Theorem plyco0 23752

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 792	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝐴‘𝑘) ≠ 0)
2		ffun 5961	. . . . . . . . . . . 12 ⊢ (𝐴:ℕ0⟶ℂ → Fun 𝐴)
3	2	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → Fun 𝐴)
4		peano2nn0 11210	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
5	4	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 + 1) ∈ ℕ0)
6		eluznn0 11633	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑘 ∈ ℕ0)
7	6	ex 449	. . . . . . . . . . . . . 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 3574	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (ℤ≥‘(𝑁 + 1)) ⊆ ℕ0)
10		fdm 5964	. . . . . . . . . . . . 13 ⊢ (𝐴:ℕ0⟶ℂ → dom 𝐴 = ℕ0)
11	10	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → dom 𝐴 = ℕ0)
12	9, 11	sseqtr4d 3605	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴)
13		funfvima2 6397	. . . . . . . . . . 11 ⊢ ((Fun 𝐴 ∧ (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1)))))
14	3, 12, 13	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘𝑘) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1)))))
15	14	ad2antrr 758	. . . . . . . . 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 11277	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
17	16	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → 𝑁 ∈ ℤ)
18	17	peano2zd 11361	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 + 1) ∈ ℤ)
19	18	ad2antrr 758	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑁 + 1) ∈ ℤ)
20		nn0z 11277	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ)
21	20	ad2antrl 760	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ∈ ℤ)
22		eluz 11577	. . . . . . . . . 10 ⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (ℤ≥‘(𝑁 + 1)) ↔ (𝑁 + 1) ≤ 𝑘))
23	19, 21, 22	syl2anc 691	. . . . . . . . 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 788	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})
25	24	eleq2d 2673	. . . . . . . . . 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 6113	. . . . . . . . . . 11 ⊢ (𝐴‘𝑘) ∈ V
27	26	elsn 4140	. . . . . . . . . 10 ⊢ ((𝐴‘𝑘) ∈ {0} ↔ (𝐴‘𝑘) = 0)
28	25, 27	syl6bb 275	. . . . . . . . 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 281	. . . . . . . 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 2795	. . . . . . 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 11178	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ)
33	32	ad2antrl 760	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ∈ ℝ)
34	18	zred 11358	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 + 1) ∈ ℝ)
35	34	ad2antrr 758	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → (𝑁 + 1) ∈ ℝ)
36	33, 35	ltnled 10063	. . . . . 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 246	. . . . 5 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 < (𝑁 + 1))
38	17	ad2antrr 758	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑁 ∈ ℤ)
39		zleltp1 11305	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑘 ≤ 𝑁 ↔ 𝑘 < (𝑁 + 1)))
40	21, 38, 39	syl2anc 691	. . . . 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 246	. . . 4 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ≤ 𝑁)
42	41	expr 641	. . 3 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
43	42	ralrimiva 2949	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
44		simpr 476	. . . . . . . 8 ⊢ ((∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))
45		eluznn0 11633	. . . . . . . 8 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1))) → 𝑛 ∈ ℕ0)
46	5, 44, 45	syl2an 493	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈ ℕ0)
47		nn0re 11178	. . . . . . . . . . . 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 11229	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈ ℝ)
52	49	ltp1d 10833	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑁 < (𝑁 + 1))
53		eluzle 11576	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → (𝑁 + 1) ≤ 𝑛)
54	53	ad2antll 761	. . . . . . . . . 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 10076	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑁 < 𝑛)
56	49, 51	ltnled 10063	. . . . . . . . 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 221	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → ¬ 𝑛 ≤ 𝑁)
58		simprl 790	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))
59		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝐴‘𝑘) = (𝐴‘𝑛))
60	59	neeq1d 2841	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → ((𝐴‘𝑘) ≠ 0 ↔ (𝐴‘𝑛) ≠ 0))
61		breq1 4586	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝑘 ≤ 𝑁 ↔ 𝑛 ≤ 𝑁))
62	60, 61	imbi12d 333	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → (((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ↔ ((𝐴‘𝑛) ≠ 0 → 𝑛 ≤ 𝑁)))
63	62	rspcva 3280	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ((𝐴‘𝑛) ≠ 0 → 𝑛 ≤ 𝑁))
64	46, 58, 63	syl2anc 691	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → ((𝐴‘𝑛) ≠ 0 → 𝑛 ≤ 𝑁))
65	64	necon1bd 2800	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (¬ 𝑛 ≤ 𝑁 → (𝐴‘𝑛) = 0))
66	57, 65	mpd 15	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → (𝐴‘𝑛) = 0)
67		ffn 5958	. . . . . . . . 9 ⊢ (𝐴:ℕ0⟶ℂ → 𝐴 Fn ℕ0)
68	67	ad2antlr 759	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝐴 Fn ℕ0)
69		fniniseg 6246	. . . . . . . 8 ⊢ (𝐴 Fn ℕ0 → (𝑛 ∈ (◡𝐴 “ {0}) ↔ (𝑛 ∈ ℕ0 ∧ (𝐴‘𝑛) = 0)))
70	68, 69	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)))
71	46, 66, 70	mpbir2and 959	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ (∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ≥‘(𝑁 + 1)))) → 𝑛 ∈ (◡𝐴 “ {0}))
72	71	expr 641	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝑛 ∈ (ℤ≥‘(𝑁 + 1)) → 𝑛 ∈ (◡𝐴 “ {0})))
73	72	ssrdv 3574	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0}))
74		funimass3 6241	. . . . . 6 ⊢ ((Fun 𝐴 ∧ (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) ⊆ {0} ↔ (ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0})))
75	3, 12, 74	syl2anc 691	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) ⊆ {0} ↔ (ℤ≥‘(𝑁 + 1)) ⊆ (◡𝐴 “ {0})))
76	75	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})))
77	73, 76	mpbird 246	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) ⊆ {0})
78	48	ltp1d 10833	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → 𝑁 < (𝑁 + 1))
79	48, 34	ltnled 10063	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁))
80	78, 79	mpbid 221	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ¬ (𝑁 + 1) ≤ 𝑁)
81	80	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ¬ (𝑁 + 1) ≤ 𝑁)
82		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑁 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑁 + 1)))
83	82	neeq1d 2841	. . . . . . . . . 10 ⊢ (𝑘 = (𝑁 + 1) → ((𝐴‘𝑘) ≠ 0 ↔ (𝐴‘(𝑁 + 1)) ≠ 0))
84		breq1 4586	. . . . . . . . . 10 ⊢ (𝑘 = (𝑁 + 1) → (𝑘 ≤ 𝑁 ↔ (𝑁 + 1) ≤ 𝑁))
85	83, 84	imbi12d 333	. . . . . . . . 9 ⊢ (𝑘 = (𝑁 + 1) → (((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ↔ ((𝐴‘(𝑁 + 1)) ≠ 0 → (𝑁 + 1) ≤ 𝑁)))
86	85	rspcva 3280	. . . . . . . 8 ⊢ (((𝑁 + 1) ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ((𝐴‘(𝑁 + 1)) ≠ 0 → (𝑁 + 1) ≤ 𝑁))
87	5, 86	sylan 487	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → ((𝐴‘(𝑁 + 1)) ≠ 0 → (𝑁 + 1) ≤ 𝑁))
88	87	necon1bd 2800	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (¬ (𝑁 + 1) ≤ 𝑁 → (𝐴‘(𝑁 + 1)) = 0))
89	81, 88	mpd 15	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴‘(𝑁 + 1)) = 0)
90		uzid 11578	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℤ → (𝑁 + 1) ∈ (ℤ≥‘(𝑁 + 1)))
91	18, 90	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝑁 + 1) ∈ (ℤ≥‘(𝑁 + 1)))
92		funfvima2 6397	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ (ℤ≥‘(𝑁 + 1)) ⊆ dom 𝐴) → ((𝑁 + 1) ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘(𝑁 + 1)) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1)))))
93	3, 12, 92	syl2anc 691	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝑁 + 1) ∈ (ℤ≥‘(𝑁 + 1)) → (𝐴‘(𝑁 + 1)) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1)))))
94	91, 93	mpd 15	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → (𝐴‘(𝑁 + 1)) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1))))
95	94	adantr 480	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴‘(𝑁 + 1)) ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1))))
96	89, 95	eqeltrrd 2689	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → 0 ∈ (𝐴 “ (ℤ≥‘(𝑁 + 1))))
97	96	snssd 4281	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → {0} ⊆ (𝐴 “ (ℤ≥‘(𝑁 + 1))))
98	77, 97	eqssd 3585	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) ∧ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})
99	43, 98	impbida 873	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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ⊆ wss 3540  {csn 4125   class class class wbr 4583  ^◡ccnv 5037  dom cdm 5038   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954  ℕ₀cn0 11169  ℤcz 11254  ℤ_≥cuz 11563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564

This theorem is referenced by:  elply2  23756  plyeq0lem  23770  coeeulem  23784  dgrlem  23789  dgrub2  23795  dgrlb  23796  coeeq2  23802  dgrle  23803  coeaddlem  23809  coemullem  23810  coe1termlem  23818  dgreq0  23825  coecj  23838  basellem2  24608