basellem3 - Metamath Proof Explorer

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Theorem basellem3 24609

Description: Lemma for basel 24616. 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 24060	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(π / 2)) → (tan‘𝐴) ∈ ℝ⁺)
2	1	adantl 481	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (tan‘𝐴) ∈ ℝ⁺)
3	2	rpreccld 11758	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / (tan‘𝐴)) ∈ ℝ⁺)
4	3	rpcnd 11750	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / (tan‘𝐴)) ∈ ℂ)
5		ax-icn 9874	. . . . . 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 11062	. . . . . . . . 9 ⊢ 2 ∈ ℕ
9		simpl 472	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑀 ∈ ℕ)
10		nnmulcl 10920	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (2 · 𝑀) ∈ ℕ)
11	8, 9, 10	sylancr 694	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (2 · 𝑀) ∈ ℕ)
12	11	peano2nnd 10914	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((2 · 𝑀) + 1) ∈ ℕ)
13	7, 12	syl5eqel 2692	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℕ)
14	13	nnnn0d 11228	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℕ₀)
15		binom 14401	. . . . 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 1318	. . . 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 12076	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (0(,)(π / 2)) → 𝐴 ∈ ℝ)
18	17	adantl 481	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝐴 ∈ ℝ)
19	18	recoscld 14713	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘𝐴) ∈ ℝ)
20	19	recnd 9947	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘𝐴) ∈ ℂ)
21	18	resincld 14712	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘𝐴) ∈ ℝ)
22	21	recnd 9947	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘𝐴) ∈ ℂ)
23		mulcl 9899	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ (sin‘𝐴) ∈ ℂ) → (i · (sin‘𝐴)) ∈ ℂ)
24	5, 22, 23	sylancr 694	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (i · (sin‘𝐴)) ∈ ℂ)
25	20, 24	addcld 9938	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((cos‘𝐴) + (i · (sin‘𝐴))) ∈ ℂ)
26		sincosq1sgn 24054	. . . . . . . . . 10 ⊢ (𝐴 ∈ (0(,)(π / 2)) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))
27	26	adantl 481	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))
28	27	simpld 474	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 0 < (sin‘𝐴))
29	28	gt0ne0d 10471	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘𝐴) ≠ 0)
30	25, 22, 29, 14	expdivd 12884	. . . . . 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 10718	. . . . . . . 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 9947	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝐴 ∈ ℂ)
33	27	simprd 478	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 0 < (cos‘𝐴))
34	33	gt0ne0d 10471	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘𝐴) ≠ 0)
35		tanval 14697	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴)))
36	32, 34, 35	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴)))
37	36	oveq2d 6565	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / (tan‘𝐴)) = (1 / ((sin‘𝐴) / (cos‘𝐴))))
38	22, 20, 29, 34	recdivd 10697	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / ((sin‘𝐴) / (cos‘𝐴))) = ((cos‘𝐴) / (sin‘𝐴)))
39	37, 38	eqtr2d 2645	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((cos‘𝐴) / (sin‘𝐴)) = (1 / (tan‘𝐴)))
40	6, 22, 29	divcan4d 10686	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((i · (sin‘𝐴)) / (sin‘𝐴)) = i)
41	39, 40	oveq12d 6567	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((cos‘𝐴) / (sin‘𝐴)) + ((i · (sin‘𝐴)) / (sin‘𝐴))) = ((1 / (tan‘𝐴)) + i))
42	31, 41	eqtrd 2644	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((cos‘𝐴) + (i · (sin‘𝐴))) / (sin‘𝐴)) = ((1 / (tan‘𝐴)) + i))
43	42	oveq1d 6564	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((((cos‘𝐴) + (i · (sin‘𝐴))) / (sin‘𝐴))↑𝑁) = (((1 / (tan‘𝐴)) + i)↑𝑁))
44	13	nnzd 11357	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℤ)
45		demoivre 14769	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴)))))
46	32, 44, 45	syl2anc 691	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴)))))
47	46	oveq1d 6564	. . . . . 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 2652	. . . . 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 10912	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℝ)
50	49, 18	remulcld 9949	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑁 · 𝐴) ∈ ℝ)
51	50	recoscld 14713	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘(𝑁 · 𝐴)) ∈ ℝ)
52	51	recnd 9947	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘(𝑁 · 𝐴)) ∈ ℂ)
53	50	resincld 14712	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘(𝑁 · 𝐴)) ∈ ℝ)
54	53	recnd 9947	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘(𝑁 · 𝐴)) ∈ ℂ)
55		mulcl 9899	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ (sin‘(𝑁 · 𝐴)) ∈ ℂ) → (i · (sin‘(𝑁 · 𝐴))) ∈ ℂ)
56	5, 54, 55	sylancr 694	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (i · (sin‘(𝑁 · 𝐴))) ∈ ℂ)
57	21, 28	elrpd 11745	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘𝐴) ∈ ℝ⁺)
58	57, 44	rpexpcld 12894	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((sin‘𝐴)↑𝑁) ∈ ℝ⁺)
59	58	rpcnd 11750	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((sin‘𝐴)↑𝑁) ∈ ℂ)
60	58	rpne0d 11753	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((sin‘𝐴)↑𝑁) ≠ 0)
61	52, 56, 59, 60	divdird 10718	. . . . 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 10715	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((i · (sin‘(𝑁 · 𝐴))) / ((sin‘𝐴)↑𝑁)) = (i · ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁))))
63	62	oveq2d 6565	. . . . 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 2648	. . . 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 2646	. . 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 6107	. 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 6557	. . . . . . 7 ⊢ (𝑚 = (𝑁 − (2 · 𝑗)) → (𝑁C𝑚) = (𝑁C(𝑁 − (2 · 𝑗))))
68		oveq2 6557	. . . . . . . . 9 ⊢ (𝑚 = (𝑁 − (2 · 𝑗)) → (𝑁 − 𝑚) = (𝑁 − (𝑁 − (2 · 𝑗))))
69	68	oveq2d 6565	. . . . . . . 8 ⊢ (𝑚 = (𝑁 − (2 · 𝑗)) → ((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) = ((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))))
70		oveq2 6557	. . . . . . . 8 ⊢ (𝑚 = (𝑁 − (2 · 𝑗)) → (i↑𝑚) = (i↑(𝑁 − (2 · 𝑗))))
71	69, 70	oveq12d 6567	. . . . . . 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 6567	. . . . . 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 6107	. . . . 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 12634	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (0...𝑀) ∈ Fin)
75		2nn0 11186	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
76		elfznn0 12302	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝑀) → 𝑘 ∈ ℕ₀)
77	76	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ ℕ₀)
78		nn0mulcl 11206	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ₀ ∧ 𝑘 ∈ ℕ₀) → (2 · 𝑘) ∈ ℕ₀)
79	75, 77, 78	sylancr 694	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ∈ ℕ₀)
80	79	nn0red 11229	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ∈ ℝ)
81	11	nnred 10912	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (2 · 𝑀) ∈ ℝ)
82	81	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑀) ∈ ℝ)
83	49	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑁 ∈ ℝ)
84		elfzle2 12216	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝑀) → 𝑘 ≤ 𝑀)
85	84	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ≤ 𝑀)
86	77	nn0red 11229	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ ℝ)
87		nnre 10904	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
88	87	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑀 ∈ ℝ)
89		2re 10967	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
90		2pos 10989	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
91	89, 90	pm3.2i 470	. . . . . . . . . . . . . 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 10755	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝑘 ≤ 𝑀 ↔ (2 · 𝑘) ≤ (2 · 𝑀)))
94	86, 88, 92, 93	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (𝑘 ≤ 𝑀 ↔ (2 · 𝑘) ≤ (2 · 𝑀)))
95	85, 94	mpbid 221	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ≤ (2 · 𝑀))
96	82	lep1d 10834	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑀) ≤ ((2 · 𝑀) + 1))
97	96, 7	syl6breqr 4625	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑀) ≤ 𝑁)
98	80, 82, 83, 95, 97	letrd 10073	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ≤ 𝑁)
99		nn0uz 11598	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
100	79, 99	syl6eleq 2698	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ∈ (ℤ_≥‘0))
101	44	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑁 ∈ ℤ)
102		elfz5 12205	. . . . . . . . . . 11 ⊢ (((2 · 𝑘) ∈ (ℤ_≥‘0) ∧ 𝑁 ∈ ℤ) → ((2 · 𝑘) ∈ (0...𝑁) ↔ (2 · 𝑘) ≤ 𝑁))
103	100, 101, 102	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → ((2 · 𝑘) ∈ (0...𝑁) ↔ (2 · 𝑘) ≤ 𝑁))
104	98, 103	mpbird 246	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ∈ (0...𝑁))
105		fznn0sub2 12315	. . . . . . . . 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 449	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑘 ∈ (0...𝑀) → (𝑁 − (2 · 𝑘)) ∈ (0...𝑁)))
108	13	nncnd 10913	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℂ)
109	108	adantr 480	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → 𝑁 ∈ ℂ)
110		2cn 10968	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
111		elfzelz 12213	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝑀) → 𝑘 ∈ ℤ)
112	111	zcnd 11359	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...𝑀) → 𝑘 ∈ ℂ)
113	112	ad2antrl 760	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → 𝑘 ∈ ℂ)
114		mulcl 9899	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (2 · 𝑘) ∈ ℂ)
115	110, 113, 114	sylancr 694	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → (2 · 𝑘) ∈ ℂ)
116	112	ssriv 3572	. . . . . . . . . . . 12 ⊢ (0...𝑀) ⊆ ℂ
117		simprr 792	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → 𝑚 ∈ (0...𝑀))
118	116, 117	sseldi 3566	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → 𝑚 ∈ ℂ)
119		mulcl 9899	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (2 · 𝑚) ∈ ℂ)
120	110, 118, 119	sylancr 694	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → (2 · 𝑚) ∈ ℂ)
121	109, 115, 120	subcanad 10314	. . . . . . . . 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 11119	. . . . . . . . . . 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 10543	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℂ ∧ 𝑚 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((2 · 𝑘) = (2 · 𝑚) ↔ 𝑘 = 𝑚))
125	113, 118, 123, 124	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → ((2 · 𝑘) = (2 · 𝑚) ↔ 𝑘 = 𝑚))
126	121, 125	bitrd 267	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → ((𝑁 − (2 · 𝑘)) = (𝑁 − (2 · 𝑚)) ↔ 𝑘 = 𝑚))
127	126	ex 449	. . . . . . 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 7881	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))):(0...𝑀)–1-1→(0...𝑁))
129		f1f1orn 6061	. . . . . 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 6557	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (2 · 𝑘) = (2 · 𝑗))
132	131	oveq2d 6565	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑁 − (2 · 𝑘)) = (𝑁 − (2 · 𝑗)))
133		eqid 2610	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) = (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))
134		ovex 6577	. . . . . . 7 ⊢ (𝑁 − (2 · 𝑗)) ∈ V
135	132, 133, 134	fvmpt 6191	. . . . . 6 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘𝑗) = (𝑁 − (2 · 𝑗)))
136	135	adantl 481	. . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘𝑗) = (𝑁 − (2 · 𝑗)))
137		f1f 6014	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))):(0...𝑀)–1-1→(0...𝑁) → (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))):(0...𝑀)⟶(0...𝑁))
138	128, 137	syl 17	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))):(0...𝑀)⟶(0...𝑁))
139		frn 5966	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))):(0...𝑀)⟶(0...𝑁) → ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) ⊆ (0...𝑁))
140	138, 139	syl 17	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) ⊆ (0...𝑁))
141	140	sselda 3568	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) → 𝑚 ∈ (0...𝑁))
142		bccl2 12972	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...𝑁) → (𝑁C𝑚) ∈ ℕ)
143	142	adantl 481	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℕ)
144	143	nncnd 10913	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℂ)
145	2	rprecred 11759	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / (tan‘𝐴)) ∈ ℝ)
146		fznn0sub 12244	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...𝑁) → (𝑁 − 𝑚) ∈ ℕ₀)
147		reexpcl 12739	. . . . . . . . . . . 12 ⊢ (((1 / (tan‘𝐴)) ∈ ℝ ∧ (𝑁 − 𝑚) ∈ ℕ₀) → ((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) ∈ ℝ)
148	145, 146, 147	syl2an 493	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) ∈ ℝ)
149	148	recnd 9947	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) ∈ ℂ)
150		elfznn0 12302	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℕ₀)
151	150	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → 𝑚 ∈ ℕ₀)
152		expcl 12740	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (i↑𝑚) ∈ ℂ)
153	5, 151, 152	sylancr 694	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (i↑𝑚) ∈ ℂ)
154	149, 153	mulcld 9939	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)) ∈ ℂ)
155	144, 154	mulcld 9939	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚))) ∈ ℂ)
156	141, 155	syldan 486	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) → ((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚))) ∈ ℂ)
157	156	imcld 13783	. . . . . 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)
158	157	recnd 9947	. . . . 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)
159	73, 74, 130, 136, 158	fsumf1o 14301	. . . 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)))))))
160		eldifi 3694	. . . . . . . 8 ⊢ (𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) → 𝑚 ∈ (0...𝑁))
161	143	nnred 10912	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℝ)
162	160, 161	sylan2 490	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → (𝑁C𝑚) ∈ ℝ)
163	160, 148	sylan2 490	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → ((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) ∈ ℝ)
164		eldif 3550	. . . . . . . . 9 ⊢ (𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) ↔ (𝑚 ∈ (0...𝑁) ∧ ¬ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))))
165		elfzelz 12213	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ)
166	165	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → 𝑚 ∈ ℤ)
167		zeo 11339	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℤ → ((𝑚 / 2) ∈ ℤ ∨ ((𝑚 + 1) / 2) ∈ ℤ))
168	166, 167	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑚 / 2) ∈ ℤ ∨ ((𝑚 + 1) / 2) ∈ ℤ))
169		i2 12827	. . . . . . . . . . . . . . . . . 18 ⊢ (i↑2) = -1
170	169	oveq1i 6559	. . . . . . . . . . . . . . . . 17 ⊢ ((i↑2)↑(𝑚 / 2)) = (-1↑(𝑚 / 2))
171		simprr 792	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (𝑚 / 2) ∈ ℤ)
172	150	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → 𝑚 ∈ ℕ₀)
173		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℝ)
174		nn0ge0 11195	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ ℕ₀ → 0 ≤ 𝑚)
175		divge0 10771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ (𝑚 / 2))
176	89, 90, 175	mpanr12 717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) → 0 ≤ (𝑚 / 2))
177	173, 174, 176	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ ℕ₀ → 0 ≤ (𝑚 / 2))
178	172, 177	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → 0 ≤ (𝑚 / 2))
179		elnn0z 11267	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 / 2) ∈ ℕ₀ ↔ ((𝑚 / 2) ∈ ℤ ∧ 0 ≤ (𝑚 / 2)))
180	171, 178, 179	sylanbrc 695	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (𝑚 / 2) ∈ ℕ₀)
181		expmul 12767	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((i ∈ ℂ ∧ 2 ∈ ℕ₀ ∧ (𝑚 / 2) ∈ ℕ₀) → (i↑(2 · (𝑚 / 2))) = ((i↑2)↑(𝑚 / 2)))
182	5, 75, 181	mp3an12 1406	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 / 2) ∈ ℕ₀ → (i↑(2 · (𝑚 / 2))) = ((i↑2)↑(𝑚 / 2)))
183	180, 182	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (i↑(2 · (𝑚 / 2))) = ((i↑2)↑(𝑚 / 2)))
184	172	nn0cnd 11230	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → 𝑚 ∈ ℂ)
185		2ne0 10990	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ≠ 0
186		divcan2 10572	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (2 · (𝑚 / 2)) = 𝑚)
187	110, 185, 186	mp3an23 1408	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℂ → (2 · (𝑚 / 2)) = 𝑚)
188	184, 187	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (2 · (𝑚 / 2)) = 𝑚)
189	188	oveq2d 6565	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (i↑(2 · (𝑚 / 2))) = (i↑𝑚))
190	183, 189	eqtr3d 2646	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → ((i↑2)↑(𝑚 / 2)) = (i↑𝑚))
191	170, 190	syl5eqr 2658	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (-1↑(𝑚 / 2)) = (i↑𝑚))
192		neg1rr 11002	. . . . . . . . . . . . . . . . 17 ⊢ -1 ∈ ℝ
193		reexpcl 12739	. . . . . . . . . . . . . . . . 17 ⊢ ((-1 ∈ ℝ ∧ (𝑚 / 2) ∈ ℕ₀) → (-1↑(𝑚 / 2)) ∈ ℝ)
194	192, 180, 193	sylancr 694	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (-1↑(𝑚 / 2)) ∈ ℝ)
195	191, 194	eqeltrrd 2689	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (i↑𝑚) ∈ ℝ)
196	195	expr 641	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑚 / 2) ∈ ℤ → (i↑𝑚) ∈ ℝ))
197	146	ad2antrl 760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − 𝑚) ∈ ℕ₀)
198		nn0re 11178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 − 𝑚) ∈ ℕ₀ → (𝑁 − 𝑚) ∈ ℝ)
199		nn0ge0 11195	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 − 𝑚) ∈ ℕ₀ → 0 ≤ (𝑁 − 𝑚))
200		divge0 10771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 − 𝑚) ∈ ℝ ∧ 0 ≤ (𝑁 − 𝑚)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ ((𝑁 − 𝑚) / 2))
201	89, 90, 200	mpanr12 717	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 − 𝑚) ∈ ℝ ∧ 0 ≤ (𝑁 − 𝑚)) → 0 ≤ ((𝑁 − 𝑚) / 2))
202	198, 199, 201	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 − 𝑚) ∈ ℕ₀ → 0 ≤ ((𝑁 − 𝑚) / 2))
203	197, 202	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 0 ≤ ((𝑁 − 𝑚) / 2))
204	197	nn0red 11229	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − 𝑚) ∈ ℝ)
205	49	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑁 ∈ ℝ)
206		peano2re 10088	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
207	205, 206	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 + 1) ∈ ℝ)
208	150	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑚 ∈ ℕ₀)
209	208, 174	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 0 ≤ 𝑚)
210	208	nn0red 11229	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑚 ∈ ℝ)
211	205, 210	subge02d 10498	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (0 ≤ 𝑚 ↔ (𝑁 − 𝑚) ≤ 𝑁))
212	209, 211	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − 𝑚) ≤ 𝑁)
213	205	ltp1d 10833	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑁 < (𝑁 + 1))
214	204, 205, 207, 212, 213	lelttrd 10074	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − 𝑚) < (𝑁 + 1))
215		2t1e2 11053	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (2 · 1) = 2
216		df-2 10956	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 = (1 + 1)
217	215, 216	eqtr2i 2633	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 + 1) = (2 · 1)
218	217	oveq2i 6560	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((2 · 𝑀) + (1 + 1)) = ((2 · 𝑀) + (2 · 1))
219	7	oveq1i 6559	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 + 1) = (((2 · 𝑀) + 1) + 1)
220	11	nncnd 10913	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (2 · 𝑀) ∈ ℂ)
221	220	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 · 𝑀) ∈ ℂ)
222		1cnd 9935	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 1 ∈ ℂ)
223	221, 222, 222	addassd 9941	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (((2 · 𝑀) + 1) + 1) = ((2 · 𝑀) + (1 + 1)))
224	219, 223	syl5eq 2656	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 + 1) = ((2 · 𝑀) + (1 + 1)))
225		2cnd 10970	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 2 ∈ ℂ)
226		nncn 10905	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
227	226	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑀 ∈ ℂ)
228	225, 227, 222	adddid 9943	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 · (𝑀 + 1)) = ((2 · 𝑀) + (2 · 1)))
229	218, 224, 228	3eqtr4a 2670	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 + 1) = (2 · (𝑀 + 1)))
230	214, 229	breqtrd 4609	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − 𝑚) < (2 · (𝑀 + 1)))
231		nnz 11276	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
232	231	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑀 ∈ ℤ)
233	232	peano2zd 11361	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑀 + 1) ∈ ℤ)
234	233	zred 11358	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑀 + 1) ∈ ℝ)
235	91	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 ∈ ℝ ∧ 0 < 2))
236		ltdivmul 10777	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 − 𝑚) ∈ ℝ ∧ (𝑀 + 1) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((𝑁 − 𝑚) / 2) < (𝑀 + 1) ↔ (𝑁 − 𝑚) < (2 · (𝑀 + 1))))
237	204, 234, 235, 236	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (((𝑁 − 𝑚) / 2) < (𝑀 + 1) ↔ (𝑁 − 𝑚) < (2 · (𝑀 + 1))))
238	230, 237	mpbird 246	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 − 𝑚) / 2) < (𝑀 + 1))
239	108	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑁 ∈ ℂ)
240	208	nn0cnd 11230	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑚 ∈ ℂ)
241	239, 240, 222	pnpcan2d 10309	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 + 1) − (𝑚 + 1)) = (𝑁 − 𝑚))
242	229	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 + 1) − (𝑚 + 1)) = ((2 · (𝑀 + 1)) − (𝑚 + 1)))
243	241, 242	eqtr3d 2646	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − 𝑚) = ((2 · (𝑀 + 1)) − (𝑚 + 1)))
244	243	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 − 𝑚) / 2) = (((2 · (𝑀 + 1)) − (𝑚 + 1)) / 2))
245	233	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑀 + 1) ∈ ℂ)
246		mulcl 9899	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((2 ∈ ℂ ∧ (𝑀 + 1) ∈ ℂ) → (2 · (𝑀 + 1)) ∈ ℂ)
247	110, 245, 246	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 · (𝑀 + 1)) ∈ ℂ)
248		peano2cn 10087	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℂ → (𝑚 + 1) ∈ ℂ)
249	240, 248	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑚 + 1) ∈ ℂ)
250	122	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 ∈ ℂ ∧ 2 ≠ 0))
251		divsubdir 10600	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((2 · (𝑀 + 1)) ∈ ℂ ∧ (𝑚 + 1) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((2 · (𝑀 + 1)) − (𝑚 + 1)) / 2) = (((2 · (𝑀 + 1)) / 2) − ((𝑚 + 1) / 2)))
252	247, 249, 250, 251	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((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)))
253	185	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 2 ≠ 0)
254	245, 225, 253	divcan3d 10685	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((2 · (𝑀 + 1)) / 2) = (𝑀 + 1))
255	254	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((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)))
256	244, 252, 255	3eqtrd 2648	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 − 𝑚) / 2) = ((𝑀 + 1) − ((𝑚 + 1) / 2)))
257		simprr 792	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑚 + 1) / 2) ∈ ℤ)
258	233, 257	zsubcld 11363	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑀 + 1) − ((𝑚 + 1) / 2)) ∈ ℤ)
259	256, 258	eqeltrd 2688	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 − 𝑚) / 2) ∈ ℤ)
260		zleltp1 11305	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 − 𝑚) / 2) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (((𝑁 − 𝑚) / 2) ≤ 𝑀 ↔ ((𝑁 − 𝑚) / 2) < (𝑀 + 1)))
261	259, 232, 260	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (((𝑁 − 𝑚) / 2) ≤ 𝑀 ↔ ((𝑁 − 𝑚) / 2) < (𝑀 + 1)))
262	238, 261	mpbird 246	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 − 𝑚) / 2) ≤ 𝑀)
263		0zd 11266	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 0 ∈ ℤ)
264		elfz 12203	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 − 𝑚) / 2) ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (((𝑁 − 𝑚) / 2) ∈ (0...𝑀) ↔ (0 ≤ ((𝑁 − 𝑚) / 2) ∧ ((𝑁 − 𝑚) / 2) ≤ 𝑀)))
265	259, 263, 232, 264	syl3anc 1318	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (((𝑁 − 𝑚) / 2) ∈ (0...𝑀) ↔ (0 ≤ ((𝑁 − 𝑚) / 2) ∧ ((𝑁 − 𝑚) / 2) ≤ 𝑀)))
266	203, 262, 265	mpbir2and 959	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 − 𝑚) / 2) ∈ (0...𝑀))
267		oveq2 6557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = ((𝑁 − 𝑚) / 2) → (2 · 𝑘) = (2 · ((𝑁 − 𝑚) / 2)))
268	267	oveq2d 6565	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = ((𝑁 − 𝑚) / 2) → (𝑁 − (2 · 𝑘)) = (𝑁 − (2 · ((𝑁 − 𝑚) / 2))))
269		ovex 6577	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 − (2 · ((𝑁 − 𝑚) / 2))) ∈ V
270	268, 133, 269	fvmpt 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 − 𝑚) / 2) ∈ (0...𝑀) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘((𝑁 − 𝑚) / 2)) = (𝑁 − (2 · ((𝑁 − 𝑚) / 2))))
271	266, 270	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))))
272	197	nn0cnd 11230	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − 𝑚) ∈ ℂ)
273	272, 225, 253	divcan2d 10682	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 · ((𝑁 − 𝑚) / 2)) = (𝑁 − 𝑚))
274	273	oveq2d 6565	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − (2 · ((𝑁 − 𝑚) / 2))) = (𝑁 − (𝑁 − 𝑚)))
275	239, 240	nncand 10276	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − (𝑁 − 𝑚)) = 𝑚)
276	271, 274, 275	3eqtrd 2648	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘((𝑁 − 𝑚) / 2)) = 𝑚)
277		ffn 5958	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))):(0...𝑀)⟶(0...𝑁) → (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) Fn (0...𝑀))
278	138, 277	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) Fn (0...𝑀))
279	278	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) Fn (0...𝑀))
280		fnfvelrn 6264	. . . . . . . . . . . . . . . . 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))))
281	279, 266, 280	syl2anc 691	. . . . . . . . . . . . . . . 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))))
282	276, 281	eqeltrrd 2689	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))
283	282	expr 641	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (((𝑚 + 1) / 2) ∈ ℤ → 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))))
284	196, 283	orim12d 879	. . . . . . . . . . . . 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))))))
285	168, 284	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((i↑𝑚) ∈ ℝ ∨ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))))
286	285	orcomd 402	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) ∨ (i↑𝑚) ∈ ℝ))
287	286	ord 391	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (¬ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) → (i↑𝑚) ∈ ℝ))
288	287	impr 647	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ¬ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → (i↑𝑚) ∈ ℝ)
289	164, 288	sylan2b 491	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → (i↑𝑚) ∈ ℝ)
290	163, 289	remulcld 9949	. . . . . . 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)
291	162, 290	remulcld 9949	. . . . . 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)
292	291	reim0d 13813	. . . . 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)
293		fzfid 12634	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (0...𝑁) ∈ Fin)
294	140, 158, 292, 293	fsumss 14303	. . . 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)))))
295	14	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑁 ∈ ℕ₀)
296		elfznn0 12302	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ₀)
297	296	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℕ₀)
298		nn0mulcl 11206	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℕ₀ ∧ 𝑗 ∈ ℕ₀) → (2 · 𝑗) ∈ ℕ₀)
299	75, 297, 298	sylancr 694	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑗) ∈ ℕ₀)
300	299	nn0zd 11356	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑗) ∈ ℤ)
301		bccl 12971	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ₀ ∧ (2 · 𝑗) ∈ ℤ) → (𝑁C(2 · 𝑗)) ∈ ℕ₀)
302	295, 300, 301	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁C(2 · 𝑗)) ∈ ℕ₀)
303	302	nn0red 11229	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁C(2 · 𝑗)) ∈ ℝ)
304		fznn0sub 12244	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 − 𝑗) ∈ ℕ₀)
305	304	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 − 𝑗) ∈ ℕ₀)
306		reexpcl 12739	. . . . . . . . . . . . . 14 ⊢ ((-1 ∈ ℝ ∧ (𝑀 − 𝑗) ∈ ℕ₀) → (-1↑(𝑀 − 𝑗)) ∈ ℝ)
307	192, 305, 306	sylancr 694	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (-1↑(𝑀 − 𝑗)) ∈ ℝ)
308	303, 307	remulcld 9949	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) ∈ ℝ)
309		2z 11286	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
310		znegcl 11289	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ ℤ → -2 ∈ ℤ)
311	309, 310	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ -2 ∈ ℤ
312		rpexpcl 12741	. . . . . . . . . . . . . . 15 ⊢ (((tan‘𝐴) ∈ ℝ⁺ ∧ -2 ∈ ℤ) → ((tan‘𝐴)↑-2) ∈ ℝ⁺)
313	2, 311, 312	sylancl 693	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((tan‘𝐴)↑-2) ∈ ℝ⁺)
314	313	rpred 11748	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((tan‘𝐴)↑-2) ∈ ℝ)
315		reexpcl 12739	. . . . . . . . . . . . 13 ⊢ ((((tan‘𝐴)↑-2) ∈ ℝ ∧ 𝑗 ∈ ℕ₀) → (((tan‘𝐴)↑-2)↑𝑗) ∈ ℝ)
316	314, 296, 315	syl2an 493	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((tan‘𝐴)↑-2)↑𝑗) ∈ ℝ)
317	308, 316	remulcld 9949	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ ℝ)
318	317	recnd 9947	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ ℂ)
319		mulcl 9899	. . . . . . . . . 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)
320	5, 318, 319	sylancr 694	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))) ∈ ℂ)
321	320	addid2d 10116	. . . . . . . 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))))
322	302	nn0cnd 11230	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁C(2 · 𝑗)) ∈ ℂ)
323	307	recnd 9947	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (-1↑(𝑀 − 𝑗)) ∈ ℂ)
324	316	recnd 9947	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((tan‘𝐴)↑-2)↑𝑗) ∈ ℂ)
325	322, 323, 324	mulassd 9942	. . . . . . . . . 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))))
326	325	oveq2d 6565	. . . . . . . . 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)))))
327	5	a1i 11	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → i ∈ ℂ)
328	323, 324	mulcld 9939	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((-1↑(𝑀 − 𝑗)) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ ℂ)
329	327, 322, 328	mul12d 10124	. . . . . . . . 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)))))
330	326, 329	eqtrd 2644	. . . . . . . 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)))))
331		bccmpl 12958	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ (2 · 𝑗) ∈ ℤ) → (𝑁C(2 · 𝑗)) = (𝑁C(𝑁 − (2 · 𝑗))))
332	295, 300, 331	syl2anc 691	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁C(2 · 𝑗)) = (𝑁C(𝑁 − (2 · 𝑗))))
333	108	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑁 ∈ ℂ)
334	299	nn0cnd 11230	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑗) ∈ ℂ)
335	333, 334	nncand 10276	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁 − (𝑁 − (2 · 𝑗))) = (2 · 𝑗))
336	335	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) = ((1 / (tan‘𝐴))↑(2 · 𝑗)))
337	2	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (tan‘𝐴) ∈ ℝ⁺)
338	337	rpcnd 11750	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (tan‘𝐴) ∈ ℂ)
339		expneg 12730	. . . . . . . . . . . . . 14 ⊢ (((tan‘𝐴) ∈ ℂ ∧ (2 · 𝑗) ∈ ℕ₀) → ((tan‘𝐴)↑-(2 · 𝑗)) = (1 / ((tan‘𝐴)↑(2 · 𝑗))))
340	338, 299, 339	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((tan‘𝐴)↑-(2 · 𝑗)) = (1 / ((tan‘𝐴)↑(2 · 𝑗))))
341	297	nn0cnd 11230	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℂ)
342		mulneg1 10345	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (-2 · 𝑗) = -(2 · 𝑗))
343	110, 341, 342	sylancr 694	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (-2 · 𝑗) = -(2 · 𝑗))
344	343	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((tan‘𝐴)↑(-2 · 𝑗)) = ((tan‘𝐴)↑-(2 · 𝑗)))
345	337	rpne0d 11753	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (tan‘𝐴) ≠ 0)
346	338, 345, 300	exprecd 12878	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((1 / (tan‘𝐴))↑(2 · 𝑗)) = (1 / ((tan‘𝐴)↑(2 · 𝑗))))
347	340, 344, 346	3eqtr4d 2654	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((tan‘𝐴)↑(-2 · 𝑗)) = ((1 / (tan‘𝐴))↑(2 · 𝑗)))
348	311	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → -2 ∈ ℤ)
349	297	nn0zd 11356	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℤ)
350		expmulz 12768	. . . . . . . . . . . . 13 ⊢ ((((tan‘𝐴) ∈ ℂ ∧ (tan‘𝐴) ≠ 0) ∧ (-2 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((tan‘𝐴)↑(-2 · 𝑗)) = (((tan‘𝐴)↑-2)↑𝑗))
351	338, 345, 348, 349, 350	syl22anc 1319	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((tan‘𝐴)↑(-2 · 𝑗)) = (((tan‘𝐴)↑-2)↑𝑗))
352	336, 347, 351	3eqtr2d 2650	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) = (((tan‘𝐴)↑-2)↑𝑗))
353	7	oveq1i 6559	. . . . . . . . . . . . . . 15 ⊢ (𝑁 − (2 · 𝑗)) = (((2 · 𝑀) + 1) − (2 · 𝑗))
354	11	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑀) ∈ ℕ)
355	354	nncnd 10913	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑀) ∈ ℂ)
356		1cnd 9935	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 1 ∈ ℂ)
357	355, 356, 334	addsubd 10292	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((2 · 𝑀) + 1) − (2 · 𝑗)) = (((2 · 𝑀) − (2 · 𝑗)) + 1))
358		2cnd 10970	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 2 ∈ ℂ)
359	226	ad2antrr 758	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑀 ∈ ℂ)
360	358, 359, 341	subdid 10365	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · (𝑀 − 𝑗)) = ((2 · 𝑀) − (2 · 𝑗)))
361	360	oveq1d 6564	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((2 · (𝑀 − 𝑗)) + 1) = (((2 · 𝑀) − (2 · 𝑗)) + 1))
362	357, 361	eqtr4d 2647	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((2 · 𝑀) + 1) − (2 · 𝑗)) = ((2 · (𝑀 − 𝑗)) + 1))
363	353, 362	syl5eq 2656	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁 − (2 · 𝑗)) = ((2 · (𝑀 − 𝑗)) + 1))
364	363	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(𝑁 − (2 · 𝑗))) = (i↑((2 · (𝑀 − 𝑗)) + 1)))
365		nn0mulcl 11206	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ₀ ∧ (𝑀 − 𝑗) ∈ ℕ₀) → (2 · (𝑀 − 𝑗)) ∈ ℕ₀)
366	75, 305, 365	sylancr 694	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · (𝑀 − 𝑗)) ∈ ℕ₀)
367		expp1 12729	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ (2 · (𝑀 − 𝑗)) ∈ ℕ₀) → (i↑((2 · (𝑀 − 𝑗)) + 1)) = ((i↑(2 · (𝑀 − 𝑗))) · i))
368	5, 366, 367	sylancr 694	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑((2 · (𝑀 − 𝑗)) + 1)) = ((i↑(2 · (𝑀 − 𝑗))) · i))
369	75	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 2 ∈ ℕ₀)
370	327, 305, 369	expmuld 12873	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(2 · (𝑀 − 𝑗))) = ((i↑2)↑(𝑀 − 𝑗)))
371	169	oveq1i 6559	. . . . . . . . . . . . . . 15 ⊢ ((i↑2)↑(𝑀 − 𝑗)) = (-1↑(𝑀 − 𝑗))
372	370, 371	syl6eq 2660	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(2 · (𝑀 − 𝑗))) = (-1↑(𝑀 − 𝑗)))
373	372	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((i↑(2 · (𝑀 − 𝑗))) · i) = ((-1↑(𝑀 − 𝑗)) · i))
374	364, 368, 373	3eqtrd 2648	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(𝑁 − (2 · 𝑗))) = ((-1↑(𝑀 − 𝑗)) · i))
375		mulcom 9901	. . . . . . . . . . . . 13 ⊢ (((-1↑(𝑀 − 𝑗)) ∈ ℂ ∧ i ∈ ℂ) → ((-1↑(𝑀 − 𝑗)) · i) = (i · (-1↑(𝑀 − 𝑗))))
376	323, 5, 375	sylancl 693	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((-1↑(𝑀 − 𝑗)) · i) = (i · (-1↑(𝑀 − 𝑗))))
377	374, 376	eqtrd 2644	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(𝑁 − (2 · 𝑗))) = (i · (-1↑(𝑀 − 𝑗))))
378	352, 377	oveq12d 6567	. . . . . . . . . 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)))))
379		mulcl 9899	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (-1↑(𝑀 − 𝑗)) ∈ ℂ) → (i · (-1↑(𝑀 − 𝑗))) ∈ ℂ)
380	5, 323, 379	sylancr 694	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i · (-1↑(𝑀 − 𝑗))) ∈ ℂ)
381	380, 324	mulcomd 9940	. . . . . . . . . 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)))))
382	327, 323, 324	mulassd 9942	. . . . . . . . . 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))))
383	378, 381, 382	3eqtr2rd 2651	. . . . . . . . 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)))))
384	332, 383	oveq12d 6567	. . . . . . . 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))))))
385	321, 330, 384	3eqtrd 2648	. . . . . . 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))))))
386	385	fveq2d 6107	. . . . . 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)))))))
387		0re 9919	. . . . . . 7 ⊢ 0 ∈ ℝ
388		crim 13703	. . . . . . 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)))
389	387, 317, 388	sylancr 694	. . . . . 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)))
390	386, 389	eqtr3d 2646	. . . . 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)))
391	390	sumeq2dv 14281	. . . 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)))
392	159, 294, 391	3eqtr3d 2652	. . 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)))
393	293, 155	fsumim 14382	. . 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)))))
394	313	rpcnd 11750	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((tan‘𝐴)↑-2) ∈ ℂ)
395		oveq1 6556	. . . . . . 7 ⊢ (𝑡 = ((tan‘𝐴)↑-2) → (𝑡↑𝑗) = (((tan‘𝐴)↑-2)↑𝑗))
396	395	oveq2d 6565	. . . . . 6 ⊢ (𝑡 = ((tan‘𝐴)↑-2) → (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗)) = (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
397	396	sumeq2sdv 14282	. . . . 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)))
398		basel.p	. . . . 5 ⊢ 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗)))
399		sumex 14266	. . . . 5 ⊢ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ V
400	397, 398, 399	fvmpt 6191	. . . 4 ⊢ (((tan‘𝐴)↑-2) ∈ ℂ → (𝑃‘((tan‘𝐴)↑-2)) = Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
401	394, 400	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)))
402	392, 393, 401	3eqtr4d 2654	. 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (ℑ‘Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁 − 𝑚)) · (i↑𝑚)))) = (𝑃‘((tan‘𝐴)↑-2)))
403	51, 58	rerpdivcld 11779	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((cos‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) ∈ ℝ)
404	53, 58	rerpdivcld 11779	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) ∈ ℝ)
405	403, 404	crimd 13820	. 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)))
406	66, 402, 405	3eqtr3d 2652	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 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 ici 9817 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ℝ⁺crp 11708 (,)cioo 12046 ...cfz 12197 ↑cexp 12722 Ccbc 12951 ℑcim 13686 Σcsu 14264 sincsin 14633 cosccos 14634 tanctan 14635 πcpi 14636

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-tan 14641 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437

This theorem is referenced by: basellem4 24610

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