plyco0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > plyco0		Structured version Visualization version GIF version

Theorem plyco0 25697

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 771	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ (𝐴 “ (ℤ_≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ₀ ∧ (𝐴‘𝑘) ≠ 0)) → (𝐴‘𝑘) ≠ 0)
2		ffun 6717	. . . . . . . . . . . 12 ⊢ (𝐴:ℕ₀⟶ℂ → Fun 𝐴)
3	2	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) → Fun 𝐴)
4		peano2nn0 12508	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
5	4	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) → (𝑁 + 1) ∈ ℕ₀)
6		eluznn0 12897	. . . . . . . . . . . . . . 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 3987	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) → (ℤ_≥‘(𝑁 + 1)) ⊆ ℕ₀)
10		fdm 6723	. . . . . . . . . . . . 13 ⊢ (𝐴:ℕ₀⟶ℂ → dom 𝐴 = ℕ₀)
11	10	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) → dom 𝐴 = ℕ₀)
12	9, 11	sseqtrrd 4022	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) → (ℤ_≥‘(𝑁 + 1)) ⊆ dom 𝐴)
13		funfvima2 7229	. . . . . . . . . . 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 724	. . . . . . . . 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 12579	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
17	16	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) → 𝑁 ∈ ℤ)
18	17	peano2zd 12665	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) → (𝑁 + 1) ∈ ℤ)
19	18	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ (𝐴 “ (ℤ_≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ₀ ∧ (𝐴‘𝑘) ≠ 0)) → (𝑁 + 1) ∈ ℤ)
20		nn0z 12579	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℤ)
21	20	ad2antrl 726	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ (𝐴 “ (ℤ_≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ₀ ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ∈ ℤ)
22		eluz 12832	. . . . . . . . . 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 767	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ (𝐴 “ (ℤ_≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ₀ ∧ (𝐴‘𝑘) ≠ 0)) → (𝐴 “ (ℤ_≥‘(𝑁 + 1))) = {0})
25	24	eleq2d 2819	. . . . . . . . . 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 6901	. . . . . . . . . . 11 ⊢ (𝐴‘𝑘) ∈ V
27	26	elsn 4642	. . . . . . . . . 10 ⊢ ((𝐴‘𝑘) ∈ {0} ↔ (𝐴‘𝑘) = 0)
28	25, 27	bitrdi 286	. . . . . . . . 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 292	. . . . . . . 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 2953	. . . . . . 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 12477	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
33	32	ad2antrl 726	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ (𝐴 “ (ℤ_≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ₀ ∧ (𝐴‘𝑘) ≠ 0)) → 𝑘 ∈ ℝ)
34	18	zred 12662	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) → (𝑁 + 1) ∈ ℝ)
35	34	ad2antrr 724	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ (𝐴 “ (ℤ_≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ₀ ∧ (𝐴‘𝑘) ≠ 0)) → (𝑁 + 1) ∈ ℝ)
36	33, 35	ltnled 11357	. . . . . 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 724	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ (𝐴 “ (ℤ_≥‘(𝑁 + 1))) = {0}) ∧ (𝑘 ∈ ℕ₀ ∧ (𝐴‘𝑘) ≠ 0)) → 𝑁 ∈ ℤ)
39		zleltp1 12609	. . . . . 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 3146	. 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 12897	. . . . . . . 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 12477	. . . . . . . . . . . 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 12529	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ (∀𝑘 ∈ ℕ₀ ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1)))) → 𝑛 ∈ ℝ)
52	49	ltp1d 12140	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ (∀𝑘 ∈ ℕ₀ ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1)))) → 𝑁 < (𝑁 + 1))
53		eluzle 12831	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘(𝑁 + 1)) → (𝑁 + 1) ≤ 𝑛)
54	53	ad2antll 727	. . . . . . . . . 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 11370	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ (∀𝑘 ∈ ℕ₀ ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1)))) → 𝑁 < 𝑛)
56	49, 51	ltnled 11357	. . . . . . . . 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 6888	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝐴‘𝑘) = (𝐴‘𝑛))
59	58	neeq1d 3000	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → ((𝐴‘𝑘) ≠ 0 ↔ (𝐴‘𝑛) ≠ 0))
60		breq1 5150	. . . . . . . . . . 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 769	. . . . . . . . . 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 3613	. . . . . . . . 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 2958	. . . . . . . 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 6714	. . . . . . . . 9 ⊢ (𝐴:ℕ₀⟶ℂ → 𝐴 Fn ℕ₀)
67	66	ad2antlr 725	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ (∀𝑘 ∈ ℕ₀ ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁) ∧ 𝑛 ∈ (ℤ_≥‘(𝑁 + 1)))) → 𝐴 Fn ℕ₀)
68		fniniseg 7058	. . . . . . . 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 711	. . . . . 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 3987	. . . 4 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ ∀𝑘 ∈ ℕ₀ ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (ℤ_≥‘(𝑁 + 1)) ⊆ (^◡𝐴 “ {0}))
73		funimass3 7052	. . . . . 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 12140	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) → 𝑁 < (𝑁 + 1))
78	48, 34	ltnled 11357	. . . . . . . 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 6888	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑁 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑁 + 1)))
82	81	neeq1d 3000	. . . . . . . . . 10 ⊢ (𝑘 = (𝑁 + 1) → ((𝐴‘𝑘) ≠ 0 ↔ (𝐴‘(𝑁 + 1)) ≠ 0))
83		breq1 5150	. . . . . . . . . 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 3610	. . . . . . . 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 2958	. . . . . 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 12833	. . . . . . . 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 7229	. . . . . . . 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 2834	. . . 4 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ ∀𝑘 ∈ ℕ₀ ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → 0 ∈ (𝐴 “ (ℤ_≥‘(𝑁 + 1))))
96	95	snssd 4811	. . 3 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ ∀𝑘 ∈ ℕ₀ ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → {0} ⊆ (𝐴 “ (ℤ_≥‘(𝑁 + 1))))
97	76, 96	eqssd 3998	. 2 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) ∧ ∀𝑘 ∈ ℕ₀ ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) → (𝐴 “ (ℤ_≥‘(𝑁 + 1))) = {0})
98	43, 97	impbida 799	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴:ℕ₀⟶ℂ) → ((𝐴 “ (ℤ_≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ₀ ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ⊆ wss 3947 {csn 4627 class class class wbr 5147 ^◡ccnv 5674 dom cdm 5675 “ cima 5678 Fun wfun 6534 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 < clt 11244 ≤ cle 11245 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819

This theorem is referenced by: elply2 25701 plyeq0lem 25715 coeeulem 25729 dgrlem 25734 dgrub2 25740 dgrlb 25741 coeeq2 25747 dgrle 25748 coeaddlem 25754 coemullem 25755 coe1termlem 25763 dgreq0 25770 coecj 25783 basellem2 26575

W3C validator