basellem3 - Metamath Proof Explorer

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Theorem basellem3 26584

Description: Lemma for basel 26591. Using the binomial theorem and de Moivre's formula, we have the identity e↑i𝑁𝑥 / (sin𝑥)↑𝑛 = Σ𝑚 ∈ (0...𝑁)(𝑁C𝑚)(i↑𝑚)(cot𝑥)↑(𝑁 − 𝑚), so taking imaginary parts yields sin(𝑁𝑥) / (sin𝑥)↑𝑁 = Σ𝑗 ∈ (0...𝑀)(𝑁C2𝑗)(-1)↑(𝑀 − 𝑗) (cot𝑥)↑(-2𝑗) = 𝑃((cot𝑥)↑2), where 𝑁 = 2𝑀 + 1. (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
basellem3	⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑃‘((tan‘𝐴)↑-2)) = ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)))

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

Proof of Theorem basellem3 Dummy variables 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tanrpcl 26013	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(π / 2)) → (tan‘𝐴) ∈ ℝ⁺)
2	1	adantl 482	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (tan‘𝐴) ∈ ℝ⁺)
3	2	rpreccld 13025	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / (tan‘𝐴)) ∈ ℝ⁺)
4	3	rpcnd 13017	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / (tan‘𝐴)) ∈ ℂ)
5		ax-icn 11168	. . . . . 6 ⊢ i ∈ ℂ
6	5	a1i 11	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → i ∈ ℂ)
7		basel.n	. . . . . . 7 ⊢ 𝑁 = ((2 · 𝑀) + 1)
8		2nn 12284	. . . . . . . . 9 ⊢ 2 ∈ ℕ
9		simpl 483	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑀 ∈ ℕ)
10		nnmulcl 12235	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (2 · 𝑀) ∈ ℕ)
11	8, 9, 10	sylancr 587	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (2 · 𝑀) ∈ ℕ)
12	11	peano2nnd 12228	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((2 · 𝑀) + 1) ∈ ℕ)
13	7, 12	eqeltrid 2837	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℕ)
14	13	nnnn0d 12531	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℕ₀)
15		binom 15775	. . . . 5 ⊢ (((1 / (tan‘𝐴)) ∈ ℂ ∧ i ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → (((1 / (tan‘𝐴)) + i)↑𝑁) = Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚))))
16	4, 6, 14, 15	syl3anc 1371	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((1 / (tan‘𝐴)) + i)↑𝑁) = Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚))))
17		elioore 13353	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (0(,)(π / 2)) → 𝐴 ∈ ℝ)
18	17	adantl 482	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝐴 ∈ ℝ)
19	18	recoscld 16086	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘𝐴) ∈ ℝ)
20	19	recnd 11241	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘𝐴) ∈ ℂ)
21	18	resincld 16085	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘𝐴) ∈ ℝ)
22	21	recnd 11241	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘𝐴) ∈ ℂ)
23		mulcl 11193	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ (sin‘𝐴) ∈ ℂ) → (i · (sin‘𝐴)) ∈ ℂ)
24	5, 22, 23	sylancr 587	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (i · (sin‘𝐴)) ∈ ℂ)
25	20, 24	addcld 11232	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((cos‘𝐴) + (i · (sin‘𝐴))) ∈ ℂ)
26		sincosq1sgn 26007	. . . . . . . . . 10 ⊢ (𝐴 ∈ (0(,)(π / 2)) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))
27	26	adantl 482	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))
28	27	simpld 495	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 0 < (sin‘𝐴))
29	28	gt0ne0d 11777	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘𝐴) ≠ 0)
30	25, 22, 29, 14	expdivd 14124	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((((cos‘𝐴) + (i · (sin‘𝐴))) / (sin‘𝐴))↑𝑁) = ((((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) / ((sin‘𝐴)↑𝑁)))
31	20, 24, 22, 29	divdird 12027	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((cos‘𝐴) + (i · (sin‘𝐴))) / (sin‘𝐴)) = (((cos‘𝐴) / (sin‘𝐴)) + ((i · (sin‘𝐴)) / (sin‘𝐴))))
32	18	recnd 11241	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝐴 ∈ ℂ)
33	27	simprd 496	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 0 < (cos‘𝐴))
34	33	gt0ne0d 11777	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘𝐴) ≠ 0)
35		tanval 16070	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴)))
36	32, 34, 35	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴)))
37	36	oveq2d 7424	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / (tan‘𝐴)) = (1 / ((sin‘𝐴) / (cos‘𝐴))))
38	22, 20, 29, 34	recdivd 12006	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / ((sin‘𝐴) / (cos‘𝐴))) = ((cos‘𝐴) / (sin‘𝐴)))
39	37, 38	eqtr2d 2773	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((cos‘𝐴) / (sin‘𝐴)) = (1 / (tan‘𝐴)))
40	6, 22, 29	divcan4d 11995	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((i · (sin‘𝐴)) / (sin‘𝐴)) = i)
41	39, 40	oveq12d 7426	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((cos‘𝐴) / (sin‘𝐴)) + ((i · (sin‘𝐴)) / (sin‘𝐴))) = ((1 / (tan‘𝐴)) + i))
42	31, 41	eqtrd 2772	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((cos‘𝐴) + (i · (sin‘𝐴))) / (sin‘𝐴)) = ((1 / (tan‘𝐴)) + i))
43	42	oveq1d 7423	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((((cos‘𝐴) + (i · (sin‘𝐴))) / (sin‘𝐴))↑𝑁) = (((1 / (tan‘𝐴)) + i)↑𝑁))
44	13	nnzd 12584	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℤ)
45		demoivre 16142	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴)))))
46	32, 44, 45	syl2anc 584	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴)))))
47	46	oveq1d 7423	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) / ((sin‘𝐴)↑𝑁)) = (((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴)))) / ((sin‘𝐴)↑𝑁)))
48	30, 43, 47	3eqtr3d 2780	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((1 / (tan‘𝐴)) + i)↑𝑁) = (((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴)))) / ((sin‘𝐴)↑𝑁)))
49	13	nnred 12226	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℝ)
50	49, 18	remulcld 11243	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑁 · 𝐴) ∈ ℝ)
51	50	recoscld 16086	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘(𝑁 · 𝐴)) ∈ ℝ)
52	51	recnd 11241	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘(𝑁 · 𝐴)) ∈ ℂ)
53	50	resincld 16085	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘(𝑁 · 𝐴)) ∈ ℝ)
54	53	recnd 11241	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘(𝑁 · 𝐴)) ∈ ℂ)
55		mulcl 11193	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ (sin‘(𝑁 · 𝐴)) ∈ ℂ) → (i · (sin‘(𝑁 · 𝐴))) ∈ ℂ)
56	5, 54, 55	sylancr 587	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (i · (sin‘(𝑁 · 𝐴))) ∈ ℂ)
57	21, 28	elrpd 13012	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘𝐴) ∈ ℝ⁺)
58	57, 44	rpexpcld 14209	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((sin‘𝐴)↑𝑁) ∈ ℝ⁺)
59	58	rpcnd 13017	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((sin‘𝐴)↑𝑁) ∈ ℂ)
60	58	rpne0d 13020	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((sin‘𝐴)↑𝑁) ≠ 0)
61	52, 56, 59, 60	divdird 12027	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴)))) / ((sin‘𝐴)↑𝑁)) = (((cos‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) + ((i · (sin‘(𝑁 · 𝐴))) / ((sin‘𝐴)↑𝑁))))
62	6, 54, 59, 60	divassd 12024	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((i · (sin‘(𝑁 · 𝐴))) / ((sin‘𝐴)↑𝑁)) = (i · ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁))))
63	62	oveq2d 7424	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((cos‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) + ((i · (sin‘(𝑁 · 𝐴))) / ((sin‘𝐴)↑𝑁))) = (((cos‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) + (i · ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)))))
64	48, 61, 63	3eqtrd 2776	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((1 / (tan‘𝐴)) + i)↑𝑁) = (((cos‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) + (i · ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)))))
65	16, 64	eqtr3d 2774	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚))) = (((cos‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) + (i · ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)))))
66	65	fveq2d 6895	. 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (ℑ‘Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)))) = (ℑ‘(((cos‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) + (i · ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁))))))
67		oveq2 7416	. . . . . . 7 ⊢ (𝑚 = (𝑁 − (2 · 𝑗)) → (𝑁C𝑚) = (𝑁C(𝑁 − (2 · 𝑗))))
68		oveq2 7416	. . . . . . . . 9 ⊢ (𝑚 = (𝑁 − (2 · 𝑗)) → (𝑁 − 𝑚) = (𝑁 − (𝑁 − (2 · 𝑗))))
69	68	oveq2d 7424	. . . . . . . 8 ⊢ (𝑚 = (𝑁 − (2 · 𝑗)) → ((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) = ((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))))
70		oveq2 7416	. . . . . . . 8 ⊢ (𝑚 = (𝑁 − (2 · 𝑗)) → (i↑𝑚) = (i↑(𝑁 − (2 · 𝑗))))
71	69, 70	oveq12d 7426	. . . . . . 7 ⊢ (𝑚 = (𝑁 − (2 · 𝑗)) → (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)) = (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗)))))
72	67, 71	oveq12d 7426	. . . . . 6 ⊢ (𝑚 = (𝑁 − (2 · 𝑗)) → ((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚))) = ((𝑁C(𝑁 − (2 · 𝑗))) · (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗))))))
73	72	fveq2d 6895	. . . . 5 ⊢ (𝑚 = (𝑁 − (2 · 𝑗)) → (ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)))) = (ℑ‘((𝑁C(𝑁 − (2 · 𝑗))) · (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗)))))))
74		fzfid 13937	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (0...𝑀) ∈ Fin)
75		2nn0 12488	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
76		elfznn0 13593	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝑀) → 𝑘 ∈ ℕ₀)
77	76	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ ℕ₀)
78		nn0mulcl 12507	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (2 · 𝑘) ∈ ℕ₀)
79	75, 77, 78	sylancr 587	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ∈ ℕ₀)
80	79	nn0red 12532	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ∈ ℝ)
81	11	nnred 12226	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (2 · 𝑀) ∈ ℝ)
82	81	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑀) ∈ ℝ)
83	49	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑁 ∈ ℝ)
84		elfzle2 13504	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝑀) → 𝑘 ≤ 𝑀)
85	84	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ≤ 𝑀)
86	77	nn0red 12532	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ ℝ)
87		nnre 12218	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
88	87	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑀 ∈ ℝ)
89		2re 12285	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
90		2pos 12314	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
91	89, 90	pm3.2i 471	. . . . . . . . . . . . . 14 ⊢ (2 ∈ ℝ ∧ 0 < 2)
92	91	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 ∈ ℝ ∧ 0 < 2))
93		lemul2 12066	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝑘 ≤ 𝑀 ↔ (2 · 𝑘) ≤ (2 · 𝑀)))
94	86, 88, 92, 93	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (𝑘 ≤ 𝑀 ↔ (2 · 𝑘) ≤ (2 · 𝑀)))
95	85, 94	mpbid 231	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ≤ (2 · 𝑀))
96	82	lep1d 12144	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑀) ≤ ((2 · 𝑀) + 1))
97	96, 7	breqtrrdi 5190	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑀) ≤ 𝑁)
98	80, 82, 83, 95, 97	letrd 11370	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ≤ 𝑁)
99		nn0uz 12863	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
100	79, 99	eleqtrdi 2843	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ∈ (ℤ_≥‘0))
101	44	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑁 ∈ ℤ)
102		elfz5 13492	. . . . . . . . . . 11 ⊢ (((2 · 𝑘) ∈ (ℤ_≥‘0) ∧ 𝑁 ∈ ℤ) → ((2 · 𝑘) ∈ (0...𝑁) ↔ (2 · 𝑘) ≤ 𝑁))
103	100, 101, 102	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → ((2 · 𝑘) ∈ (0...𝑁) ↔ (2 · 𝑘) ≤ 𝑁))
104	98, 103	mpbird 256	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ∈ (0...𝑁))
105		fznn0sub2 13607	. . . . . . . . 9 ⊢ ((2 · 𝑘) ∈ (0...𝑁) → (𝑁 − (2 · 𝑘)) ∈ (0...𝑁))
106	104, 105	syl 17	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (𝑁 − (2 · 𝑘)) ∈ (0...𝑁))
107	106	ex 413	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑘 ∈ (0...𝑀) → (𝑁 − (2 · 𝑘)) ∈ (0...𝑁)))
108	13	nncnd 12227	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℂ)
109	108	adantr 481	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → 𝑁 ∈ ℂ)
110		2cn 12286	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
111		elfzelz 13500	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝑀) → 𝑘 ∈ ℤ)
112	111	zcnd 12666	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...𝑀) → 𝑘 ∈ ℂ)
113	112	ad2antrl 726	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → 𝑘 ∈ ℂ)
114		mulcl 11193	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (2 · 𝑘) ∈ ℂ)
115	110, 113, 114	sylancr 587	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → (2 · 𝑘) ∈ ℂ)
116	112	ssriv 3986	. . . . . . . . . . . 12 ⊢ (0...𝑀) ⊆ ℂ
117		simprr 771	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → 𝑚 ∈ (0...𝑀))
118	116, 117	sselid 3980	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → 𝑚 ∈ ℂ)
119		mulcl 11193	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (2 · 𝑚) ∈ ℂ)
120	110, 118, 119	sylancr 587	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → (2 · 𝑚) ∈ ℂ)
121	109, 115, 120	subcanad 11613	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → ((𝑁 − (2 · 𝑘)) = (𝑁 − (2 · 𝑚)) ↔ (2 · 𝑘) = (2 · 𝑚)))
122		2cnne0 12421	. . . . . . . . . . 11 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
123	122	a1i 11	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → (2 ∈ ℂ ∧ 2 ≠ 0))
124		mulcan 11850	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℂ ∧ 𝑚 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((2 · 𝑘) = (2 · 𝑚) ↔ 𝑘 = 𝑚))
125	113, 118, 123, 124	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → ((2 · 𝑘) = (2 · 𝑚) ↔ 𝑘 = 𝑚))
126	121, 125	bitrd 278	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → ((𝑁 − (2 · 𝑘)) = (𝑁 − (2 · 𝑚)) ↔ 𝑘 = 𝑚))
127	126	ex 413	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀)) → ((𝑁 − (2 · 𝑘)) = (𝑁 − (2 · 𝑚)) ↔ 𝑘 = 𝑚)))
128	107, 127	dom2lem 8987	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))):(0...𝑀)–1-1→(0...𝑁))
129		f1f1orn 6844	. . . . . 6 ⊢ ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))):(0...𝑀)–1-1→(0...𝑁) → (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))):(0...𝑀)–1-1-onto→ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))
130	128, 129	syl 17	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))):(0...𝑀)–1-1-onto→ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))
131		oveq2 7416	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (2 · 𝑘) = (2 · 𝑗))
132	131	oveq2d 7424	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑁 − (2 · 𝑘)) = (𝑁 − (2 · 𝑗)))
133		eqid 2732	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) = (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))
134		ovex 7441	. . . . . . 7 ⊢ (𝑁 − (2 · 𝑗)) ∈ V
135	132, 133, 134	fvmpt 6998	. . . . . 6 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘𝑗) = (𝑁 − (2 · 𝑗)))
136	135	adantl 482	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘𝑗) = (𝑁 − (2 · 𝑗)))
137	106	fmpttd 7114	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))):(0...𝑀)⟶(0...𝑁))
138	137	frnd 6725	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) ⊆ (0...𝑁))
139	138	sselda 3982	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) → 𝑚 ∈ (0...𝑁))
140		bccl2 14282	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...𝑁) → (𝑁C𝑚) ∈ ℕ)
141	140	adantl 482	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℕ)
142	141	nncnd 12227	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℂ)
143	2	rprecred 13026	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / (tan‘𝐴)) ∈ ℝ)
144		fznn0sub 13532	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...𝑁) → (𝑁 − 𝑚) ∈ ℕ₀)
145		reexpcl 14043	. . . . . . . . . . . 12 ⊢ (((1 / (tan‘𝐴)) ∈ ℝ ∧ (𝑁 − 𝑚) ∈ ℕ₀) → ((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) ∈ ℝ)
146	143, 144, 145	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) ∈ ℝ)
147	146	recnd 11241	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) ∈ ℂ)
148		elfznn0 13593	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℕ₀)
149	148	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → 𝑚 ∈ ℕ₀)
150		expcl 14044	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (i↑𝑚) ∈ ℂ)
151	5, 149, 150	sylancr 587	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (i↑𝑚) ∈ ℂ)
152	147, 151	mulcld 11233	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)) ∈ ℂ)
153	142, 152	mulcld 11233	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚))) ∈ ℂ)
154	139, 153	syldan 591	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) → ((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚))) ∈ ℂ)
155	154	imcld 15141	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) → (ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)))) ∈ ℝ)
156	155	recnd 11241	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) → (ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)))) ∈ ℂ)
157	73, 74, 130, 136, 156	fsumf1o 15668	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → Σ𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))(ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)))) = Σ𝑗 ∈ (0...𝑀)(ℑ‘((𝑁C(𝑁 − (2 · 𝑗))) · (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗)))))))
158		eldifi 4126	. . . . . . . 8 ⊢ (𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) → 𝑚 ∈ (0...𝑁))
159	141	nnred 12226	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℝ)
160	158, 159	sylan2 593	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → (𝑁C𝑚) ∈ ℝ)
161	158, 146	sylan2 593	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → ((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) ∈ ℝ)
162		eldif 3958	. . . . . . . . 9 ⊢ (𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) ↔ (𝑚 ∈ (0...𝑁) ∧ ¬ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))))
163		elfzelz 13500	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ)
164	163	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → 𝑚 ∈ ℤ)
165		zeo 12647	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℤ → ((𝑚 / 2) ∈ ℤ ∨ ((𝑚 + 1) / 2) ∈ ℤ))
166	164, 165	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑚 / 2) ∈ ℤ ∨ ((𝑚 + 1) / 2) ∈ ℤ))
167		i2 14165	. . . . . . . . . . . . . . . . . 18 ⊢ (i↑2) = -1
168	167	oveq1i 7418	. . . . . . . . . . . . . . . . 17 ⊢ ((i↑2)↑(𝑚 / 2)) = (-1↑(𝑚 / 2))
169		simprr 771	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (𝑚 / 2) ∈ ℤ)
170	148	ad2antrl 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → 𝑚 ∈ ℕ₀)
171		nn0re 12480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℝ)
172		nn0ge0 12496	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ ℕ₀ → 0 ≤ 𝑚)
173		divge0 12082	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ (𝑚 / 2))
174	89, 90, 173	mpanr12 703	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) → 0 ≤ (𝑚 / 2))
175	171, 172, 174	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ ℕ₀ → 0 ≤ (𝑚 / 2))
176	170, 175	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → 0 ≤ (𝑚 / 2))
177		elnn0z 12570	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 / 2) ∈ ℕ₀ ↔ ((𝑚 / 2) ∈ ℤ ∧ 0 ≤ (𝑚 / 2)))
178	169, 176, 177	sylanbrc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (𝑚 / 2) ∈ ℕ₀)
179		expmul 14072	. . . . . . . . . . . . . . . . . . 19 ⊢ ((i ∈ ℂ ∧ 2 ∈ ℕ₀ ∧ (𝑚 / 2) ∈ ℕ₀) → (i↑(2 · (𝑚 / 2))) = ((i↑2)↑(𝑚 / 2)))
180	5, 75, 178, 179	mp3an12i 1465	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (i↑(2 · (𝑚 / 2))) = ((i↑2)↑(𝑚 / 2)))
181	170	nn0cnd 12533	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → 𝑚 ∈ ℂ)
182		2ne0 12315	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ≠ 0
183		divcan2 11879	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (2 · (𝑚 / 2)) = 𝑚)
184	110, 182, 183	mp3an23 1453	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℂ → (2 · (𝑚 / 2)) = 𝑚)
185	181, 184	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (2 · (𝑚 / 2)) = 𝑚)
186	185	oveq2d 7424	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (i↑(2 · (𝑚 / 2))) = (i↑𝑚))
187	180, 186	eqtr3d 2774	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → ((i↑2)↑(𝑚 / 2)) = (i↑𝑚))
188	168, 187	eqtr3id 2786	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (-1↑(𝑚 / 2)) = (i↑𝑚))
189		neg1rr 12326	. . . . . . . . . . . . . . . . 17 ⊢ -1 ∈ ℝ
190		reexpcl 14043	. . . . . . . . . . . . . . . . 17 ⊢ ((-1 ∈ ℝ ∧ (𝑚 / 2) ∈ ℕ₀) → (-1↑(𝑚 / 2)) ∈ ℝ)
191	189, 178, 190	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (-1↑(𝑚 / 2)) ∈ ℝ)
192	188, 191	eqeltrrd 2834	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (i↑𝑚) ∈ ℝ)
193	192	expr 457	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑚 / 2) ∈ ℤ → (i↑𝑚) ∈ ℝ))
194		0zd 12569	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 0 ∈ ℤ)
195		nnz 12578	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
196	195	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑀 ∈ ℤ)
197	108	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑁 ∈ ℂ)
198	148	ad2antrl 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑚 ∈ ℕ₀)
199	198	nn0cnd 12533	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑚 ∈ ℂ)
200		1cnd 11208	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 1 ∈ ℂ)
201	197, 199, 200	pnpcan2d 11608	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 + 1) − (𝑚 + 1)) = (𝑁 − 𝑚))
202		2t1e2 12374	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (2 · 1) = 2
203		df-2 12274	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 2 = (1 + 1)
204	202, 203	eqtr2i 2761	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1 + 1) = (2 · 1)
205	204	oveq2i 7419	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((2 · 𝑀) + (1 + 1)) = ((2 · 𝑀) + (2 · 1))
206	7	oveq1i 7418	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 + 1) = (((2 · 𝑀) + 1) + 1)
207	11	nncnd 12227	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (2 · 𝑀) ∈ ℂ)
208	207	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 · 𝑀) ∈ ℂ)
209	208, 200, 200	addassd 11235	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (((2 · 𝑀) + 1) + 1) = ((2 · 𝑀) + (1 + 1)))
210	206, 209	eqtrid 2784	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 + 1) = ((2 · 𝑀) + (1 + 1)))
211		2cnd 12289	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 2 ∈ ℂ)
212		nncn 12219	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
213	212	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑀 ∈ ℂ)
214	211, 213, 200	adddid 11237	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 · (𝑀 + 1)) = ((2 · 𝑀) + (2 · 1)))
215	205, 210, 214	3eqtr4a 2798	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 + 1) = (2 · (𝑀 + 1)))
216	215	oveq1d 7423	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 + 1) − (𝑚 + 1)) = ((2 · (𝑀 + 1)) − (𝑚 + 1)))
217	201, 216	eqtr3d 2774	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − 𝑚) = ((2 · (𝑀 + 1)) − (𝑚 + 1)))
218	217	oveq1d 7423	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 − 𝑚) / 2) = (((2 · (𝑀 + 1)) − (𝑚 + 1)) / 2))
219	196	peano2zd 12668	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑀 + 1) ∈ ℤ)
220	219	zcnd 12666	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑀 + 1) ∈ ℂ)
221		mulcl 11193	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((2 ∈ ℂ ∧ (𝑀 + 1) ∈ ℂ) → (2 · (𝑀 + 1)) ∈ ℂ)
222	110, 220, 221	sylancr 587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 · (𝑀 + 1)) ∈ ℂ)
223		peano2cn 11385	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℂ → (𝑚 + 1) ∈ ℂ)
224	199, 223	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑚 + 1) ∈ ℂ)
225	122	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 ∈ ℂ ∧ 2 ≠ 0))
226		divsubdir 11907	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2 · (𝑀 + 1)) ∈ ℂ ∧ (𝑚 + 1) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((2 · (𝑀 + 1)) − (𝑚 + 1)) / 2) = (((2 · (𝑀 + 1)) / 2) − ((𝑚 + 1) / 2)))
227	222, 224, 225, 226	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (((2 · (𝑀 + 1)) − (𝑚 + 1)) / 2) = (((2 · (𝑀 + 1)) / 2) − ((𝑚 + 1) / 2)))
228	182	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 2 ≠ 0)
229	220, 211, 228	divcan3d 11994	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((2 · (𝑀 + 1)) / 2) = (𝑀 + 1))
230	229	oveq1d 7423	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (((2 · (𝑀 + 1)) / 2) − ((𝑚 + 1) / 2)) = ((𝑀 + 1) − ((𝑚 + 1) / 2)))
231	218, 227, 230	3eqtrd 2776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 − 𝑚) / 2) = ((𝑀 + 1) − ((𝑚 + 1) / 2)))
232		simprr 771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑚 + 1) / 2) ∈ ℤ)
233	219, 232	zsubcld 12670	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑀 + 1) − ((𝑚 + 1) / 2)) ∈ ℤ)
234	231, 233	eqeltrd 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 − 𝑚) / 2) ∈ ℤ)
235	144	ad2antrl 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − 𝑚) ∈ ℕ₀)
236		nn0re 12480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 − 𝑚) ∈ ℕ₀ → (𝑁 − 𝑚) ∈ ℝ)
237		nn0ge0 12496	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 − 𝑚) ∈ ℕ₀ → 0 ≤ (𝑁 − 𝑚))
238		divge0 12082	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 − 𝑚) ∈ ℝ ∧ 0 ≤ (𝑁 − 𝑚)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ ((𝑁 − 𝑚) / 2))
239	89, 90, 238	mpanr12 703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 − 𝑚) ∈ ℝ ∧ 0 ≤ (𝑁 − 𝑚)) → 0 ≤ ((𝑁 − 𝑚) / 2))
240	236, 237, 239	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 − 𝑚) ∈ ℕ₀ → 0 ≤ ((𝑁 − 𝑚) / 2))
241	235, 240	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 0 ≤ ((𝑁 − 𝑚) / 2))
242	235	nn0red 12532	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − 𝑚) ∈ ℝ)
243	49	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑁 ∈ ℝ)
244		peano2re 11386	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
245	243, 244	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 + 1) ∈ ℝ)
246	198, 172	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 0 ≤ 𝑚)
247	198	nn0red 12532	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑚 ∈ ℝ)
248	243, 247	subge02d 11805	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (0 ≤ 𝑚 ↔ (𝑁 − 𝑚) ≤ 𝑁))
249	246, 248	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − 𝑚) ≤ 𝑁)
250	243	ltp1d 12143	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑁 < (𝑁 + 1))
251	242, 243, 245, 249, 250	lelttrd 11371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − 𝑚) < (𝑁 + 1))
252	251, 215	breqtrd 5174	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − 𝑚) < (2 · (𝑀 + 1)))
253	219	zred 12665	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑀 + 1) ∈ ℝ)
254	91	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 ∈ ℝ ∧ 0 < 2))
255		ltdivmul 12088	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 − 𝑚) ∈ ℝ ∧ (𝑀 + 1) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((𝑁 − 𝑚) / 2) < (𝑀 + 1) ↔ (𝑁 − 𝑚) < (2 · (𝑀 + 1))))
256	242, 253, 254, 255	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (((𝑁 − 𝑚) / 2) < (𝑀 + 1) ↔ (𝑁 − 𝑚) < (2 · (𝑀 + 1))))
257	252, 256	mpbird 256	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 − 𝑚) / 2) < (𝑀 + 1))
258		zleltp1 12612	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 − 𝑚) / 2) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (((𝑁 − 𝑚) / 2) ≤ 𝑀 ↔ ((𝑁 − 𝑚) / 2) < (𝑀 + 1)))
259	234, 196, 258	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (((𝑁 − 𝑚) / 2) ≤ 𝑀 ↔ ((𝑁 − 𝑚) / 2) < (𝑀 + 1)))
260	257, 259	mpbird 256	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 − 𝑚) / 2) ≤ 𝑀)
261	194, 196, 234, 241, 260	elfzd 13491	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 − 𝑚) / 2) ∈ (0...𝑀))
262		oveq2 7416	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = ((𝑁 − 𝑚) / 2) → (2 · 𝑘) = (2 · ((𝑁 − 𝑚) / 2)))
263	262	oveq2d 7424	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = ((𝑁 − 𝑚) / 2) → (𝑁 − (2 · 𝑘)) = (𝑁 − (2 · ((𝑁 − 𝑚) / 2))))
264		ovex 7441	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 − (2 · ((𝑁 − 𝑚) / 2))) ∈ V
265	263, 133, 264	fvmpt 6998	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 − 𝑚) / 2) ∈ (0...𝑀) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘((𝑁 − 𝑚) / 2)) = (𝑁 − (2 · ((𝑁 − 𝑚) / 2))))
266	261, 265	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘((𝑁 − 𝑚) / 2)) = (𝑁 − (2 · ((𝑁 − 𝑚) / 2))))
267	235	nn0cnd 12533	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − 𝑚) ∈ ℂ)
268	267, 211, 228	divcan2d 11991	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 · ((𝑁 − 𝑚) / 2)) = (𝑁 − 𝑚))
269	268	oveq2d 7424	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − (2 · ((𝑁 − 𝑚) / 2))) = (𝑁 − (𝑁 − 𝑚)))
270	197, 199	nncand 11575	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − (𝑁 − 𝑚)) = 𝑚)
271	266, 269, 270	3eqtrd 2776	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘((𝑁 − 𝑚) / 2)) = 𝑚)
272	137	ffnd 6718	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) Fn (0...𝑀))
273		fnfvelrn 7082	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) Fn (0...𝑀) ∧ ((𝑁 − 𝑚) / 2) ∈ (0...𝑀)) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘((𝑁 − 𝑚) / 2)) ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))
274	272, 261, 273	syl2an2r 683	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘((𝑁 − 𝑚) / 2)) ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))
275	271, 274	eqeltrrd 2834	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))
276	275	expr 457	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (((𝑚 + 1) / 2) ∈ ℤ → 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))))
277	193, 276	orim12d 963	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (((𝑚 / 2) ∈ ℤ ∨ ((𝑚 + 1) / 2) ∈ ℤ) → ((i↑𝑚) ∈ ℝ ∨ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))))
278	166, 277	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((i↑𝑚) ∈ ℝ ∨ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))))
279	278	orcomd 869	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) ∨ (i↑𝑚) ∈ ℝ))
280	279	ord 862	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (¬ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) → (i↑𝑚) ∈ ℝ))
281	280	impr 455	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ¬ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → (i↑𝑚) ∈ ℝ)
282	162, 281	sylan2b 594	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → (i↑𝑚) ∈ ℝ)
283	161, 282	remulcld 11243	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)) ∈ ℝ)
284	160, 283	remulcld 11243	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → ((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚))) ∈ ℝ)
285	284	reim0d 15171	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → (ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)))) = 0)
286		fzfid 13937	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (0...𝑁) ∈ Fin)
287	138, 156, 285, 286	fsumss 15670	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → Σ𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))(ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)))) = Σ𝑚 ∈ (0...𝑁)(ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)))))
288		elfznn0 13593	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ₀)
289	288	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℕ₀)
290		nn0mulcl 12507	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℕ₀ ∧ 𝑗 ∈ ℕ₀) → (2 · 𝑗) ∈ ℕ₀)
291	75, 289, 290	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑗) ∈ ℕ₀)
292	291	nn0zd 12583	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑗) ∈ ℤ)
293		bccl 14281	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ₀ ∧ (2 · 𝑗) ∈ ℤ) → (𝑁C(2 · 𝑗)) ∈ ℕ₀)
294	14, 292, 293	syl2an2r 683	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁C(2 · 𝑗)) ∈ ℕ₀)
295	294	nn0red 12532	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁C(2 · 𝑗)) ∈ ℝ)
296		fznn0sub 13532	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 − 𝑗) ∈ ℕ₀)
297	296	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 − 𝑗) ∈ ℕ₀)
298		reexpcl 14043	. . . . . . . . . . . . . 14 ⊢ ((-1 ∈ ℝ ∧ (𝑀 − 𝑗) ∈ ℕ₀) → (-1↑(𝑀 − 𝑗)) ∈ ℝ)
299	189, 297, 298	sylancr 587	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (-1↑(𝑀 − 𝑗)) ∈ ℝ)
300	295, 299	remulcld 11243	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) ∈ ℝ)
301		2z 12593	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
302		znegcl 12596	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ ℤ → -2 ∈ ℤ)
303	301, 302	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ -2 ∈ ℤ
304		rpexpcl 14045	. . . . . . . . . . . . . . 15 ⊢ (((tan‘𝐴) ∈ ℝ⁺ ∧ -2 ∈ ℤ) → ((tan‘𝐴)↑-2) ∈ ℝ⁺)
305	2, 303, 304	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((tan‘𝐴)↑-2) ∈ ℝ⁺)
306	305	rpred 13015	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((tan‘𝐴)↑-2) ∈ ℝ)
307		reexpcl 14043	. . . . . . . . . . . . 13 ⊢ ((((tan‘𝐴)↑-2) ∈ ℝ ∧ 𝑗 ∈ ℕ₀) → (((tan‘𝐴)↑-2)↑𝑗) ∈ ℝ)
308	306, 288, 307	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((tan‘𝐴)↑-2)↑𝑗) ∈ ℝ)
309	300, 308	remulcld 11243	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ ℝ)
310	309	recnd 11241	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ ℂ)
311		mulcl 11193	. . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ ℂ) → (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))) ∈ ℂ)
312	5, 310, 311	sylancr 587	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))) ∈ ℂ)
313	312	addlidd 11414	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (0 + (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))) = (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))))
314	294	nn0cnd 12533	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁C(2 · 𝑗)) ∈ ℂ)
315	299	recnd 11241	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (-1↑(𝑀 − 𝑗)) ∈ ℂ)
316	308	recnd 11241	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((tan‘𝐴)↑-2)↑𝑗) ∈ ℂ)
317	314, 315, 316	mulassd 11236	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) = ((𝑁C(2 · 𝑗)) · ((-1↑(𝑀 − 𝑗)) · (((tan‘𝐴)↑-2)↑𝑗))))
318	317	oveq2d 7424	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))) = (i · ((𝑁C(2 · 𝑗)) · ((-1↑(𝑀 − 𝑗)) · (((tan‘𝐴)↑-2)↑𝑗)))))
319	5	a1i 11	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → i ∈ ℂ)
320	315, 316	mulcld 11233	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((-1↑(𝑀 − 𝑗)) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ ℂ)
321	319, 314, 320	mul12d 11422	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i · ((𝑁C(2 · 𝑗)) · ((-1↑(𝑀 − 𝑗)) · (((tan‘𝐴)↑-2)↑𝑗)))) = ((𝑁C(2 · 𝑗)) · (i · ((-1↑(𝑀 − 𝑗)) · (((tan‘𝐴)↑-2)↑𝑗)))))
322	318, 321	eqtrd 2772	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))) = ((𝑁C(2 · 𝑗)) · (i · ((-1↑(𝑀 − 𝑗)) · (((tan‘𝐴)↑-2)↑𝑗)))))
323		bccmpl 14268	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ (2 · 𝑗) ∈ ℤ) → (𝑁C(2 · 𝑗)) = (𝑁C(𝑁 − (2 · 𝑗))))
324	14, 292, 323	syl2an2r 683	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁C(2 · 𝑗)) = (𝑁C(𝑁 − (2 · 𝑗))))
325	108	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑁 ∈ ℂ)
326	291	nn0cnd 12533	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑗) ∈ ℂ)
327	325, 326	nncand 11575	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁 − (𝑁 − (2 · 𝑗))) = (2 · 𝑗))
328	327	oveq2d 7424	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) = ((1 / (tan‘𝐴))↑(2 · 𝑗)))
329	2	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (tan‘𝐴) ∈ ℝ⁺)
330	329	rpcnd 13017	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (tan‘𝐴) ∈ ℂ)
331		expneg 14034	. . . . . . . . . . . . . 14 ⊢ (((tan‘𝐴) ∈ ℂ ∧ (2 · 𝑗) ∈ ℕ₀) → ((tan‘𝐴)↑-(2 · 𝑗)) = (1 / ((tan‘𝐴)↑(2 · 𝑗))))
332	330, 291, 331	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((tan‘𝐴)↑-(2 · 𝑗)) = (1 / ((tan‘𝐴)↑(2 · 𝑗))))
333	289	nn0cnd 12533	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℂ)
334		mulneg1 11649	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (-2 · 𝑗) = -(2 · 𝑗))
335	110, 333, 334	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (-2 · 𝑗) = -(2 · 𝑗))
336	335	oveq2d 7424	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((tan‘𝐴)↑(-2 · 𝑗)) = ((tan‘𝐴)↑-(2 · 𝑗)))
337	329	rpne0d 13020	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (tan‘𝐴) ≠ 0)
338	330, 337, 292	exprecd 14118	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((1 / (tan‘𝐴))↑(2 · 𝑗)) = (1 / ((tan‘𝐴)↑(2 · 𝑗))))
339	332, 336, 338	3eqtr4d 2782	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((tan‘𝐴)↑(-2 · 𝑗)) = ((1 / (tan‘𝐴))↑(2 · 𝑗)))
340	303	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → -2 ∈ ℤ)
341	289	nn0zd 12583	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℤ)
342		expmulz 14073	. . . . . . . . . . . . 13 ⊢ ((((tan‘𝐴) ∈ ℂ ∧ (tan‘𝐴) ≠ 0) ∧ (-2 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((tan‘𝐴)↑(-2 · 𝑗)) = (((tan‘𝐴)↑-2)↑𝑗))
343	330, 337, 340, 341, 342	syl22anc 837	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((tan‘𝐴)↑(-2 · 𝑗)) = (((tan‘𝐴)↑-2)↑𝑗))
344	328, 339, 343	3eqtr2d 2778	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) = (((tan‘𝐴)↑-2)↑𝑗))
345	7	oveq1i 7418	. . . . . . . . . . . . . . 15 ⊢ (𝑁 − (2 · 𝑗)) = (((2 · 𝑀) + 1) − (2 · 𝑗))
346	11	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑀) ∈ ℕ)
347	346	nncnd 12227	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑀) ∈ ℂ)
348		1cnd 11208	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 1 ∈ ℂ)
349	347, 348, 326	addsubd 11591	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((2 · 𝑀) + 1) − (2 · 𝑗)) = (((2 · 𝑀) − (2 · 𝑗)) + 1))
350		2cnd 12289	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 2 ∈ ℂ)
351	212	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑀 ∈ ℂ)
352	350, 351, 333	subdid 11669	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · (𝑀 − 𝑗)) = ((2 · 𝑀) − (2 · 𝑗)))
353	352	oveq1d 7423	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((2 · (𝑀 − 𝑗)) + 1) = (((2 · 𝑀) − (2 · 𝑗)) + 1))
354	349, 353	eqtr4d 2775	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((2 · 𝑀) + 1) − (2 · 𝑗)) = ((2 · (𝑀 − 𝑗)) + 1))
355	345, 354	eqtrid 2784	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁 − (2 · 𝑗)) = ((2 · (𝑀 − 𝑗)) + 1))
356	355	oveq2d 7424	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(𝑁 − (2 · 𝑗))) = (i↑((2 · (𝑀 − 𝑗)) + 1)))
357		nn0mulcl 12507	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ₀ ∧ (𝑀 − 𝑗) ∈ ℕ₀) → (2 · (𝑀 − 𝑗)) ∈ ℕ₀)
358	75, 297, 357	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · (𝑀 − 𝑗)) ∈ ℕ₀)
359		expp1 14033	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ (2 · (𝑀 − 𝑗)) ∈ ℕ₀) → (i↑((2 · (𝑀 − 𝑗)) + 1)) = ((i↑(2 · (𝑀 − 𝑗))) · i))
360	5, 358, 359	sylancr 587	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑((2 · (𝑀 − 𝑗)) + 1)) = ((i↑(2 · (𝑀 − 𝑗))) · i))
361	75	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 2 ∈ ℕ₀)
362	319, 297, 361	expmuld 14113	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(2 · (𝑀 − 𝑗))) = ((i↑2)↑(𝑀 − 𝑗)))
363	167	oveq1i 7418	. . . . . . . . . . . . . . 15 ⊢ ((i↑2)↑(𝑀 − 𝑗)) = (-1↑(𝑀 − 𝑗))
364	362, 363	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(2 · (𝑀 − 𝑗))) = (-1↑(𝑀 − 𝑗)))
365	364	oveq1d 7423	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((i↑(2 · (𝑀 − 𝑗))) · i) = ((-1↑(𝑀 − 𝑗)) · i))
366	356, 360, 365	3eqtrd 2776	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(𝑁 − (2 · 𝑗))) = ((-1↑(𝑀 − 𝑗)) · i))
367		mulcom 11195	. . . . . . . . . . . . 13 ⊢ (((-1↑(𝑀 − 𝑗)) ∈ ℂ ∧ i ∈ ℂ) → ((-1↑(𝑀 − 𝑗)) · i) = (i · (-1↑(𝑀 − 𝑗))))
368	315, 5, 367	sylancl 586	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((-1↑(𝑀 − 𝑗)) · i) = (i · (-1↑(𝑀 − 𝑗))))
369	366, 368	eqtrd 2772	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(𝑁 − (2 · 𝑗))) = (i · (-1↑(𝑀 − 𝑗))))
370	344, 369	oveq12d 7426	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗)))) = ((((tan‘𝐴)↑-2)↑𝑗) · (i · (-1↑(𝑀 − 𝑗)))))
371		mulcl 11193	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (-1↑(𝑀 − 𝑗)) ∈ ℂ) → (i · (-1↑(𝑀 − 𝑗))) ∈ ℂ)
372	5, 315, 371	sylancr 587	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i · (-1↑(𝑀 − 𝑗))) ∈ ℂ)
373	372, 316	mulcomd 11234	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((i · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) = ((((tan‘𝐴)↑-2)↑𝑗) · (i · (-1↑(𝑀 − 𝑗)))))
374	319, 315, 316	mulassd 11236	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((i · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) = (i · ((-1↑(𝑀 − 𝑗)) · (((tan‘𝐴)↑-2)↑𝑗))))
375	370, 373, 374	3eqtr2rd 2779	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i · ((-1↑(𝑀 − 𝑗)) · (((tan‘𝐴)↑-2)↑𝑗))) = (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗)))))
376	324, 375	oveq12d 7426	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑁C(2 · 𝑗)) · (i · ((-1↑(𝑀 − 𝑗)) · (((tan‘𝐴)↑-2)↑𝑗)))) = ((𝑁C(𝑁 − (2 · 𝑗))) · (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗))))))
377	313, 322, 376	3eqtrd 2776	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (0 + (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))) = ((𝑁C(𝑁 − (2 · 𝑗))) · (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗))))))
378	377	fveq2d 6895	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (ℑ‘(0 + (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))))) = (ℑ‘((𝑁C(𝑁 − (2 · 𝑗))) · (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗)))))))
379		0re 11215	. . . . . . 7 ⊢ 0 ∈ ℝ
380		crim 15061	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ ℝ) → (ℑ‘(0 + (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))))) = (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
381	379, 309, 380	sylancr 587	. . . . . 6 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (ℑ‘(0 + (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))))) = (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
382	378, 381	eqtr3d 2774	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (ℑ‘((𝑁C(𝑁 − (2 · 𝑗))) · (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗)))))) = (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
383	382	sumeq2dv 15648	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → Σ𝑗 ∈ (0...𝑀)(ℑ‘((𝑁C(𝑁 − (2 · 𝑗))) · (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗)))))) = Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
384	157, 287, 383	3eqtr3d 2780	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → Σ𝑚 ∈ (0...𝑁)(ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)))) = Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
385	286, 153	fsumim 15754	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (ℑ‘Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)))) = Σ𝑚 ∈ (0...𝑁)(ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)))))
386	305	rpcnd 13017	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((tan‘𝐴)↑-2) ∈ ℂ)
387		oveq1 7415	. . . . . . 7 ⊢ (𝑡 = ((tan‘𝐴)↑-2) → (𝑡↑𝑗) = (((tan‘𝐴)↑-2)↑𝑗))
388	387	oveq2d 7424	. . . . . 6 ⊢ (𝑡 = ((tan‘𝐴)↑-2) → (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗)) = (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
389	388	sumeq2sdv 15649	. . . . 5 ⊢ (𝑡 = ((tan‘𝐴)↑-2) → Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗)) = Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
390		basel.p	. . . . 5 ⊢ 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗)))
391		sumex 15633	. . . . 5 ⊢ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ V
392	389, 390, 391	fvmpt 6998	. . . 4 ⊢ (((tan‘𝐴)↑-2) ∈ ℂ → (𝑃‘((tan‘𝐴)↑-2)) = Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
393	386, 392	syl 17	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑃‘((tan‘𝐴)↑-2)) = Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
394	384, 385, 393	3eqtr4d 2782	. 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (ℑ‘Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)))) = (𝑃‘((tan‘𝐴)↑-2)))
395	51, 58	rerpdivcld 13046	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((cos‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) ∈ ℝ)
396	53, 58	rerpdivcld 13046	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) ∈ ℝ)
397	395, 396	crimd 15178	. 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (ℑ‘(((cos‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) + (i · ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁))))) = ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)))
398	66, 394, 397	3eqtr3d 2780	1 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑃‘((tan‘𝐴)↑-2)) = ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∖ cdif 3945 class class class wbr 5148 ↦ cmpt 5231 ran crn 5677 Fn wfn 6538 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7408 ℂcc 11107 ℝcr 11108 0cc0 11109 1c1 11110 ici 11111 + caddc 11112 · cmul 11114 < clt 11247 ≤ cle 11248 − cmin 11443 -cneg 11444 / cdiv 11870 ℕcn 12211 2c2 12266 ℕ₀cn0 12471 ℤcz 12557 ℤ_≥cuz 12821 ℝ⁺crp 12973 (,)cioo 13323 ...cfz 13483 ↑cexp 14026 Ccbc 14261 ℑcim 15044 Σcsu 15631 sincsin 16006 cosccos 16007 tanctan 16008 πcpi 16009

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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189

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-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ioc 13328 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-seq 13966 df-exp 14027 df-fac 14233 df-bc 14262 df-hash 14290 df-shft 15013 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-limsup 15414 df-clim 15431 df-rlim 15432 df-sum 15632 df-ef 16010 df-sin 16012 df-cos 16013 df-tan 16014 df-pi 16015 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-mulg 18950 df-cntz 19180 df-cmn 19649 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-lp 22639 df-perf 22640 df-cn 22730 df-cnp 22731 df-haus 22818 df-tx 23065 df-hmeo 23258 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-xms 23825 df-ms 23826 df-tms 23827 df-cncf 24393 df-limc 25382 df-dv 25383

This theorem is referenced by: basellem4 26585

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