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Theorem basellem2 26999

Description: Lemma for basel 27007. 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		ssidd 3973	. . . 4 ⊢ (𝑀 ∈ ℕ → ℂ ⊆ ℂ)
3		nnnn0 12456	. . . 4 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
4		elfznn0 13588	. . . . . . 7 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ₀)
5		oveq2 7398	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (2 · 𝑛) = (2 · 𝑗))
6	5	oveq2d 7406	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑁C(2 · 𝑛)) = (𝑁C(2 · 𝑗)))
7		oveq2 7398	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (𝑀 − 𝑛) = (𝑀 − 𝑗))
8	7	oveq2d 7406	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (-1↑(𝑀 − 𝑛)) = (-1↑(𝑀 − 𝑗)))
9	6, 8	oveq12d 7408	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))))
10		eqid 2730	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))) = (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))
11		ovex 7423	. . . . . . . 8 ⊢ ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) ∈ V
12	9, 10, 11	fvmpt 6971	. . . . . . 7 ⊢ (𝑗 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))))
13	4, 12	syl 17	. . . . . 6 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))))
14	13	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))))
15		basel.n	. . . . . . . . . . . 12 ⊢ 𝑁 = ((2 · 𝑀) + 1)
16		2nn 12266	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ
17		nnmulcl 12217	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (2 · 𝑀) ∈ ℕ)
18	16, 17	mpan 690	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ ℕ)
19	18	peano2nnd 12210	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ → ((2 · 𝑀) + 1) ∈ ℕ)
20	15, 19	eqeltrid 2833	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑁 ∈ ℕ)
21	20	nnnn0d 12510	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑁 ∈ ℕ₀)
22		2z 12572	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
23		nn0z 12561	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
24		zmulcl 12589	. . . . . . . . . . 11 ⊢ ((2 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2 · 𝑛) ∈ ℤ)
25	22, 23, 24	sylancr 587	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → (2 · 𝑛) ∈ ℤ)
26		bccl 14294	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ (2 · 𝑛) ∈ ℤ) → (𝑁C(2 · 𝑛)) ∈ ℕ₀)
27	21, 25, 26	syl2an 596	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (𝑁C(2 · 𝑛)) ∈ ℕ₀)
28	27	nn0cnd 12512	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (𝑁C(2 · 𝑛)) ∈ ℂ)
29		neg1cn 12178	. . . . . . . . 9 ⊢ -1 ∈ ℂ
30		neg1ne0 12180	. . . . . . . . 9 ⊢ -1 ≠ 0
31		nnz 12557	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
32		zsubcl 12582	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑀 − 𝑛) ∈ ℤ)
33	31, 23, 32	syl2an 596	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (𝑀 − 𝑛) ∈ ℤ)
34		expclz 14056	. . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ -1 ≠ 0 ∧ (𝑀 − 𝑛) ∈ ℤ) → (-1↑(𝑀 − 𝑛)) ∈ ℂ)
35	29, 30, 33, 34	mp3an12i 1467	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (-1↑(𝑀 − 𝑛)) ∈ ℂ)
36	28, 35	mulcld 11201	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))) ∈ ℂ)
37	36	fmpttd 7090	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))):ℕ₀⟶ℂ)
38		ffvelcdm 7056	. . . . . 6 ⊢ (((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))):ℕ₀⟶ℂ ∧ 𝑗 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ∈ ℂ)
39	37, 4, 38	syl2an 596	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ∈ ℂ)
40	14, 39	eqeltrrd 2830	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ (0...𝑀)) → ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) ∈ ℂ)
41	2, 3, 40	elplyd 26114	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗))) ∈ (Poly‘ℂ))
42	1, 41	eqeltrid 2833	. 2 ⊢ (𝑀 ∈ ℕ → 𝑃 ∈ (Poly‘ℂ))
43		nnre 12200	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
44		nn0re 12458	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ₀ → 𝑗 ∈ ℝ)
45		ltnle 11260	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑀 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑀))
46	43, 44, 45	syl2an 596	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (𝑀 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑀))
47	12	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))))
48	21	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 𝑁 ∈ ℕ₀)
49		nn0z 12561	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ₀ → 𝑗 ∈ ℤ)
50	49	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 𝑗 ∈ ℤ)
51		zmulcl 12589	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (2 · 𝑗) ∈ ℤ)
52	22, 50, 51	sylancr 587	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (2 · 𝑗) ∈ ℤ)
53		ax-1cn 11133	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
54	53	2timesi 12326	. . . . . . . . . . . . . . . 16 ⊢ (2 · 1) = (1 + 1)
55	54	oveq2i 7401	. . . . . . . . . . . . . . 15 ⊢ ((2 · 𝑀) + (2 · 1)) = ((2 · 𝑀) + (1 + 1))
56		2cnd 12271	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 2 ∈ ℂ)
57		nncn 12201	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
58	57	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 𝑀 ∈ ℂ)
59	53	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 1 ∈ ℂ)
60	56, 58, 59	adddid 11205	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (2 · (𝑀 + 1)) = ((2 · 𝑀) + (2 · 1)))
61	15	oveq1i 7400	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 + 1) = (((2 · 𝑀) + 1) + 1)
62	18	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (2 · 𝑀) ∈ ℕ)
63	62	nncnd 12209	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (2 · 𝑀) ∈ ℂ)
64	63, 59, 59	addassd 11203	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (((2 · 𝑀) + 1) + 1) = ((2 · 𝑀) + (1 + 1)))
65	61, 64	eqtrid 2777	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (𝑁 + 1) = ((2 · 𝑀) + (1 + 1)))
66	55, 60, 65	3eqtr4a 2791	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (2 · (𝑀 + 1)) = (𝑁 + 1))
67		zltp1le 12590	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑀 < 𝑗 ↔ (𝑀 + 1) ≤ 𝑗))
68	31, 49, 67	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (𝑀 < 𝑗 ↔ (𝑀 + 1) ≤ 𝑗))
69	68	biimpa 476	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (𝑀 + 1) ≤ 𝑗)
70	43	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 𝑀 ∈ ℝ)
71		peano2re 11354	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
72	70, 71	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (𝑀 + 1) ∈ ℝ)
73	44	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 𝑗 ∈ ℝ)
74		2re 12267	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
75		2pos 12296	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 2
76	74, 75	pm3.2i 470	. . . . . . . . . . . . . . . . 17 ⊢ (2 ∈ ℝ ∧ 0 < 2)
77	76	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (2 ∈ ℝ ∧ 0 < 2))
78		lemul2 12042	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 + 1) ∈ ℝ ∧ 𝑗 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((𝑀 + 1) ≤ 𝑗 ↔ (2 · (𝑀 + 1)) ≤ (2 · 𝑗)))
79	72, 73, 77, 78	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → ((𝑀 + 1) ≤ 𝑗 ↔ (2 · (𝑀 + 1)) ≤ (2 · 𝑗)))
80	69, 79	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (2 · (𝑀 + 1)) ≤ (2 · 𝑗))
81	66, 80	eqbrtrrd 5134	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (𝑁 + 1) ≤ (2 · 𝑗))
82	20	nnzd 12563	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 𝑁 ∈ ℤ)
83	82	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 𝑁 ∈ ℤ)
84		zltp1le 12590	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℤ ∧ (2 · 𝑗) ∈ ℤ) → (𝑁 < (2 · 𝑗) ↔ (𝑁 + 1) ≤ (2 · 𝑗)))
85	83, 52, 84	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (𝑁 < (2 · 𝑗) ↔ (𝑁 + 1) ≤ (2 · 𝑗)))
86	81, 85	mpbird 257	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → 𝑁 < (2 · 𝑗))
87	86	olcd 874	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → ((2 · 𝑗) < 0 ∨ 𝑁 < (2 · 𝑗)))
88		bcval4 14279	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ (2 · 𝑗) ∈ ℤ ∧ ((2 · 𝑗) < 0 ∨ 𝑁 < (2 · 𝑗))) → (𝑁C(2 · 𝑗)) = 0)
89	48, 52, 87, 88	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (𝑁C(2 · 𝑗)) = 0)
90	89	oveq1d 7405	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) = (0 · (-1↑(𝑀 − 𝑗))))
91		zsubcl 12582	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑀 − 𝑗) ∈ ℤ)
92	31, 49, 91	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (𝑀 − 𝑗) ∈ ℤ)
93		expclz 14056	. . . . . . . . . . . 12 ⊢ ((-1 ∈ ℂ ∧ -1 ≠ 0 ∧ (𝑀 − 𝑗) ∈ ℤ) → (-1↑(𝑀 − 𝑗)) ∈ ℂ)
94	29, 30, 92, 93	mp3an12i 1467	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (-1↑(𝑀 − 𝑗)) ∈ ℂ)
95	94	adantr 480	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (-1↑(𝑀 − 𝑗)) ∈ ℂ)
96	95	mul02d 11379	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → (0 · (-1↑(𝑀 − 𝑗))) = 0)
97	47, 90, 96	3eqtrd 2769	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) ∧ 𝑀 < 𝑗) → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = 0)
98	97	ex 412	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (𝑀 < 𝑗 → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = 0))
99	46, 98	sylbird 260	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (¬ 𝑗 ≤ 𝑀 → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = 0))
100	99	necon1ad 2943	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ₀) → (((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ≠ 0 → 𝑗 ≤ 𝑀))
101	100	ralrimiva 3126	. . . 4 ⊢ (𝑀 ∈ ℕ → ∀𝑗 ∈ ℕ₀ (((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ≠ 0 → 𝑗 ≤ 𝑀))
102		plyco0 26104	. . . . 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)))
103	3, 37, 102	syl2anc 584	. . . 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)))
104	101, 103	mpbird 257	. . 3 ⊢ (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))) “ (ℤ_≥‘(𝑀 + 1))) = {0})
105	13	oveq1d 7405	. . . . . . 7 ⊢ (𝑗 ∈ (0...𝑀) → (((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) · (𝑡↑𝑗)) = (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗)))
106	105	sumeq2i 15671	. . . . . 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))
107	106	mpteq2i 5206	. . . . 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)))
108	1, 107	eqtr4i 2756	. . . 4 ⊢ 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) · (𝑡↑𝑗)))
109	108	a1i 11	. . 3 ⊢ (𝑀 ∈ ℕ → 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) · (𝑡↑𝑗))))
110		oveq2 7398	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (2 · 𝑛) = (2 · 𝑀))
111	110	oveq2d 7406	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → (𝑁C(2 · 𝑛)) = (𝑁C(2 · 𝑀)))
112		oveq2 7398	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (𝑀 − 𝑛) = (𝑀 − 𝑀))
113	112	oveq2d 7406	. . . . . . . 8 ⊢ (𝑛 = 𝑀 → (-1↑(𝑀 − 𝑛)) = (-1↑(𝑀 − 𝑀)))
114	111, 113	oveq12d 7408	. . . . . . 7 ⊢ (𝑛 = 𝑀 → ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))) = ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀))))
115		ovex 7423	. . . . . . 7 ⊢ ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀))) ∈ V
116	114, 10, 115	fvmpt 6971	. . . . . 6 ⊢ (𝑀 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑀) = ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀))))
117	3, 116	syl 17	. . . . 5 ⊢ (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑀) = ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀))))
118	57	subidd 11528	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑀 − 𝑀) = 0)
119	118	oveq2d 7406	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (-1↑(𝑀 − 𝑀)) = (-1↑0))
120		exp0 14037	. . . . . . . 8 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
121	29, 120	ax-mp 5	. . . . . . 7 ⊢ (-1↑0) = 1
122	119, 121	eqtrdi 2781	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (-1↑(𝑀 − 𝑀)) = 1)
123	122	oveq2d 7406	. . . . 5 ⊢ (𝑀 ∈ ℕ → ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀))) = ((𝑁C(2 · 𝑀)) · 1))
124	18	nnred 12208	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ ℝ)
125	124	lep1d 12121	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ≤ ((2 · 𝑀) + 1))
126	125, 15	breqtrrdi 5152	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ≤ 𝑁)
127	18	nnnn0d 12510	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ ℕ₀)
128		nn0uz 12842	. . . . . . . . . . 11 ⊢ ℕ₀ = (ℤ_≥‘0)
129	127, 128	eleqtrdi 2839	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ (ℤ_≥‘0))
130		elfz5 13484	. . . . . . . . . 10 ⊢ (((2 · 𝑀) ∈ (ℤ_≥‘0) ∧ 𝑁 ∈ ℤ) → ((2 · 𝑀) ∈ (0...𝑁) ↔ (2 · 𝑀) ≤ 𝑁))
131	129, 82, 130	syl2anc 584	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → ((2 · 𝑀) ∈ (0...𝑁) ↔ (2 · 𝑀) ≤ 𝑁))
132	126, 131	mpbird 257	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ (0...𝑁))
133		bccl2 14295	. . . . . . . 8 ⊢ ((2 · 𝑀) ∈ (0...𝑁) → (𝑁C(2 · 𝑀)) ∈ ℕ)
134	132, 133	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (𝑁C(2 · 𝑀)) ∈ ℕ)
135	134	nncnd 12209	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑁C(2 · 𝑀)) ∈ ℂ)
136	135	mulridd 11198	. . . . 5 ⊢ (𝑀 ∈ ℕ → ((𝑁C(2 · 𝑀)) · 1) = (𝑁C(2 · 𝑀)))
137	117, 123, 136	3eqtrd 2769	. . . 4 ⊢ (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑀) = (𝑁C(2 · 𝑀)))
138	134	nnne0d 12243	. . . 4 ⊢ (𝑀 ∈ ℕ → (𝑁C(2 · 𝑀)) ≠ 0)
139	137, 138	eqnetrd 2993	. . 3 ⊢ (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑀) ≠ 0)
140	42, 3, 37, 104, 109, 139	dgreq 26156	. 2 ⊢ (𝑀 ∈ ℕ → (deg‘𝑃) = 𝑀)
141	42, 3, 37, 104, 109	coeeq 26139	. 2 ⊢ (𝑀 ∈ ℕ → (coeff‘𝑃) = (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))))
142	42, 140, 141	3jca 1128	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 {csn 4592 class class class wbr 5110 ↦ cmpt 5191 “ cima 5644 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 < clt 11215 ≤ cle 11216 − cmin 11412 -cneg 11413 ℕcn 12193 2c2 12248 ℕ₀cn0 12449 ℤcz 12536 ℤ_≥cuz 12800 ...cfz 13475 ↑cexp 14033 Ccbc 14274 Σcsu 15659 Polycply 26096 coeffccoe 26098 degcdgr 26099

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-rp 12959 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-fac 14246 df-bc 14275 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462 df-sum 15660 df-0p 25578 df-ply 26100 df-coe 26102 df-dgr 26103

This theorem is referenced by: basellem4 27001 basellem5 27002

W3C validator