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Theorem basellem3 27060
Description: Lemma for basel 27067. Using the binomial theorem and de Moivre's formula, we have the identity e↑i𝑁𝑥 / (sin𝑥)↑𝑛 = Σ𝑚 ∈ (0...𝑁)(𝑁C𝑚)(i↑𝑚)(cot𝑥)↑(𝑁𝑚), so taking imaginary parts yields sin(𝑁𝑥) / (sin𝑥)↑𝑁 = Σ𝑗 ∈ (0...𝑀)(𝑁C2𝑗)(-1)↑(𝑀𝑗) (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↑(𝑀𝑗))) · (𝑡𝑗)))
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.
StepHypRef Expression
1 tanrpcl 26484 . . . . . . . 8 (𝐴 ∈ (0(,)(π / 2)) → (tan‘𝐴) ∈ ℝ+)
21adantl 480 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (tan‘𝐴) ∈ ℝ+)
32rpreccld 13061 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / (tan‘𝐴)) ∈ ℝ+)
43rpcnd 13053 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / (tan‘𝐴)) ∈ ℂ)
5 ax-icn 11199 . . . . . 6 i ∈ ℂ
65a1i 11 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → i ∈ ℂ)
7 basel.n . . . . . . 7 𝑁 = ((2 · 𝑀) + 1)
8 2nn 12318 . . . . . . . . 9 2 ∈ ℕ
9 simpl 481 . . . . . . . . 9 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑀 ∈ ℕ)
10 nnmulcl 12269 . . . . . . . . 9 ((2 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (2 · 𝑀) ∈ ℕ)
118, 9, 10sylancr 585 . . . . . . . 8 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (2 · 𝑀) ∈ ℕ)
1211peano2nnd 12262 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((2 · 𝑀) + 1) ∈ ℕ)
137, 12eqeltrid 2829 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℕ)
1413nnnn0d 12565 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℕ0)
15 binom 15812 . . . . 5 (((1 / (tan‘𝐴)) ∈ ℂ ∧ i ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (((1 / (tan‘𝐴)) + i)↑𝑁) = Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚))))
164, 6, 14, 15syl3anc 1368 . . . 4 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((1 / (tan‘𝐴)) + i)↑𝑁) = Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚))))
17 elioore 13389 . . . . . . . . . . 11 (𝐴 ∈ (0(,)(π / 2)) → 𝐴 ∈ ℝ)
1817adantl 480 . . . . . . . . . 10 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝐴 ∈ ℝ)
1918recoscld 16124 . . . . . . . . 9 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘𝐴) ∈ ℝ)
2019recnd 11274 . . . . . . . 8 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘𝐴) ∈ ℂ)
2118resincld 16123 . . . . . . . . . 10 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘𝐴) ∈ ℝ)
2221recnd 11274 . . . . . . . . 9 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘𝐴) ∈ ℂ)
23 mulcl 11224 . . . . . . . . 9 ((i ∈ ℂ ∧ (sin‘𝐴) ∈ ℂ) → (i · (sin‘𝐴)) ∈ ℂ)
245, 22, 23sylancr 585 . . . . . . . 8 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (i · (sin‘𝐴)) ∈ ℂ)
2520, 24addcld 11265 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((cos‘𝐴) + (i · (sin‘𝐴))) ∈ ℂ)
26 sincosq1sgn 26478 . . . . . . . . . 10 (𝐴 ∈ (0(,)(π / 2)) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))
2726adantl 480 . . . . . . . . 9 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))
2827simpld 493 . . . . . . . 8 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 0 < (sin‘𝐴))
2928gt0ne0d 11810 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘𝐴) ≠ 0)
3025, 22, 29, 14expdivd 14160 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((((cos‘𝐴) + (i · (sin‘𝐴))) / (sin‘𝐴))↑𝑁) = ((((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) / ((sin‘𝐴)↑𝑁)))
3120, 24, 22, 29divdird 12061 . . . . . . . 8 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((cos‘𝐴) + (i · (sin‘𝐴))) / (sin‘𝐴)) = (((cos‘𝐴) / (sin‘𝐴)) + ((i · (sin‘𝐴)) / (sin‘𝐴))))
3218recnd 11274 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝐴 ∈ ℂ)
3327simprd 494 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 0 < (cos‘𝐴))
3433gt0ne0d 11810 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘𝐴) ≠ 0)
35 tanval 16108 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴)))
3632, 34, 35syl2anc 582 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴)))
3736oveq2d 7435 . . . . . . . . . 10 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / (tan‘𝐴)) = (1 / ((sin‘𝐴) / (cos‘𝐴))))
3822, 20, 29, 34recdivd 12040 . . . . . . . . . 10 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / ((sin‘𝐴) / (cos‘𝐴))) = ((cos‘𝐴) / (sin‘𝐴)))
3937, 38eqtr2d 2766 . . . . . . . . 9 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((cos‘𝐴) / (sin‘𝐴)) = (1 / (tan‘𝐴)))
406, 22, 29divcan4d 12029 . . . . . . . . 9 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((i · (sin‘𝐴)) / (sin‘𝐴)) = i)
4139, 40oveq12d 7437 . . . . . . . 8 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((cos‘𝐴) / (sin‘𝐴)) + ((i · (sin‘𝐴)) / (sin‘𝐴))) = ((1 / (tan‘𝐴)) + i))
4231, 41eqtrd 2765 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((cos‘𝐴) + (i · (sin‘𝐴))) / (sin‘𝐴)) = ((1 / (tan‘𝐴)) + i))
4342oveq1d 7434 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((((cos‘𝐴) + (i · (sin‘𝐴))) / (sin‘𝐴))↑𝑁) = (((1 / (tan‘𝐴)) + i)↑𝑁))
4413nnzd 12618 . . . . . . . 8 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℤ)
45 demoivre 16180 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴)))))
4632, 44, 45syl2anc 582 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((cos‘𝐴) + (i · (sin‘𝐴)))↑𝑁) = ((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴)))))
4746oveq1d 7434 . . . . . 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)))
4830, 43, 473eqtr3d 2773 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (((1 / (tan‘𝐴)) + i)↑𝑁) = (((cos‘(𝑁 · 𝐴)) + (i · (sin‘(𝑁 · 𝐴)))) / ((sin‘𝐴)↑𝑁)))
4913nnred 12260 . . . . . . . . 9 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℝ)
5049, 18remulcld 11276 . . . . . . . 8 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑁 · 𝐴) ∈ ℝ)
5150recoscld 16124 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘(𝑁 · 𝐴)) ∈ ℝ)
5251recnd 11274 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (cos‘(𝑁 · 𝐴)) ∈ ℂ)
5350resincld 16123 . . . . . . . 8 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘(𝑁 · 𝐴)) ∈ ℝ)
5453recnd 11274 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘(𝑁 · 𝐴)) ∈ ℂ)
55 mulcl 11224 . . . . . . 7 ((i ∈ ℂ ∧ (sin‘(𝑁 · 𝐴)) ∈ ℂ) → (i · (sin‘(𝑁 · 𝐴))) ∈ ℂ)
565, 54, 55sylancr 585 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (i · (sin‘(𝑁 · 𝐴))) ∈ ℂ)
5721, 28elrpd 13048 . . . . . . . 8 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (sin‘𝐴) ∈ ℝ+)
5857, 44rpexpcld 14245 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((sin‘𝐴)↑𝑁) ∈ ℝ+)
5958rpcnd 13053 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((sin‘𝐴)↑𝑁) ∈ ℂ)
6058rpne0d 13056 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((sin‘𝐴)↑𝑁) ≠ 0)
6152, 56, 59, 60divdird 12061 . . . . 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))))
626, 54, 59, 60divassd 12058 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((i · (sin‘(𝑁 · 𝐴))) / ((sin‘𝐴)↑𝑁)) = (i · ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁))))
6362oveq2d 7435 . . . . 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)))))
6448, 61, 633eqtrd 2769 . . . 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)))))
6516, 64eqtr3d 2767 . . 3 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚))) = (((cos‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) + (i · ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)))))
6665fveq2d 6900 . 2 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (ℑ‘Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)))) = (ℑ‘(((cos‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) + (i · ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁))))))
67 oveq2 7427 . . . . . . 7 (𝑚 = (𝑁 − (2 · 𝑗)) → (𝑁C𝑚) = (𝑁C(𝑁 − (2 · 𝑗))))
68 oveq2 7427 . . . . . . . . 9 (𝑚 = (𝑁 − (2 · 𝑗)) → (𝑁𝑚) = (𝑁 − (𝑁 − (2 · 𝑗))))
6968oveq2d 7435 . . . . . . . 8 (𝑚 = (𝑁 − (2 · 𝑗)) → ((1 / (tan‘𝐴))↑(𝑁𝑚)) = ((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))))
70 oveq2 7427 . . . . . . . 8 (𝑚 = (𝑁 − (2 · 𝑗)) → (i↑𝑚) = (i↑(𝑁 − (2 · 𝑗))))
7169, 70oveq12d 7437 . . . . . . 7 (𝑚 = (𝑁 − (2 · 𝑗)) → (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)) = (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗)))))
7267, 71oveq12d 7437 . . . . . 6 (𝑚 = (𝑁 − (2 · 𝑗)) → ((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚))) = ((𝑁C(𝑁 − (2 · 𝑗))) · (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗))))))
7372fveq2d 6900 . . . . 5 (𝑚 = (𝑁 − (2 · 𝑗)) → (ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)))) = (ℑ‘((𝑁C(𝑁 − (2 · 𝑗))) · (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗)))))))
74 fzfid 13974 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (0...𝑀) ∈ Fin)
75 2nn0 12522 . . . . . . . . . . . . 13 2 ∈ ℕ0
76 elfznn0 13629 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...𝑀) → 𝑘 ∈ ℕ0)
7776adantl 480 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ ℕ0)
78 nn0mulcl 12541 . . . . . . . . . . . . 13 ((2 ∈ ℕ0𝑘 ∈ ℕ0) → (2 · 𝑘) ∈ ℕ0)
7975, 77, 78sylancr 585 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ∈ ℕ0)
8079nn0red 12566 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ∈ ℝ)
8111nnred 12260 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (2 · 𝑀) ∈ ℝ)
8281adantr 479 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑀) ∈ ℝ)
8349adantr 479 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑁 ∈ ℝ)
84 elfzle2 13540 . . . . . . . . . . . . 13 (𝑘 ∈ (0...𝑀) → 𝑘𝑀)
8584adantl 480 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘𝑀)
8677nn0red 12566 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑘 ∈ ℝ)
87 nnre 12252 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
8887ad2antrr 724 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑀 ∈ ℝ)
89 2re 12319 . . . . . . . . . . . . . . 15 2 ∈ ℝ
90 2pos 12348 . . . . . . . . . . . . . . 15 0 < 2
9189, 90pm3.2i 469 . . . . . . . . . . . . . 14 (2 ∈ ℝ ∧ 0 < 2)
9291a1i 11 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 ∈ ℝ ∧ 0 < 2))
93 lemul2 12100 . . . . . . . . . . . . 13 ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝑘𝑀 ↔ (2 · 𝑘) ≤ (2 · 𝑀)))
9486, 88, 92, 93syl3anc 1368 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (𝑘𝑀 ↔ (2 · 𝑘) ≤ (2 · 𝑀)))
9585, 94mpbid 231 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ≤ (2 · 𝑀))
9682lep1d 12178 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑀) ≤ ((2 · 𝑀) + 1))
9796, 7breqtrrdi 5191 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑀) ≤ 𝑁)
9880, 82, 83, 95, 97letrd 11403 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ≤ 𝑁)
99 nn0uz 12897 . . . . . . . . . . . 12 0 = (ℤ‘0)
10079, 99eleqtrdi 2835 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ∈ (ℤ‘0))
10144adantr 479 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → 𝑁 ∈ ℤ)
102 elfz5 13528 . . . . . . . . . . 11 (((2 · 𝑘) ∈ (ℤ‘0) ∧ 𝑁 ∈ ℤ) → ((2 · 𝑘) ∈ (0...𝑁) ↔ (2 · 𝑘) ≤ 𝑁))
103100, 101, 102syl2anc 582 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → ((2 · 𝑘) ∈ (0...𝑁) ↔ (2 · 𝑘) ≤ 𝑁))
10498, 103mpbird 256 . . . . . . . . 9 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (2 · 𝑘) ∈ (0...𝑁))
105 fznn0sub2 13643 . . . . . . . . 9 ((2 · 𝑘) ∈ (0...𝑁) → (𝑁 − (2 · 𝑘)) ∈ (0...𝑁))
106104, 105syl 17 . . . . . . . 8 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑘 ∈ (0...𝑀)) → (𝑁 − (2 · 𝑘)) ∈ (0...𝑁))
107106ex 411 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑘 ∈ (0...𝑀) → (𝑁 − (2 · 𝑘)) ∈ (0...𝑁)))
10813nncnd 12261 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → 𝑁 ∈ ℂ)
109108adantr 479 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → 𝑁 ∈ ℂ)
110 2cn 12320 . . . . . . . . . . 11 2 ∈ ℂ
111 elfzelz 13536 . . . . . . . . . . . . 13 (𝑘 ∈ (0...𝑀) → 𝑘 ∈ ℤ)
112111zcnd 12700 . . . . . . . . . . . 12 (𝑘 ∈ (0...𝑀) → 𝑘 ∈ ℂ)
113112ad2antrl 726 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → 𝑘 ∈ ℂ)
114 mulcl 11224 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (2 · 𝑘) ∈ ℂ)
115110, 113, 114sylancr 585 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → (2 · 𝑘) ∈ ℂ)
116112ssriv 3980 . . . . . . . . . . . 12 (0...𝑀) ⊆ ℂ
117 simprr 771 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → 𝑚 ∈ (0...𝑀))
118116, 117sselid 3974 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → 𝑚 ∈ ℂ)
119 mulcl 11224 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (2 · 𝑚) ∈ ℂ)
120110, 118, 119sylancr 585 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → (2 · 𝑚) ∈ ℂ)
121109, 115, 120subcanad 11646 . . . . . . . . 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 12455 . . . . . . . . . . 11 (2 ∈ ℂ ∧ 2 ≠ 0)
123122a1i 11 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → (2 ∈ ℂ ∧ 2 ≠ 0))
124 mulcan 11883 . . . . . . . . . 10 ((𝑘 ∈ ℂ ∧ 𝑚 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((2 · 𝑘) = (2 · 𝑚) ↔ 𝑘 = 𝑚))
125113, 118, 123, 124syl3anc 1368 . . . . . . . . 9 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → ((2 · 𝑘) = (2 · 𝑚) ↔ 𝑘 = 𝑚))
126121, 125bitrd 278 . . . . . . . 8 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀))) → ((𝑁 − (2 · 𝑘)) = (𝑁 − (2 · 𝑚)) ↔ 𝑘 = 𝑚))
127126ex 411 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((𝑘 ∈ (0...𝑀) ∧ 𝑚 ∈ (0...𝑀)) → ((𝑁 − (2 · 𝑘)) = (𝑁 − (2 · 𝑚)) ↔ 𝑘 = 𝑚)))
128107, 127dom2lem 9013 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))):(0...𝑀)–1-1→(0...𝑁))
129 f1f1orn 6849 . . . . . 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))))
130128, 129syl 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 7427 . . . . . . . 8 (𝑘 = 𝑗 → (2 · 𝑘) = (2 · 𝑗))
132131oveq2d 7435 . . . . . . 7 (𝑘 = 𝑗 → (𝑁 − (2 · 𝑘)) = (𝑁 − (2 · 𝑗)))
133 eqid 2725 . . . . . . 7 (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) = (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))
134 ovex 7452 . . . . . . 7 (𝑁 − (2 · 𝑗)) ∈ V
135132, 133, 134fvmpt 7004 . . . . . 6 (𝑗 ∈ (0...𝑀) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘𝑗) = (𝑁 − (2 · 𝑗)))
136135adantl 480 . . . . 5 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘𝑗) = (𝑁 − (2 · 𝑗)))
137106fmpttd 7124 . . . . . . . . . 10 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))):(0...𝑀)⟶(0...𝑁))
138137frnd 6731 . . . . . . . . 9 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) ⊆ (0...𝑁))
139138sselda 3976 . . . . . . . 8 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) → 𝑚 ∈ (0...𝑁))
140 bccl2 14318 . . . . . . . . . . 11 (𝑚 ∈ (0...𝑁) → (𝑁C𝑚) ∈ ℕ)
141140adantl 480 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℕ)
142141nncnd 12261 . . . . . . . . 9 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℂ)
1432rprecred 13062 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (1 / (tan‘𝐴)) ∈ ℝ)
144 fznn0sub 13568 . . . . . . . . . . . 12 (𝑚 ∈ (0...𝑁) → (𝑁𝑚) ∈ ℕ0)
145 reexpcl 14079 . . . . . . . . . . . 12 (((1 / (tan‘𝐴)) ∈ ℝ ∧ (𝑁𝑚) ∈ ℕ0) → ((1 / (tan‘𝐴))↑(𝑁𝑚)) ∈ ℝ)
146143, 144, 145syl2an 594 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((1 / (tan‘𝐴))↑(𝑁𝑚)) ∈ ℝ)
147146recnd 11274 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((1 / (tan‘𝐴))↑(𝑁𝑚)) ∈ ℂ)
148 elfznn0 13629 . . . . . . . . . . . 12 (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℕ0)
149148adantl 480 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → 𝑚 ∈ ℕ0)
150 expcl 14080 . . . . . . . . . . 11 ((i ∈ ℂ ∧ 𝑚 ∈ ℕ0) → (i↑𝑚) ∈ ℂ)
1515, 149, 150sylancr 585 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (i↑𝑚) ∈ ℂ)
152147, 151mulcld 11266 . . . . . . . . 9 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)) ∈ ℂ)
153142, 152mulcld 11266 . . . . . . . 8 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚))) ∈ ℂ)
154139, 153syldan 589 . . . . . . 7 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) → ((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚))) ∈ ℂ)
155154imcld 15178 . . . . . 6 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) → (ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)))) ∈ ℝ)
156155recnd 11274 . . . . 5 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) → (ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)))) ∈ ℂ)
15773, 74, 130, 136, 156fsumf1o 15705 . . . 4 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → Σ𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))(ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)))) = Σ𝑗 ∈ (0...𝑀)(ℑ‘((𝑁C(𝑁 − (2 · 𝑗))) · (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗)))))))
158 eldifi 4123 . . . . . . . 8 (𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) → 𝑚 ∈ (0...𝑁))
159141nnred 12260 . . . . . . . 8 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑁C𝑚) ∈ ℝ)
160158, 159sylan2 591 . . . . . . 7 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → (𝑁C𝑚) ∈ ℝ)
161158, 146sylan2 591 . . . . . . . 8 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → ((1 / (tan‘𝐴))↑(𝑁𝑚)) ∈ ℝ)
162 eldif 3954 . . . . . . . . 9 (𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))) ↔ (𝑚 ∈ (0...𝑁) ∧ ¬ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))))
163 elfzelz 13536 . . . . . . . . . . . . . . 15 (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ)
164163adantl 480 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → 𝑚 ∈ ℤ)
165 zeo 12681 . . . . . . . . . . . . . 14 (𝑚 ∈ ℤ → ((𝑚 / 2) ∈ ℤ ∨ ((𝑚 + 1) / 2) ∈ ℤ))
166164, 165syl 17 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑚 / 2) ∈ ℤ ∨ ((𝑚 + 1) / 2) ∈ ℤ))
167 i2 14201 . . . . . . . . . . . . . . . . . 18 (i↑2) = -1
168167oveq1i 7429 . . . . . . . . . . . . . . . . 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)
170148ad2antrl 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → 𝑚 ∈ ℕ0)
171 nn0re 12514 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ0𝑚 ∈ ℝ)
172 nn0ge0 12530 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ ℕ0 → 0 ≤ 𝑚)
173 divge0 12116 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ (𝑚 / 2))
17489, 90, 173mpanr12 703 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) → 0 ≤ (𝑚 / 2))
175171, 172, 174syl2anc 582 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ0 → 0 ≤ (𝑚 / 2))
176170, 175syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → 0 ≤ (𝑚 / 2))
177 elnn0z 12604 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 / 2) ∈ ℕ0 ↔ ((𝑚 / 2) ∈ ℤ ∧ 0 ≤ (𝑚 / 2)))
178169, 176, 177sylanbrc 581 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (𝑚 / 2) ∈ ℕ0)
179 expmul 14108 . . . . . . . . . . . . . . . . . . 19 ((i ∈ ℂ ∧ 2 ∈ ℕ0 ∧ (𝑚 / 2) ∈ ℕ0) → (i↑(2 · (𝑚 / 2))) = ((i↑2)↑(𝑚 / 2)))
1805, 75, 178, 179mp3an12i 1461 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (i↑(2 · (𝑚 / 2))) = ((i↑2)↑(𝑚 / 2)))
181170nn0cnd 12567 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → 𝑚 ∈ ℂ)
182 2ne0 12349 . . . . . . . . . . . . . . . . . . . . 21 2 ≠ 0
183 divcan2 11913 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (2 · (𝑚 / 2)) = 𝑚)
184110, 182, 183mp3an23 1449 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℂ → (2 · (𝑚 / 2)) = 𝑚)
185181, 184syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (2 · (𝑚 / 2)) = 𝑚)
186185oveq2d 7435 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (i↑(2 · (𝑚 / 2))) = (i↑𝑚))
187180, 186eqtr3d 2767 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → ((i↑2)↑(𝑚 / 2)) = (i↑𝑚))
188168, 187eqtr3id 2779 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (-1↑(𝑚 / 2)) = (i↑𝑚))
189 neg1rr 12360 . . . . . . . . . . . . . . . . 17 -1 ∈ ℝ
190 reexpcl 14079 . . . . . . . . . . . . . . . . 17 ((-1 ∈ ℝ ∧ (𝑚 / 2) ∈ ℕ0) → (-1↑(𝑚 / 2)) ∈ ℝ)
191189, 178, 190sylancr 585 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (-1↑(𝑚 / 2)) ∈ ℝ)
192188, 191eqeltrrd 2826 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ (𝑚 / 2) ∈ ℤ)) → (i↑𝑚) ∈ ℝ)
193192expr 455 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((𝑚 / 2) ∈ ℤ → (i↑𝑚) ∈ ℝ))
194 0zd 12603 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 0 ∈ ℤ)
195 nnz 12612 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
196195ad2antrr 724 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑀 ∈ ℤ)
197108adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑁 ∈ ℂ)
198148ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑚 ∈ ℕ0)
199198nn0cnd 12567 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑚 ∈ ℂ)
200 1cnd 11241 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 1 ∈ ℂ)
201197, 199, 200pnpcan2d 11641 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 + 1) − (𝑚 + 1)) = (𝑁𝑚))
202 2t1e2 12408 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (2 · 1) = 2
203 df-2 12308 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 = (1 + 1)
204202, 203eqtr2i 2754 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (1 + 1) = (2 · 1)
205204oveq2i 7430 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((2 · 𝑀) + (1 + 1)) = ((2 · 𝑀) + (2 · 1))
2067oveq1i 7429 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 + 1) = (((2 · 𝑀) + 1) + 1)
20711nncnd 12261 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (2 · 𝑀) ∈ ℂ)
208207adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 · 𝑀) ∈ ℂ)
209208, 200, 200addassd 11268 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (((2 · 𝑀) + 1) + 1) = ((2 · 𝑀) + (1 + 1)))
210206, 209eqtrid 2777 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 + 1) = ((2 · 𝑀) + (1 + 1)))
211 2cnd 12323 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 2 ∈ ℂ)
212 nncn 12253 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
213212ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑀 ∈ ℂ)
214211, 213, 200adddid 11270 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 · (𝑀 + 1)) = ((2 · 𝑀) + (2 · 1)))
215205, 210, 2143eqtr4a 2791 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 + 1) = (2 · (𝑀 + 1)))
216215oveq1d 7434 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁 + 1) − (𝑚 + 1)) = ((2 · (𝑀 + 1)) − (𝑚 + 1)))
217201, 216eqtr3d 2767 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁𝑚) = ((2 · (𝑀 + 1)) − (𝑚 + 1)))
218217oveq1d 7434 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁𝑚) / 2) = (((2 · (𝑀 + 1)) − (𝑚 + 1)) / 2))
219196peano2zd 12702 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑀 + 1) ∈ ℤ)
220219zcnd 12700 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑀 + 1) ∈ ℂ)
221 mulcl 11224 . . . . . . . . . . . . . . . . . . . . . . 23 ((2 ∈ ℂ ∧ (𝑀 + 1) ∈ ℂ) → (2 · (𝑀 + 1)) ∈ ℂ)
222110, 220, 221sylancr 585 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 · (𝑀 + 1)) ∈ ℂ)
223 peano2cn 11418 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ ℂ → (𝑚 + 1) ∈ ℂ)
224199, 223syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑚 + 1) ∈ ℂ)
225122a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 ∈ ℂ ∧ 2 ≠ 0))
226 divsubdir 11941 . . . . . . . . . . . . . . . . . . . . . 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)))
227222, 224, 225, 226syl3anc 1368 . . . . . . . . . . . . . . . . . . . . 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)))
228182a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 2 ≠ 0)
229220, 211, 228divcan3d 12028 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((2 · (𝑀 + 1)) / 2) = (𝑀 + 1))
230229oveq1d 7434 . . . . . . . . . . . . . . . . . . . . 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)))
231218, 227, 2303eqtrd 2769 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁𝑚) / 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)
233219, 232zsubcld 12704 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑀 + 1) − ((𝑚 + 1) / 2)) ∈ ℤ)
234231, 233eqeltrd 2825 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁𝑚) / 2) ∈ ℤ)
235144ad2antrl 726 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁𝑚) ∈ ℕ0)
236 nn0re 12514 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁𝑚) ∈ ℕ0 → (𝑁𝑚) ∈ ℝ)
237 nn0ge0 12530 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁𝑚) ∈ ℕ0 → 0 ≤ (𝑁𝑚))
238 divge0 12116 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑁𝑚) ∈ ℝ ∧ 0 ≤ (𝑁𝑚)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ ((𝑁𝑚) / 2))
23989, 90, 238mpanr12 703 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁𝑚) ∈ ℝ ∧ 0 ≤ (𝑁𝑚)) → 0 ≤ ((𝑁𝑚) / 2))
240236, 237, 239syl2anc 582 . . . . . . . . . . . . . . . . . . . 20 ((𝑁𝑚) ∈ ℕ0 → 0 ≤ ((𝑁𝑚) / 2))
241235, 240syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 0 ≤ ((𝑁𝑚) / 2))
242235nn0red 12566 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁𝑚) ∈ ℝ)
24349adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑁 ∈ ℝ)
244 peano2re 11419 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
245243, 244syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 + 1) ∈ ℝ)
246198, 172syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 0 ≤ 𝑚)
247198nn0red 12566 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑚 ∈ ℝ)
248243, 247subge02d 11838 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (0 ≤ 𝑚 ↔ (𝑁𝑚) ≤ 𝑁))
249246, 248mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁𝑚) ≤ 𝑁)
250243ltp1d 12177 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑁 < (𝑁 + 1))
251242, 243, 245, 249, 250lelttrd 11404 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁𝑚) < (𝑁 + 1))
252251, 215breqtrd 5175 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁𝑚) < (2 · (𝑀 + 1)))
253219zred 12699 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑀 + 1) ∈ ℝ)
25491a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 ∈ ℝ ∧ 0 < 2))
255 ltdivmul 12122 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁𝑚) ∈ ℝ ∧ (𝑀 + 1) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((𝑁𝑚) / 2) < (𝑀 + 1) ↔ (𝑁𝑚) < (2 · (𝑀 + 1))))
256242, 253, 254, 255syl3anc 1368 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (((𝑁𝑚) / 2) < (𝑀 + 1) ↔ (𝑁𝑚) < (2 · (𝑀 + 1))))
257252, 256mpbird 256 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁𝑚) / 2) < (𝑀 + 1))
258 zleltp1 12646 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁𝑚) / 2) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (((𝑁𝑚) / 2) ≤ 𝑀 ↔ ((𝑁𝑚) / 2) < (𝑀 + 1)))
259234, 196, 258syl2anc 582 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (((𝑁𝑚) / 2) ≤ 𝑀 ↔ ((𝑁𝑚) / 2) < (𝑀 + 1)))
260257, 259mpbird 256 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁𝑚) / 2) ≤ 𝑀)
261194, 196, 234, 241, 260elfzd 13527 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑁𝑚) / 2) ∈ (0...𝑀))
262 oveq2 7427 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = ((𝑁𝑚) / 2) → (2 · 𝑘) = (2 · ((𝑁𝑚) / 2)))
263262oveq2d 7435 . . . . . . . . . . . . . . . . . . 19 (𝑘 = ((𝑁𝑚) / 2) → (𝑁 − (2 · 𝑘)) = (𝑁 − (2 · ((𝑁𝑚) / 2))))
264 ovex 7452 . . . . . . . . . . . . . . . . . . 19 (𝑁 − (2 · ((𝑁𝑚) / 2))) ∈ V
265263, 133, 264fvmpt 7004 . . . . . . . . . . . . . . . . . 18 (((𝑁𝑚) / 2) ∈ (0...𝑀) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘((𝑁𝑚) / 2)) = (𝑁 − (2 · ((𝑁𝑚) / 2))))
266261, 265syl 17 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘((𝑁𝑚) / 2)) = (𝑁 − (2 · ((𝑁𝑚) / 2))))
267235nn0cnd 12567 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁𝑚) ∈ ℂ)
268267, 211, 228divcan2d 12025 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (2 · ((𝑁𝑚) / 2)) = (𝑁𝑚))
269268oveq2d 7435 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − (2 · ((𝑁𝑚) / 2))) = (𝑁 − (𝑁𝑚)))
270197, 199nncand 11608 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → (𝑁 − (𝑁𝑚)) = 𝑚)
271266, 269, 2703eqtrd 2769 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘((𝑁𝑚) / 2)) = 𝑚)
272137ffnd 6724 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) Fn (0...𝑀))
273 fnfvelrn 7089 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) Fn (0...𝑀) ∧ ((𝑁𝑚) / 2) ∈ (0...𝑀)) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘((𝑁𝑚) / 2)) ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))
274272, 261, 273syl2an2r 683 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → ((𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))‘((𝑁𝑚) / 2)) ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))
275271, 274eqeltrrd 2826 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ((𝑚 + 1) / 2) ∈ ℤ)) → 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))
276275expr 455 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (((𝑚 + 1) / 2) ∈ ℤ → 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))))
277193, 276orim12d 962 . . . . . . . . . . . . 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))))))
278166, 277mpd 15 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → ((i↑𝑚) ∈ ℝ ∨ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))))
279278orcomd 869 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) ∨ (i↑𝑚) ∈ ℝ))
280279ord 862 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ (0...𝑁)) → (¬ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))) → (i↑𝑚) ∈ ℝ))
281280impr 453 . . . . . . . . 9 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ (𝑚 ∈ (0...𝑁) ∧ ¬ 𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → (i↑𝑚) ∈ ℝ)
282162, 281sylan2b 592 . . . . . . . 8 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → (i↑𝑚) ∈ ℝ)
283161, 282remulcld 11276 . . . . . . 7 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)) ∈ ℝ)
284160, 283remulcld 11276 . . . . . 6 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → ((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚))) ∈ ℝ)
285284reim0d 15208 . . . . 5 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑚 ∈ ((0...𝑁) ∖ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘))))) → (ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)))) = 0)
286 fzfid 13974 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (0...𝑁) ∈ Fin)
287138, 156, 285, 286fsumss 15707 . . . 4 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → Σ𝑚 ∈ ran (𝑘 ∈ (0...𝑀) ↦ (𝑁 − (2 · 𝑘)))(ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)))) = Σ𝑚 ∈ (0...𝑁)(ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)))))
288 elfznn0 13629 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0)
289288adantl 480 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℕ0)
290 nn0mulcl 12541 . . . . . . . . . . . . . . . . 17 ((2 ∈ ℕ0𝑗 ∈ ℕ0) → (2 · 𝑗) ∈ ℕ0)
29175, 289, 290sylancr 585 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑗) ∈ ℕ0)
292291nn0zd 12617 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑗) ∈ ℤ)
293 bccl 14317 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ (2 · 𝑗) ∈ ℤ) → (𝑁C(2 · 𝑗)) ∈ ℕ0)
29414, 292, 293syl2an2r 683 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁C(2 · 𝑗)) ∈ ℕ0)
295294nn0red 12566 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁C(2 · 𝑗)) ∈ ℝ)
296 fznn0sub 13568 . . . . . . . . . . . . . . 15 (𝑗 ∈ (0...𝑀) → (𝑀𝑗) ∈ ℕ0)
297296adantl 480 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑀𝑗) ∈ ℕ0)
298 reexpcl 14079 . . . . . . . . . . . . . 14 ((-1 ∈ ℝ ∧ (𝑀𝑗) ∈ ℕ0) → (-1↑(𝑀𝑗)) ∈ ℝ)
299189, 297, 298sylancr 585 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (-1↑(𝑀𝑗)) ∈ ℝ)
300295, 299remulcld 11276 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) ∈ ℝ)
301 2z 12627 . . . . . . . . . . . . . . . 16 2 ∈ ℤ
302 znegcl 12630 . . . . . . . . . . . . . . . 16 (2 ∈ ℤ → -2 ∈ ℤ)
303301, 302ax-mp 5 . . . . . . . . . . . . . . 15 -2 ∈ ℤ
304 rpexpcl 14081 . . . . . . . . . . . . . . 15 (((tan‘𝐴) ∈ ℝ+ ∧ -2 ∈ ℤ) → ((tan‘𝐴)↑-2) ∈ ℝ+)
3052, 303, 304sylancl 584 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((tan‘𝐴)↑-2) ∈ ℝ+)
306305rpred 13051 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((tan‘𝐴)↑-2) ∈ ℝ)
307 reexpcl 14079 . . . . . . . . . . . . 13 ((((tan‘𝐴)↑-2) ∈ ℝ ∧ 𝑗 ∈ ℕ0) → (((tan‘𝐴)↑-2)↑𝑗) ∈ ℝ)
308306, 288, 307syl2an 594 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((tan‘𝐴)↑-2)↑𝑗) ∈ ℝ)
309300, 308remulcld 11276 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ ℝ)
310309recnd 11274 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ ℂ)
311 mulcl 11224 . . . . . . . . . 10 ((i ∈ ℂ ∧ (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ ℂ) → (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))) ∈ ℂ)
3125, 310, 311sylancr 585 . . . . . . . . 9 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))) ∈ ℂ)
313312addlidd 11447 . . . . . . . 8 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (0 + (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))) = (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))))
314294nn0cnd 12567 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁C(2 · 𝑗)) ∈ ℂ)
315299recnd 11274 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (-1↑(𝑀𝑗)) ∈ ℂ)
316308recnd 11274 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((tan‘𝐴)↑-2)↑𝑗) ∈ ℂ)
317314, 315, 316mulassd 11269 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) = ((𝑁C(2 · 𝑗)) · ((-1↑(𝑀𝑗)) · (((tan‘𝐴)↑-2)↑𝑗))))
318317oveq2d 7435 . . . . . . . . 9 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))) = (i · ((𝑁C(2 · 𝑗)) · ((-1↑(𝑀𝑗)) · (((tan‘𝐴)↑-2)↑𝑗)))))
3195a1i 11 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → i ∈ ℂ)
320315, 316mulcld 11266 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((-1↑(𝑀𝑗)) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ ℂ)
321319, 314, 320mul12d 11455 . . . . . . . . 9 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i · ((𝑁C(2 · 𝑗)) · ((-1↑(𝑀𝑗)) · (((tan‘𝐴)↑-2)↑𝑗)))) = ((𝑁C(2 · 𝑗)) · (i · ((-1↑(𝑀𝑗)) · (((tan‘𝐴)↑-2)↑𝑗)))))
322318, 321eqtrd 2765 . . . . . . . 8 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))) = ((𝑁C(2 · 𝑗)) · (i · ((-1↑(𝑀𝑗)) · (((tan‘𝐴)↑-2)↑𝑗)))))
323 bccmpl 14304 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (2 · 𝑗) ∈ ℤ) → (𝑁C(2 · 𝑗)) = (𝑁C(𝑁 − (2 · 𝑗))))
32414, 292, 323syl2an2r 683 . . . . . . . . 9 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁C(2 · 𝑗)) = (𝑁C(𝑁 − (2 · 𝑗))))
325108adantr 479 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑁 ∈ ℂ)
326291nn0cnd 12567 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑗) ∈ ℂ)
327325, 326nncand 11608 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁 − (𝑁 − (2 · 𝑗))) = (2 · 𝑗))
328327oveq2d 7435 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) = ((1 / (tan‘𝐴))↑(2 · 𝑗)))
3292adantr 479 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (tan‘𝐴) ∈ ℝ+)
330329rpcnd 13053 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (tan‘𝐴) ∈ ℂ)
331 expneg 14070 . . . . . . . . . . . . . 14 (((tan‘𝐴) ∈ ℂ ∧ (2 · 𝑗) ∈ ℕ0) → ((tan‘𝐴)↑-(2 · 𝑗)) = (1 / ((tan‘𝐴)↑(2 · 𝑗))))
332330, 291, 331syl2anc 582 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((tan‘𝐴)↑-(2 · 𝑗)) = (1 / ((tan‘𝐴)↑(2 · 𝑗))))
333289nn0cnd 12567 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℂ)
334 mulneg1 11682 . . . . . . . . . . . . . . 15 ((2 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (-2 · 𝑗) = -(2 · 𝑗))
335110, 333, 334sylancr 585 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (-2 · 𝑗) = -(2 · 𝑗))
336335oveq2d 7435 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((tan‘𝐴)↑(-2 · 𝑗)) = ((tan‘𝐴)↑-(2 · 𝑗)))
337329rpne0d 13056 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (tan‘𝐴) ≠ 0)
338330, 337, 292exprecd 14154 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((1 / (tan‘𝐴))↑(2 · 𝑗)) = (1 / ((tan‘𝐴)↑(2 · 𝑗))))
339332, 336, 3383eqtr4d 2775 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((tan‘𝐴)↑(-2 · 𝑗)) = ((1 / (tan‘𝐴))↑(2 · 𝑗)))
340303a1i 11 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → -2 ∈ ℤ)
341289nn0zd 12617 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℤ)
342 expmulz 14109 . . . . . . . . . . . . 13 ((((tan‘𝐴) ∈ ℂ ∧ (tan‘𝐴) ≠ 0) ∧ (-2 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((tan‘𝐴)↑(-2 · 𝑗)) = (((tan‘𝐴)↑-2)↑𝑗))
343330, 337, 340, 341, 342syl22anc 837 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((tan‘𝐴)↑(-2 · 𝑗)) = (((tan‘𝐴)↑-2)↑𝑗))
344328, 339, 3433eqtr2d 2771 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) = (((tan‘𝐴)↑-2)↑𝑗))
3457oveq1i 7429 . . . . . . . . . . . . . . 15 (𝑁 − (2 · 𝑗)) = (((2 · 𝑀) + 1) − (2 · 𝑗))
34611adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑀) ∈ ℕ)
347346nncnd 12261 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · 𝑀) ∈ ℂ)
348 1cnd 11241 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 1 ∈ ℂ)
349347, 348, 326addsubd 11624 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((2 · 𝑀) + 1) − (2 · 𝑗)) = (((2 · 𝑀) − (2 · 𝑗)) + 1))
350 2cnd 12323 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 2 ∈ ℂ)
351212ad2antrr 724 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 𝑀 ∈ ℂ)
352350, 351, 333subdid 11702 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · (𝑀𝑗)) = ((2 · 𝑀) − (2 · 𝑗)))
353352oveq1d 7434 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((2 · (𝑀𝑗)) + 1) = (((2 · 𝑀) − (2 · 𝑗)) + 1))
354349, 353eqtr4d 2768 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (((2 · 𝑀) + 1) − (2 · 𝑗)) = ((2 · (𝑀𝑗)) + 1))
355345, 354eqtrid 2777 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑁 − (2 · 𝑗)) = ((2 · (𝑀𝑗)) + 1))
356355oveq2d 7435 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(𝑁 − (2 · 𝑗))) = (i↑((2 · (𝑀𝑗)) + 1)))
357 nn0mulcl 12541 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ0 ∧ (𝑀𝑗) ∈ ℕ0) → (2 · (𝑀𝑗)) ∈ ℕ0)
35875, 297, 357sylancr 585 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (2 · (𝑀𝑗)) ∈ ℕ0)
359 expp1 14069 . . . . . . . . . . . . . 14 ((i ∈ ℂ ∧ (2 · (𝑀𝑗)) ∈ ℕ0) → (i↑((2 · (𝑀𝑗)) + 1)) = ((i↑(2 · (𝑀𝑗))) · i))
3605, 358, 359sylancr 585 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑((2 · (𝑀𝑗)) + 1)) = ((i↑(2 · (𝑀𝑗))) · i))
36175a1i 11 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → 2 ∈ ℕ0)
362319, 297, 361expmuld 14149 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(2 · (𝑀𝑗))) = ((i↑2)↑(𝑀𝑗)))
363167oveq1i 7429 . . . . . . . . . . . . . . 15 ((i↑2)↑(𝑀𝑗)) = (-1↑(𝑀𝑗))
364362, 363eqtrdi 2781 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(2 · (𝑀𝑗))) = (-1↑(𝑀𝑗)))
365364oveq1d 7434 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((i↑(2 · (𝑀𝑗))) · i) = ((-1↑(𝑀𝑗)) · i))
366356, 360, 3653eqtrd 2769 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(𝑁 − (2 · 𝑗))) = ((-1↑(𝑀𝑗)) · i))
367 mulcom 11226 . . . . . . . . . . . . 13 (((-1↑(𝑀𝑗)) ∈ ℂ ∧ i ∈ ℂ) → ((-1↑(𝑀𝑗)) · i) = (i · (-1↑(𝑀𝑗))))
368315, 5, 367sylancl 584 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((-1↑(𝑀𝑗)) · i) = (i · (-1↑(𝑀𝑗))))
369366, 368eqtrd 2765 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i↑(𝑁 − (2 · 𝑗))) = (i · (-1↑(𝑀𝑗))))
370344, 369oveq12d 7437 . . . . . . . . . 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↑(𝑀𝑗)))))
371 mulcl 11224 . . . . . . . . . . . 12 ((i ∈ ℂ ∧ (-1↑(𝑀𝑗)) ∈ ℂ) → (i · (-1↑(𝑀𝑗))) ∈ ℂ)
3725, 315, 371sylancr 585 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i · (-1↑(𝑀𝑗))) ∈ ℂ)
373372, 316mulcomd 11267 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((i · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) = ((((tan‘𝐴)↑-2)↑𝑗) · (i · (-1↑(𝑀𝑗)))))
374319, 315, 316mulassd 11269 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((i · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) = (i · ((-1↑(𝑀𝑗)) · (((tan‘𝐴)↑-2)↑𝑗))))
375370, 373, 3743eqtr2rd 2772 . . . . . . . . 9 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (i · ((-1↑(𝑀𝑗)) · (((tan‘𝐴)↑-2)↑𝑗))) = (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗)))))
376324, 375oveq12d 7437 . . . . . . . 8 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑁C(2 · 𝑗)) · (i · ((-1↑(𝑀𝑗)) · (((tan‘𝐴)↑-2)↑𝑗)))) = ((𝑁C(𝑁 − (2 · 𝑗))) · (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗))))))
377313, 322, 3763eqtrd 2769 . . . . . . 7 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (0 + (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))) = ((𝑁C(𝑁 − (2 · 𝑗))) · (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗))))))
378377fveq2d 6900 . . . . . 6 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (ℑ‘(0 + (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))))) = (ℑ‘((𝑁C(𝑁 − (2 · 𝑗))) · (((1 / (tan‘𝐴))↑(𝑁 − (𝑁 − (2 · 𝑗)))) · (i↑(𝑁 − (2 · 𝑗)))))))
379 0re 11248 . . . . . . 7 0 ∈ ℝ
380 crim 15098 . . . . . . 7 ((0 ∈ ℝ ∧ (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ ℝ) → (ℑ‘(0 + (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))))) = (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
381379, 309, 380sylancr 585 . . . . . 6 (((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) ∧ 𝑗 ∈ (0...𝑀)) → (ℑ‘(0 + (i · (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗))))) = (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
382378, 381eqtr3d 2767 . . . . 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↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
383382sumeq2dv 15685 . . . 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↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
384157, 287, 3833eqtr3d 2773 . . 3 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → Σ𝑚 ∈ (0...𝑁)(ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)))) = Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
385286, 153fsumim 15791 . . 3 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (ℑ‘Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)))) = Σ𝑚 ∈ (0...𝑁)(ℑ‘((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)))))
386305rpcnd 13053 . . . 4 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((tan‘𝐴)↑-2) ∈ ℂ)
387 oveq1 7426 . . . . . . 7 (𝑡 = ((tan‘𝐴)↑-2) → (𝑡𝑗) = (((tan‘𝐴)↑-2)↑𝑗))
388387oveq2d 7435 . . . . . 6 (𝑡 = ((tan‘𝐴)↑-2) → (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)) = (((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
389388sumeq2sdv 15686 . . . . 5 (𝑡 = ((tan‘𝐴)↑-2) → Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)) = Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
390 basel.p . . . . 5 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (𝑡𝑗)))
391 sumex 15670 . . . . 5 Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)) ∈ V
392389, 390, 391fvmpt 7004 . . . 4 (((tan‘𝐴)↑-2) ∈ ℂ → (𝑃‘((tan‘𝐴)↑-2)) = Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
393386, 392syl 17 . . 3 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (𝑃‘((tan‘𝐴)↑-2)) = Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀𝑗))) · (((tan‘𝐴)↑-2)↑𝑗)))
394384, 385, 3933eqtr4d 2775 . 2 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → (ℑ‘Σ𝑚 ∈ (0...𝑁)((𝑁C𝑚) · (((1 / (tan‘𝐴))↑(𝑁𝑚)) · (i↑𝑚)))) = (𝑃‘((tan‘𝐴)↑-2)))
39551, 58rerpdivcld 13082 . . 3 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((cos‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) ∈ ℝ)
39653, 58rerpdivcld 13082 . . 3 ((𝑀 ∈ ℕ ∧ 𝐴 ∈ (0(,)(π / 2))) → ((sin‘(𝑁 · 𝐴)) / ((sin‘𝐴)↑𝑁)) ∈ ℝ)
397395, 396crimd 15215 . 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)))
39866, 394, 3973eqtr3d 2773 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 394  wo 845   = wceq 1533  wcel 2098  wne 2929  cdif 3941   class class class wbr 5149  cmpt 5232  ran crn 5679   Fn wfn 6544  1-1wf1 6546  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  cc 11138  cr 11139  0cc0 11140  1c1 11141  ici 11142   + caddc 11143   · cmul 11145   < clt 11280  cle 11281  cmin 11476  -cneg 11477   / cdiv 11903  cn 12245  2c2 12300  0cn0 12505  cz 12591  cuz 12855  +crp 13009  (,)cioo 13359  ...cfz 13519  cexp 14062  Ccbc 14297  cim 15081  Σcsu 15668  sincsin 16043  cosccos 16044  tanctan 16045  πcpi 16046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218  ax-addf 11219
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9388  df-fi 9436  df-sup 9467  df-inf 9468  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-uz 12856  df-q 12966  df-rp 13010  df-xneg 13127  df-xadd 13128  df-xmul 13129  df-ioo 13363  df-ioc 13364  df-ico 13365  df-icc 13366  df-fz 13520  df-fzo 13663  df-fl 13793  df-seq 14003  df-exp 14063  df-fac 14269  df-bc 14298  df-hash 14326  df-shft 15050  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-limsup 15451  df-clim 15468  df-rlim 15469  df-sum 15669  df-ef 16047  df-sin 16049  df-cos 16050  df-tan 16051  df-pi 16052  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-starv 17251  df-sca 17252  df-vsca 17253  df-ip 17254  df-tset 17255  df-ple 17256  df-ds 17258  df-unif 17259  df-hom 17260  df-cco 17261  df-rest 17407  df-topn 17408  df-0g 17426  df-gsum 17427  df-topgen 17428  df-pt 17429  df-prds 17432  df-xrs 17487  df-qtop 17492  df-imas 17493  df-xps 17495  df-mre 17569  df-mrc 17570  df-acs 17572  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-submnd 18744  df-mulg 19032  df-cntz 19280  df-cmn 19749  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-fbas 21293  df-fg 21294  df-cnfld 21297  df-top 22840  df-topon 22857  df-topsp 22879  df-bases 22893  df-cld 22967  df-ntr 22968  df-cls 22969  df-nei 23046  df-lp 23084  df-perf 23085  df-cn 23175  df-cnp 23176  df-haus 23263  df-tx 23510  df-hmeo 23703  df-fil 23794  df-fm 23886  df-flim 23887  df-flf 23888  df-xms 24270  df-ms 24271  df-tms 24272  df-cncf 24842  df-limc 25839  df-dv 25840
This theorem is referenced by:  basellem4  27061
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