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Theorem basellem2 24808

Description: Lemma for basel 24816. Show that 𝑃 is a polynomial of degree 𝑀, and compute its coefficient function. (Contributed by Mario Carneiro, 30-Jul-2014.)

**Hypotheses**
Ref	Expression
basel.n	⊢ 𝑁 = ((2 · 𝑀) + 1)
basel.p	⊢ 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗)))

**Assertion**
Ref	Expression
basellem2	⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (Poly‘ℂ) ∧ (deg‘𝑃) = 𝑀 ∧ (coeff‘𝑃) = (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))))

Distinct variable groups: 𝑡,𝑗,𝑛,𝑀 𝑗,𝑁,𝑛,𝑡 𝑃,𝑛 Allowed substitution hints: 𝑃(𝑡,𝑗)

**Proof of Theorem basellem2**
Step	Hyp	Ref	Expression
1		basel.p	. . 3 ⊢ 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗)))
2		ssid 3624	. . . . 5 ⊢ ℂ ⊆ ℂ
3	2	a1i 11	. . . 4 ⊢ (𝑀 ∈ ℕ → ℂ ⊆ ℂ)
4		nnnn0 11299	. . . 4 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
5		elfznn0 12433	. . . . . . 7 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ₀)
6		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (2 · 𝑛) = (2 · 𝑗))
7	6	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑁C(2 · 𝑛)) = (𝑁C(2 · 𝑗)))
8		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (𝑀 − 𝑛) = (𝑀 − 𝑗))
9	8	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (-1↑(𝑀 − 𝑛)) = (-1↑(𝑀 − 𝑗)))
10	7, 9	oveq12d 6668	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))))
11		eqid 2622	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))) = (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))
12		ovex 6678	. . . . . . . 8 ⊢ ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) ∈ V
13	10, 11, 12	fvmpt 6282	. . . . . . 7 ⊢ (𝑗 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))))
14	5, 13	syl 17	. . . . . 6 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))))
15	14	adantl 482	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))))
16		basel.n	. . . . . . . . . . . 12 ⊢ 𝑁 = ((2 · 𝑀) + 1)
17		2nn 11185	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ
18		nnmulcl 11043	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (2 · 𝑀) ∈ ℕ)
19	17, 18	mpan 706	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ ℕ)
20	19	peano2nnd 11037	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ → ((2 · 𝑀) + 1) ∈ ℕ)
21	16, 20	syl5eqel 2705	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑁 ∈ ℕ)
22	21	nnnn0d 11351	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑁 ∈ ℕ₀)
23		2z 11409	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
24		nn0z 11400	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
25		zmulcl 11426	. . . . . . . . . . 11 ⊢ ((2 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2 · 𝑛) ∈ ℤ)
26	23, 24, 25	sylancr 695	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → (2 · 𝑛) ∈ ℤ)
27		bccl 13109	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ (2 · 𝑛) ∈ ℤ) → (𝑁C(2 · 𝑛)) ∈ ℕ₀)
28	22, 26, 27	syl2an 494	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (𝑁C(2 · 𝑛)) ∈ ℕ₀)
29	28	nn0cnd 11353	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (𝑁C(2 · 𝑛)) ∈ ℂ)
30		nnz 11399	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
31		zsubcl 11419	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑀 − 𝑛) ∈ ℤ)
32	30, 24, 31	syl2an 494	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (𝑀 − 𝑛) ∈ ℤ)
33		neg1cn 11124	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
34		neg1ne0 11126	. . . . . . . . . 10 ⊢ -1 ≠ 0
35		expclz 12885	. . . . . . . . . 10 ⊢ ((-1 ∈ ℂ ∧ -1 ≠ 0 ∧ (𝑀 − 𝑛) ∈ ℤ) → (-1↑(𝑀 − 𝑛)) ∈ ℂ)
36	33, 34, 35	mp3an12 1414	. . . . . . . . 9 ⊢ ((𝑀 − 𝑛) ∈ ℤ → (-1↑(𝑀 − 𝑛)) ∈ ℂ)
37	32, 36	syl 17	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (-1↑(𝑀 − 𝑛)) ∈ ℂ)
38	29, 37	mulcld 10060	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))) ∈ ℂ)
39	38, 11	fmptd 6385	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))):ℕ₀⟶ℂ)
40		ffvelrn 6357	. . . . . 6 ⊢ (((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))):ℕ₀⟶ℂ ∧ 𝑗 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ∈ ℂ)
41	39, 5, 40	syl2an 494	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ∈ ℂ)
42	15, 41	eqeltrrd 2702	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ (0...𝑀)) → ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) ∈ ℂ)
43	3, 4, 42	elplyd 23958	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗))) ∈ (Poly‘ℂ))
44	1, 43	syl5eqel 2705	. 2 ⊢ (𝑀 ∈ ℕ → 𝑃 ∈ (Poly‘ℂ))
45		nnre 11027	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
46		nn0re 11301	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ₀ → 𝑗 ∈ ℝ)
47		ltnle 10117	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑀 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑀))
48	45, 46, 47	syl2an 494	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (𝑀 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑀))
49	13	ad2antlr 763	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))))
50	22	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 𝑁 ∈ ℕ₀)
51		nn0z 11400	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ₀ → 𝑗 ∈ ℤ)
52	51	ad2antlr 763	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 𝑗 ∈ ℤ)
53		zmulcl 11426	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (2 · 𝑗) ∈ ℤ)
54	23, 52, 53	sylancr 695	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (2 · 𝑗) ∈ ℤ)
55		ax-1cn 9994	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
56	55	2timesi 11147	. . . . . . . . . . . . . . . 16 ⊢ (2 · 1) = (1 + 1)
57	56	oveq2i 6661	. . . . . . . . . . . . . . 15 ⊢ ((2 · 𝑀) + (2 · 1)) = ((2 · 𝑀) + (1 + 1))
58		2cnd 11093	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 2 ∈ ℂ)
59		nncn 11028	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
60	59	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 𝑀 ∈ ℂ)
61	55	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 1 ∈ ℂ)
62	58, 60, 61	adddid 10064	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (2 · (𝑀 + 1)) = ((2 · 𝑀) + (2 · 1)))
63	16	oveq1i 6660	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 + 1) = (((2 · 𝑀) + 1) + 1)
64	19	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (2 · 𝑀) ∈ ℕ)
65	64	nncnd 11036	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (2 · 𝑀) ∈ ℂ)
66	65, 61, 61	addassd 10062	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (((2 · 𝑀) + 1) + 1) = ((2 · 𝑀) + (1 + 1)))
67	63, 66	syl5eq 2668	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (𝑁 + 1) = ((2 · 𝑀) + (1 + 1)))
68	57, 62, 67	3eqtr4a 2682	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (2 · (𝑀 + 1)) = (𝑁 + 1))
69		zltp1le 11427	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑀 < 𝑗 ↔ (𝑀 + 1) ≤ 𝑗))
70	30, 51, 69	syl2an 494	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (𝑀 < 𝑗 ↔ (𝑀 + 1) ≤ 𝑗))
71	70	biimpa 501	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (𝑀 + 1) ≤ 𝑗)
72	45	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 𝑀 ∈ ℝ)
73		peano2re 10209	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
74	72, 73	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (𝑀 + 1) ∈ ℝ)
75	46	ad2antlr 763	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 𝑗 ∈ ℝ)
76		2re 11090	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
77		2pos 11112	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 2
78	76, 77	pm3.2i 471	. . . . . . . . . . . . . . . . 17 ⊢ (2 ∈ ℝ ∧ 0 < 2)
79	78	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (2 ∈ ℝ ∧ 0 < 2))
80		lemul2 10876	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 + 1) ∈ ℝ ∧ 𝑗 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((𝑀 + 1) ≤ 𝑗 ↔ (2 · (𝑀 + 1)) ≤ (2 · 𝑗)))
81	74, 75, 79, 80	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → ((𝑀 + 1) ≤ 𝑗 ↔ (2 · (𝑀 + 1)) ≤ (2 · 𝑗)))
82	71, 81	mpbid 222	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (2 · (𝑀 + 1)) ≤ (2 · 𝑗))
83	68, 82	eqbrtrrd 4677	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (𝑁 + 1) ≤ (2 · 𝑗))
84	21	nnzd 11481	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 𝑁 ∈ ℤ)
85	84	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 𝑁 ∈ ℤ)
86		zltp1le 11427	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℤ ∧ (2 · 𝑗) ∈ ℤ) → (𝑁 < (2 · 𝑗) ↔ (𝑁 + 1) ≤ (2 · 𝑗)))
87	85, 54, 86	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (𝑁 < (2 · 𝑗) ↔ (𝑁 + 1) ≤ (2 · 𝑗)))
88	83, 87	mpbird 247	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 𝑁 < (2 · 𝑗))
89	88	olcd 408	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → ((2 · 𝑗) < 0 ∨ 𝑁 < (2 · 𝑗)))
90		bcval4 13094	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ (2 · 𝑗) ∈ ℤ ∧ ((2 · 𝑗) < 0 ∨ 𝑁 < (2 · 𝑗))) → (𝑁C(2 · 𝑗)) = 0)
91	50, 54, 89, 90	syl3anc 1326	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (𝑁C(2 · 𝑗)) = 0)
92	91	oveq1d 6665	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) = (0 · (-1↑(𝑀 − 𝑗))))
93		zsubcl 11419	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑀 − 𝑗) ∈ ℤ)
94	30, 51, 93	syl2an 494	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (𝑀 − 𝑗) ∈ ℤ)
95		expclz 12885	. . . . . . . . . . . . 13 ⊢ ((-1 ∈ ℂ ∧ -1 ≠ 0 ∧ (𝑀 − 𝑗) ∈ ℤ) → (-1↑(𝑀 − 𝑗)) ∈ ℂ)
96	33, 34, 95	mp3an12 1414	. . . . . . . . . . . 12 ⊢ ((𝑀 − 𝑗) ∈ ℤ → (-1↑(𝑀 − 𝑗)) ∈ ℂ)
97	94, 96	syl 17	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (-1↑(𝑀 − 𝑗)) ∈ ℂ)
98	97	adantr 481	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (-1↑(𝑀 − 𝑗)) ∈ ℂ)
99	98	mul02d 10234	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (0 · (-1↑(𝑀 − 𝑗))) = 0)
100	49, 92, 99	3eqtrd 2660	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = 0)
101	100	ex 450	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (𝑀 < 𝑗 → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = 0))
102	48, 101	sylbird 250	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (¬ 𝑗 ≤ 𝑀 → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = 0))
103	102	necon1ad 2811	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ≠ 0 → 𝑗 ≤ 𝑀))
104	103	ralrimiva 2966	. . . 4 ⊢ (𝑀 ∈ ℕ → ∀𝑗 ∈ ℕ₀ (((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ≠ 0 → 𝑗 ≤ 𝑀))
105		plyco0 23948	. . . . 5 ⊢ ((𝑀 ∈ ℕ₀ ∧ (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))):ℕ₀⟶ℂ) → (((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))) “ (ℤ_≥‘(𝑀 + 1))) = {0} ↔ ∀𝑗 ∈ ℕ₀ (((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ≠ 0 → 𝑗 ≤ 𝑀)))
106	4, 39, 105	syl2anc 693	. . . 4 ⊢ (𝑀 ∈ ℕ → (((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))) “ (ℤ_≥‘(𝑀 + 1))) = {0} ↔ ∀𝑗 ∈ ℕ₀ (((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ≠ 0 → 𝑗 ≤ 𝑀)))
107	104, 106	mpbird 247	. . 3 ⊢ (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))) “ (ℤ_≥‘(𝑀 + 1))) = {0})
108	14	oveq1d 6665	. . . . . . 7 ⊢ (𝑗 ∈ (0...𝑀) → (((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) · (𝑡↑𝑗)) = (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗)))
109	108	sumeq2i 14429	. . . . . 6 ⊢ Σ𝑗 ∈ (0...𝑀)(((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) · (𝑡↑𝑗)) = Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗))
110	109	mpteq2i 4741	. . . . 5 ⊢ (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) · (𝑡↑𝑗))) = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗)))
111	1, 110	eqtr4i 2647	. . . 4 ⊢ 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) · (𝑡↑𝑗)))
112	111	a1i 11	. . 3 ⊢ (𝑀 ∈ ℕ → 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) · (𝑡↑𝑗))))
113		oveq2 6658	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (2 · 𝑛) = (2 · 𝑀))
114	113	oveq2d 6666	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → (𝑁C(2 · 𝑛)) = (𝑁C(2 · 𝑀)))
115		oveq2 6658	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (𝑀 − 𝑛) = (𝑀 − 𝑀))
116	115	oveq2d 6666	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → (-1↑(𝑀 − 𝑛)) = (-1↑(𝑀 − 𝑀)))
117	114, 116	oveq12d 6668	. . . . . . 7 ⊢ (𝑛 = 𝑀 → ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))) = ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀))))
118		ovex 6678	. . . . . . 7 ⊢ ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀))) ∈ V
119	117, 11, 118	fvmpt 6282	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑀) = ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀))))
120	4, 119	syl 17	. . . . 5 ⊢ (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑀) = ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀))))
121	59	subidd 10380	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑀 − 𝑀) = 0)
122	121	oveq2d 6666	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (-1↑(𝑀 − 𝑀)) = (-1↑0))
123		exp0 12864	. . . . . . . 8 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
124	33, 123	ax-mp 5	. . . . . . 7 ⊢ (-1↑0) = 1
125	122, 124	syl6eq 2672	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (-1↑(𝑀 − 𝑀)) = 1)
126	125	oveq2d 6666	. . . . 5 ⊢ (𝑀 ∈ ℕ → ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀))) = ((𝑁C(2 · 𝑀)) · 1))
127	19	nnred 11035	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ ℝ)
128	127	lep1d 10955	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ≤ ((2 · 𝑀) + 1))
129	128, 16	syl6breqr 4695	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ≤ 𝑁)
130	19	nnnn0d 11351	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ ℕ₀)
131		nn0uz 11722	. . . . . . . . . . 11 ⊢ ℕ₀ = (ℤ_≥‘0)
132	130, 131	syl6eleq 2711	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ (ℤ_≥‘0))
133		elfz5 12334	. . . . . . . . . 10 ⊢ (((2 · 𝑀) ∈ (ℤ_≥‘0) ∧ 𝑁 ∈ ℤ) → ((2 · 𝑀) ∈ (0...𝑁) ↔ (2 · 𝑀) ≤ 𝑁))
134	132, 84, 133	syl2anc 693	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → ((2 · 𝑀) ∈ (0...𝑁) ↔ (2 · 𝑀) ≤ 𝑁))
135	129, 134	mpbird 247	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ (0...𝑁))
136		bccl2 13110	. . . . . . . 8 ⊢ ((2 · 𝑀) ∈ (0...𝑁) → (𝑁C(2 · 𝑀)) ∈ ℕ)
137	135, 136	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (𝑁C(2 · 𝑀)) ∈ ℕ)
138	137	nncnd 11036	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑁C(2 · 𝑀)) ∈ ℂ)
139	138	mulid1d 10057	. . . . 5 ⊢ (𝑀 ∈ ℕ → ((𝑁C(2 · 𝑀)) · 1) = (𝑁C(2 · 𝑀)))
140	120, 126, 139	3eqtrd 2660	. . . 4 ⊢ (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑀) = (𝑁C(2 · 𝑀)))
141	137	nnne0d 11065	. . . 4 ⊢ (𝑀 ∈ ℕ → (𝑁C(2 · 𝑀)) ≠ 0)
142	140, 141	eqnetrd 2861	. . 3 ⊢ (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑀) ≠ 0)
143	44, 4, 39, 107, 112, 142	dgreq 24000	. 2 ⊢ (𝑀 ∈ ℕ → (deg‘𝑃) = 𝑀)
144	44, 4, 39, 107, 112	coeeq 23983	. 2 ⊢ (𝑀 ∈ ℕ → (coeff‘𝑃) = (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))))
145	44, 143, 144	3jca 1242	1 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (Poly‘ℂ) ∧ (deg‘𝑃) = 𝑀 ∧ (coeff‘𝑃) = (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ⊆ wss 3574 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 -cneg 10267 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 ↑cexp 12860 Ccbc 13089 Σcsu 14416 Polycply 23940 coeffccoe 23942 degcdgr 23943

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-0p 23437 df-ply 23944 df-coe 23946 df-dgr 23947

This theorem is referenced by: basellem4 24810 basellem5 24811

W3C validator