊢ (𝐴:ℕ0⟶ℂ → 𝐴 Fn ℕ0)
67	66	ad2antlr 728	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝐴 Fn ℕ0)
68		fniniseg 7012	. . . . . . . 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 714	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈ (◡𝐴 “ {0}))
71	70	expr 456	. . . . 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 7006	. . . . . 6 ⊢ ((Fun 𝐴 ∧ (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) ⊆ {0} ↔ (ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0})))
74	3, 12, 73	syl2anc 585	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) ⊆ {0} ↔ (ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0})))
75	74	adantr 480	. . . 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 257	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) ⊆ {0})
77	48	ltp1d 12086	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → 𝑁 < (𝑁 + 1))
78	48, 34	ltnled 11293	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁))
79	77, 78	mpbid 232	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ¬ (𝑁 + 1) ≤ 𝑁)
80	79	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ¬ (𝑁 + 1) ≤ 𝑁)
81		fveq2 6840	. . . . . . . . . . 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 5088	. . . . . . . . . 10 ⊢ (𝑘 = (𝑁 + 1) → (𝑘 ≤ 𝑁 ↔ (𝑁 + 1) ≤ 𝑁))
84	82, 83	imbi12d 344	. . . . . . . . 9 ⊢ (𝑘 = (𝑁 + 1) → (((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ↔ ((𝐴‘(𝑁 + 1)) ≠ 0 → (𝑁 + 1) ≤ 𝑁)))
85	84	rspcva 3562	. . . . . . . 8 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ((𝐴‘(𝑁 + 1)) ≠ 0 → (𝑁 + 1) ≤ 𝑁))
86	5, 85	sylan 581	. . . . . . 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 12803	. . . . . . . 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 7186	. . . . . . . 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 585	. . . . . . 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 480	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴‘(𝑁 + 1)) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1))))
95	88, 94	eqeltrrd 2837	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → 0 ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1))))
96	95	snssd 4730	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → {0} ⊆ (𝐴 “ (ℤ≥‘(𝑁 + 1))))
97	76, 96	eqssd 3939	. 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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051   ⊆ wss 3889  {csn 4567   class class class wbr 5085  ^◡ccnv 5630  dom cdm 5631   “ cima 5634  Fun wfun 6492   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  ℂcc 11036  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   < clt 11179   ≤ cle 11180  ℕ₀cn0 12437  ℤcz 12524  ℤ_≥cuz 12788

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 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789

This theorem is referenced by:  elply2  26161  plyeq0lem  26175  coeeulem  26189  dgrlem  26194  dgrub2  26200  dgrlb  26201  coeeq2  26207  dgrle  26208  coeaddlem  26214  coemullem  26215  coe1termlem  26223  dgreq0  26230  coecj  26243  coecjOLD  26245  basellem2  27045