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Theorem basellem4 24810

Description: Lemma for basel 24816. By basellem3 24809, the expression 𝑃((cot𝑥)↑2) = sin(𝑁𝑥) / (sin𝑥)↑𝑁 goes to zero whenever 𝑥 = 𝑛π / 𝑁 for some 𝑛 ∈ (1...𝑀), so this function enumerates 𝑀 distinct roots of a degree- 𝑀 polynomial, which must therefore be all the roots by fta1 24063. (Contributed by Mario Carneiro, 28-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)))
basel.t	⊢ 𝑇 = (𝑛 ∈ (1...𝑀) ↦ ((tan‘((𝑛 · π) / 𝑁))↑-2))

**Assertion**
Ref	Expression
basellem4	⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)–1-1-onto→(^◡𝑃 “ {0}))

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

Proof of Theorem basellem4 Dummy variables 𝑘 𝑚 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		basel.n	. . . . . . . . 9 ⊢ 𝑁 = ((2 · 𝑀) + 1)
2	1	basellem1 24807	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((𝑛 · π) / 𝑁) ∈ (0(,)(π / 2)))
3		tanrpcl 24256	. . . . . . . 8 ⊢ (((𝑛 · π) / 𝑁) ∈ (0(,)(π / 2)) → (tan‘((𝑛 · π) / 𝑁)) ∈ ℝ⁺)
4	2, 3	syl 17	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (tan‘((𝑛 · π) / 𝑁)) ∈ ℝ⁺)
5		2z 11409	. . . . . . . 8 ⊢ 2 ∈ ℤ
6		znegcl 11412	. . . . . . . 8 ⊢ (2 ∈ ℤ → -2 ∈ ℤ)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ -2 ∈ ℤ
8		rpexpcl 12879	. . . . . . 7 ⊢ (((tan‘((𝑛 · π) / 𝑁)) ∈ ℝ⁺ ∧ -2 ∈ ℤ) → ((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℝ⁺)
9	4, 7, 8	sylancl 694	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℝ⁺)
10	9	rpcnd 11874	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℂ)
11		basel.p	. . . . . . . 8 ⊢ 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗)))
12	1, 11	basellem3 24809	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ ((𝑛 · π) / 𝑁) ∈ (0(,)(π / 2))) → (𝑃‘((tan‘((𝑛 · π) / 𝑁))↑-2)) = ((sin‘(𝑁 · ((𝑛 · π) / 𝑁))) / ((sin‘((𝑛 · π) / 𝑁))↑𝑁)))
13	2, 12	syldan 487	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑃‘((tan‘((𝑛 · π) / 𝑁))↑-2)) = ((sin‘(𝑁 · ((𝑛 · π) / 𝑁))) / ((sin‘((𝑛 · π) / 𝑁))↑𝑁)))
14		elfzelz 12342	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℤ)
15	14	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑛 ∈ ℤ)
16	15	zred 11482	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑛 ∈ ℝ)
17		pire 24210	. . . . . . . . . . . 12 ⊢ π ∈ ℝ
18		remulcl 10021	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℝ ∧ π ∈ ℝ) → (𝑛 · π) ∈ ℝ)
19	16, 17, 18	sylancl 694	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑛 · π) ∈ ℝ)
20	19	recnd 10068	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑛 · π) ∈ ℂ)
21		2nn 11185	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
22		nnmulcl 11043	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (2 · 𝑀) ∈ ℕ)
23	21, 22	mpan 706	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ ℕ)
24	23	peano2nnd 11037	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → ((2 · 𝑀) + 1) ∈ ℕ)
25	1, 24	syl5eqel 2705	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ → 𝑁 ∈ ℕ)
26	25	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℕ)
27	26	nncnd 11036	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℂ)
28	26	nnne0d 11065	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ≠ 0)
29	20, 27, 28	divcan2d 10803	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑁 · ((𝑛 · π) / 𝑁)) = (𝑛 · π))
30	29	fveq2d 6195	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘(𝑁 · ((𝑛 · π) / 𝑁))) = (sin‘(𝑛 · π)))
31		sinkpi 24271	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → (sin‘(𝑛 · π)) = 0)
32	15, 31	syl 17	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘(𝑛 · π)) = 0)
33	30, 32	eqtrd 2656	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘(𝑁 · ((𝑛 · π) / 𝑁))) = 0)
34	33	oveq1d 6665	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((sin‘(𝑁 · ((𝑛 · π) / 𝑁))) / ((sin‘((𝑛 · π) / 𝑁))↑𝑁)) = (0 / ((sin‘((𝑛 · π) / 𝑁))↑𝑁)))
35	19, 26	nndivred 11069	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((𝑛 · π) / 𝑁) ∈ ℝ)
36	35	resincld 14873	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ∈ ℝ)
37	36	recnd 10068	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ∈ ℂ)
38	26	nnnn0d 11351	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℕ₀)
39	37, 38	expcld 13008	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((sin‘((𝑛 · π) / 𝑁))↑𝑁) ∈ ℂ)
40		sincosq1sgn 24250	. . . . . . . . . . 11 ⊢ (((𝑛 · π) / 𝑁) ∈ (0(,)(π / 2)) → (0 < (sin‘((𝑛 · π) / 𝑁)) ∧ 0 < (cos‘((𝑛 · π) / 𝑁))))
41	2, 40	syl 17	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (0 < (sin‘((𝑛 · π) / 𝑁)) ∧ 0 < (cos‘((𝑛 · π) / 𝑁))))
42	41	simpld 475	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 0 < (sin‘((𝑛 · π) / 𝑁)))
43	42	gt0ne0d 10592	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ≠ 0)
44	26	nnzd 11481	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℤ)
45	37, 43, 44	expne0d 13014	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((sin‘((𝑛 · π) / 𝑁))↑𝑁) ≠ 0)
46	39, 45	div0d 10800	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (0 / ((sin‘((𝑛 · π) / 𝑁))↑𝑁)) = 0)
47	13, 34, 46	3eqtrd 2660	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑃‘((tan‘((𝑛 · π) / 𝑁))↑-2)) = 0)
48	1, 11	basellem2 24808	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (Poly‘ℂ) ∧ (deg‘𝑃) = 𝑀 ∧ (coeff‘𝑃) = (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))))
49	48	simp1d 1073	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑃 ∈ (Poly‘ℂ))
50		plyf 23954	. . . . . . . 8 ⊢ (𝑃 ∈ (Poly‘ℂ) → 𝑃:ℂ⟶ℂ)
51		ffn 6045	. . . . . . . 8 ⊢ (𝑃:ℂ⟶ℂ → 𝑃 Fn ℂ)
52	49, 50, 51	3syl 18	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑃 Fn ℂ)
53	52	adantr 481	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑃 Fn ℂ)
54		fniniseg 6338	. . . . . 6 ⊢ (𝑃 Fn ℂ → (((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ (^◡𝑃 “ {0}) ↔ (((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℂ ∧ (𝑃‘((tan‘((𝑛 · π) / 𝑁))↑-2)) = 0)))
55	53, 54	syl 17	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ (^◡𝑃 “ {0}) ↔ (((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℂ ∧ (𝑃‘((tan‘((𝑛 · π) / 𝑁))↑-2)) = 0)))
56	10, 47, 55	mpbir2and 957	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ (^◡𝑃 “ {0}))
57		basel.t	. . . 4 ⊢ 𝑇 = (𝑛 ∈ (1...𝑀) ↦ ((tan‘((𝑛 · π) / 𝑁))↑-2))
58	56, 57	fmptd 6385	. . 3 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)⟶(^◡𝑃 “ {0}))
59		fveq2 6191	. . . . . 6 ⊢ (𝑘 = 𝑚 → (𝑇‘𝑘) = (𝑇‘𝑚))
60		fveq2 6191	. . . . . 6 ⊢ (𝑘 = 𝑥 → (𝑇‘𝑘) = (𝑇‘𝑥))
61		fveq2 6191	. . . . . 6 ⊢ (𝑘 = 𝑦 → (𝑇‘𝑘) = (𝑇‘𝑦))
62	14	zred 11482	. . . . . . 7 ⊢ (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℝ)
63	62	ssriv 3607	. . . . . 6 ⊢ (1...𝑀) ⊆ ℝ
64	9	rpred 11872	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℝ)
65	64, 57	fmptd 6385	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)⟶ℝ)
66	65	ffvelrnda 6359	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (1...𝑀)) → (𝑇‘𝑘) ∈ ℝ)
67		simplr 792	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑘 < 𝑚)
68	63	sseli 3599	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℝ)
69	68	ad2antrl 764	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑘 ∈ ℝ)
70	63	sseli 3599	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℝ)
71	70	ad2antll 765	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑚 ∈ ℝ)
72	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → π ∈ ℝ)
73		pipos 24212	. . . . . . . . . . . . . . . 16 ⊢ 0 < π
74	73	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 0 < π)
75		ltmul1 10873	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ ∧ (π ∈ ℝ ∧ 0 < π)) → (𝑘 < 𝑚 ↔ (𝑘 · π) < (𝑚 · π)))
76	69, 71, 72, 74, 75	syl112anc 1330	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑘 < 𝑚 ↔ (𝑘 · π) < (𝑚 · π)))
77	67, 76	mpbid 222	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑘 · π) < (𝑚 · π))
78		remulcl 10021	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ π ∈ ℝ) → (𝑘 · π) ∈ ℝ)
79	69, 17, 78	sylancl 694	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑘 · π) ∈ ℝ)
80		remulcl 10021	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℝ ∧ π ∈ ℝ) → (𝑚 · π) ∈ ℝ)
81	71, 17, 80	sylancl 694	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑚 · π) ∈ ℝ)
82	25	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑁 ∈ ℕ)
83	82	nnred 11035	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑁 ∈ ℝ)
84	82	nngt0d 11064	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 0 < 𝑁)
85		ltdiv1 10887	. . . . . . . . . . . . . 14 ⊢ (((𝑘 · π) ∈ ℝ ∧ (𝑚 · π) ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑘 · π) < (𝑚 · π) ↔ ((𝑘 · π) / 𝑁) < ((𝑚 · π) / 𝑁)))
86	79, 81, 83, 84, 85	syl112anc 1330	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑘 · π) < (𝑚 · π) ↔ ((𝑘 · π) / 𝑁) < ((𝑚 · π) / 𝑁)))
87	77, 86	mpbid 222	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑘 · π) / 𝑁) < ((𝑚 · π) / 𝑁))
88		neghalfpirx 24218	. . . . . . . . . . . . . . 15 ⊢ -(π / 2) ∈ ℝ^*
89		pirp 24213	. . . . . . . . . . . . . . . . 17 ⊢ π ∈ ℝ⁺
90		rphalfcl 11858	. . . . . . . . . . . . . . . . 17 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
91		rpge0 11845	. . . . . . . . . . . . . . . . 17 ⊢ ((π / 2) ∈ ℝ⁺ → 0 ≤ (π / 2))
92	89, 90, 91	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ 0 ≤ (π / 2)
93		halfpire 24216	. . . . . . . . . . . . . . . . 17 ⊢ (π / 2) ∈ ℝ
94		le0neg2 10537	. . . . . . . . . . . . . . . . 17 ⊢ ((π / 2) ∈ ℝ → (0 ≤ (π / 2) ↔ -(π / 2) ≤ 0))
95	93, 94	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (0 ≤ (π / 2) ↔ -(π / 2) ≤ 0)
96	92, 95	mpbi 220	. . . . . . . . . . . . . . 15 ⊢ -(π / 2) ≤ 0
97		iooss1 12210	. . . . . . . . . . . . . . 15 ⊢ ((-(π / 2) ∈ ℝ^* ∧ -(π / 2) ≤ 0) → (0(,)(π / 2)) ⊆ (-(π / 2)(,)(π / 2)))
98	88, 96, 97	mp2an 708	. . . . . . . . . . . . . 14 ⊢ (0(,)(π / 2)) ⊆ (-(π / 2)(,)(π / 2))
99	1	basellem1 24807	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (1...𝑀)) → ((𝑘 · π) / 𝑁) ∈ (0(,)(π / 2)))
100	99	ad2ant2r 783	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑘 · π) / 𝑁) ∈ (0(,)(π / 2)))
101	98, 100	sseldi 3601	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑘 · π) / 𝑁) ∈ (-(π / 2)(,)(π / 2)))
102	1	basellem1 24807	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑚 ∈ (1...𝑀)) → ((𝑚 · π) / 𝑁) ∈ (0(,)(π / 2)))
103	102	ad2ant2rl 785	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑚 · π) / 𝑁) ∈ (0(,)(π / 2)))
104	98, 103	sseldi 3601	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑚 · π) / 𝑁) ∈ (-(π / 2)(,)(π / 2)))
105		tanord 24284	. . . . . . . . . . . . 13 ⊢ ((((𝑘 · π) / 𝑁) ∈ (-(π / 2)(,)(π / 2)) ∧ ((𝑚 · π) / 𝑁) ∈ (-(π / 2)(,)(π / 2))) → (((𝑘 · π) / 𝑁) < ((𝑚 · π) / 𝑁) ↔ (tan‘((𝑘 · π) / 𝑁)) < (tan‘((𝑚 · π) / 𝑁))))
106	101, 104, 105	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (((𝑘 · π) / 𝑁) < ((𝑚 · π) / 𝑁) ↔ (tan‘((𝑘 · π) / 𝑁)) < (tan‘((𝑚 · π) / 𝑁))))
107	87, 106	mpbid 222	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑘 · π) / 𝑁)) < (tan‘((𝑚 · π) / 𝑁)))
108		tanrpcl 24256	. . . . . . . . . . . . 13 ⊢ (((𝑘 · π) / 𝑁) ∈ (0(,)(π / 2)) → (tan‘((𝑘 · π) / 𝑁)) ∈ ℝ⁺)
109	100, 108	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑘 · π) / 𝑁)) ∈ ℝ⁺)
110		tanrpcl 24256	. . . . . . . . . . . . 13 ⊢ (((𝑚 · π) / 𝑁) ∈ (0(,)(π / 2)) → (tan‘((𝑚 · π) / 𝑁)) ∈ ℝ⁺)
111	103, 110	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑚 · π) / 𝑁)) ∈ ℝ⁺)
112		rprege0 11847	. . . . . . . . . . . . 13 ⊢ ((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ⁺ → ((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ ∧ 0 ≤ (tan‘((𝑘 · π) / 𝑁))))
113		rprege0 11847	. . . . . . . . . . . . 13 ⊢ ((tan‘((𝑚 · π) / 𝑁)) ∈ ℝ⁺ → ((tan‘((𝑚 · π) / 𝑁)) ∈ ℝ ∧ 0 ≤ (tan‘((𝑚 · π) / 𝑁))))
114		lt2sq 12937	. . . . . . . . . . . . 13 ⊢ ((((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ ∧ 0 ≤ (tan‘((𝑘 · π) / 𝑁))) ∧ ((tan‘((𝑚 · π) / 𝑁)) ∈ ℝ ∧ 0 ≤ (tan‘((𝑚 · π) / 𝑁)))) → ((tan‘((𝑘 · π) / 𝑁)) < (tan‘((𝑚 · π) / 𝑁)) ↔ ((tan‘((𝑘 · π) / 𝑁))↑2) < ((tan‘((𝑚 · π) / 𝑁))↑2)))
115	112, 113, 114	syl2an 494	. . . . . . . . . . . 12 ⊢ (((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ⁺ ∧ (tan‘((𝑚 · π) / 𝑁)) ∈ ℝ⁺) → ((tan‘((𝑘 · π) / 𝑁)) < (tan‘((𝑚 · π) / 𝑁)) ↔ ((tan‘((𝑘 · π) / 𝑁))↑2) < ((tan‘((𝑚 · π) / 𝑁))↑2)))
116	109, 111, 115	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑘 · π) / 𝑁)) < (tan‘((𝑚 · π) / 𝑁)) ↔ ((tan‘((𝑘 · π) / 𝑁))↑2) < ((tan‘((𝑚 · π) / 𝑁))↑2)))
117	107, 116	mpbid 222	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑘 · π) / 𝑁))↑2) < ((tan‘((𝑚 · π) / 𝑁))↑2))
118		rpexpcl 12879	. . . . . . . . . . . 12 ⊢ (((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ⁺ ∧ 2 ∈ ℤ) → ((tan‘((𝑘 · π) / 𝑁))↑2) ∈ ℝ⁺)
119	109, 5, 118	sylancl 694	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑘 · π) / 𝑁))↑2) ∈ ℝ⁺)
120		rpexpcl 12879	. . . . . . . . . . . 12 ⊢ (((tan‘((𝑚 · π) / 𝑁)) ∈ ℝ⁺ ∧ 2 ∈ ℤ) → ((tan‘((𝑚 · π) / 𝑁))↑2) ∈ ℝ⁺)
121	111, 5, 120	sylancl 694	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑚 · π) / 𝑁))↑2) ∈ ℝ⁺)
122	119, 121	ltrecd 11890	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (((tan‘((𝑘 · π) / 𝑁))↑2) < ((tan‘((𝑚 · π) / 𝑁))↑2) ↔ (1 / ((tan‘((𝑚 · π) / 𝑁))↑2)) < (1 / ((tan‘((𝑘 · π) / 𝑁))↑2))))
123	117, 122	mpbid 222	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (1 / ((tan‘((𝑚 · π) / 𝑁))↑2)) < (1 / ((tan‘((𝑘 · π) / 𝑁))↑2)))
124		oveq1 6657	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝑛 · π) = (𝑚 · π))
125	124	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝑛 · π) / 𝑁) = ((𝑚 · π) / 𝑁))
126	125	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (tan‘((𝑛 · π) / 𝑁)) = (tan‘((𝑚 · π) / 𝑁)))
127	126	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((tan‘((𝑛 · π) / 𝑁))↑-2) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
128		ovex 6678	. . . . . . . . . . . 12 ⊢ ((tan‘((𝑚 · π) / 𝑁))↑-2) ∈ V
129	127, 57, 128	fvmpt 6282	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...𝑀) → (𝑇‘𝑚) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
130	129	ad2antll 765	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑚) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
131	111	rpcnd 11874	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑚 · π) / 𝑁)) ∈ ℂ)
132		2nn0 11309	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
133		expneg 12868	. . . . . . . . . . 11 ⊢ (((tan‘((𝑚 · π) / 𝑁)) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((tan‘((𝑚 · π) / 𝑁))↑-2) = (1 / ((tan‘((𝑚 · π) / 𝑁))↑2)))
134	131, 132, 133	sylancl 694	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑚 · π) / 𝑁))↑-2) = (1 / ((tan‘((𝑚 · π) / 𝑁))↑2)))
135	130, 134	eqtrd 2656	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑚) = (1 / ((tan‘((𝑚 · π) / 𝑁))↑2)))
136		oveq1 6657	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝑛 · π) = (𝑘 · π))
137	136	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → ((𝑛 · π) / 𝑁) = ((𝑘 · π) / 𝑁))
138	137	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (tan‘((𝑛 · π) / 𝑁)) = (tan‘((𝑘 · π) / 𝑁)))
139	138	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((tan‘((𝑛 · π) / 𝑁))↑-2) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
140		ovex 6678	. . . . . . . . . . . 12 ⊢ ((tan‘((𝑘 · π) / 𝑁))↑-2) ∈ V
141	139, 57, 140	fvmpt 6282	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑀) → (𝑇‘𝑘) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
142	141	ad2antrl 764	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑘) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
143	109	rpcnd 11874	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑘 · π) / 𝑁)) ∈ ℂ)
144		expneg 12868	. . . . . . . . . . 11 ⊢ (((tan‘((𝑘 · π) / 𝑁)) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((tan‘((𝑘 · π) / 𝑁))↑-2) = (1 / ((tan‘((𝑘 · π) / 𝑁))↑2)))
145	143, 132, 144	sylancl 694	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑘 · π) / 𝑁))↑-2) = (1 / ((tan‘((𝑘 · π) / 𝑁))↑2)))
146	142, 145	eqtrd 2656	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑘) = (1 / ((tan‘((𝑘 · π) / 𝑁))↑2)))
147	123, 135, 146	3brtr4d 4685	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑚) < (𝑇‘𝑘))
148	147	an32s 846	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) ∧ 𝑘 < 𝑚) → (𝑇‘𝑚) < (𝑇‘𝑘))
149	148	ex 450	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑘 < 𝑚 → (𝑇‘𝑚) < (𝑇‘𝑘)))
150	59, 60, 61, 63, 66, 149	eqord2 10559	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑀))) → (𝑥 = 𝑦 ↔ (𝑇‘𝑥) = (𝑇‘𝑦)))
151	150	biimprd 238	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑀))) → ((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
152	151	ralrimivva 2971	. . 3 ⊢ (𝑀 ∈ ℕ → ∀𝑥 ∈ (1...𝑀)∀𝑦 ∈ (1...𝑀)((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
153		dff13 6512	. . 3 ⊢ (𝑇:(1...𝑀)–1-1→(^◡𝑃 “ {0}) ↔ (𝑇:(1...𝑀)⟶(^◡𝑃 “ {0}) ∧ ∀𝑥 ∈ (1...𝑀)∀𝑦 ∈ (1...𝑀)((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦)))
154	58, 152, 153	sylanbrc 698	. 2 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)–1-1→(^◡𝑃 “ {0}))
155	48	simp2d 1074	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (deg‘𝑃) = 𝑀)
156		nnne0 11053	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ≠ 0)
157	155, 156	eqnetrd 2861	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (deg‘𝑃) ≠ 0)
158		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑃 = 0_𝑝 → (deg‘𝑃) = (deg‘0_𝑝))
159		dgr0 24018	. . . . . . . . . 10 ⊢ (deg‘0_𝑝) = 0
160	158, 159	syl6eq 2672	. . . . . . . . 9 ⊢ (𝑃 = 0_𝑝 → (deg‘𝑃) = 0)
161	160	necon3i 2826	. . . . . . . 8 ⊢ ((deg‘𝑃) ≠ 0 → 𝑃 ≠ 0_𝑝)
162	157, 161	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑃 ≠ 0_𝑝)
163		eqid 2622	. . . . . . . 8 ⊢ (^◡𝑃 “ {0}) = (^◡𝑃 “ {0})
164	163	fta1 24063	. . . . . . 7 ⊢ ((𝑃 ∈ (Poly‘ℂ) ∧ 𝑃 ≠ 0_𝑝) → ((^◡𝑃 “ {0}) ∈ Fin ∧ (#‘(^◡𝑃 “ {0})) ≤ (deg‘𝑃)))
165	49, 162, 164	syl2anc 693	. . . . . 6 ⊢ (𝑀 ∈ ℕ → ((^◡𝑃 “ {0}) ∈ Fin ∧ (#‘(^◡𝑃 “ {0})) ≤ (deg‘𝑃)))
166	165	simpld 475	. . . . 5 ⊢ (𝑀 ∈ ℕ → (^◡𝑃 “ {0}) ∈ Fin)
167		f1domg 7975	. . . . 5 ⊢ ((^◡𝑃 “ {0}) ∈ Fin → (𝑇:(1...𝑀)–1-1→(^◡𝑃 “ {0}) → (1...𝑀) ≼ (^◡𝑃 “ {0})))
168	166, 154, 167	sylc 65	. . . 4 ⊢ (𝑀 ∈ ℕ → (1...𝑀) ≼ (^◡𝑃 “ {0}))
169	165	simprd 479	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (#‘(^◡𝑃 “ {0})) ≤ (deg‘𝑃))
170		nnnn0 11299	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
171		hashfz1 13134	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → (#‘(1...𝑀)) = 𝑀)
172	170, 171	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (#‘(1...𝑀)) = 𝑀)
173	155, 172	eqtr4d 2659	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (deg‘𝑃) = (#‘(1...𝑀)))
174	169, 173	breqtrd 4679	. . . . 5 ⊢ (𝑀 ∈ ℕ → (#‘(^◡𝑃 “ {0})) ≤ (#‘(1...𝑀)))
175		fzfid 12772	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (1...𝑀) ∈ Fin)
176		hashdom 13168	. . . . . 6 ⊢ (((^◡𝑃 “ {0}) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((#‘(^◡𝑃 “ {0})) ≤ (#‘(1...𝑀)) ↔ (^◡𝑃 “ {0}) ≼ (1...𝑀)))
177	166, 175, 176	syl2anc 693	. . . . 5 ⊢ (𝑀 ∈ ℕ → ((#‘(^◡𝑃 “ {0})) ≤ (#‘(1...𝑀)) ↔ (^◡𝑃 “ {0}) ≼ (1...𝑀)))
178	174, 177	mpbid 222	. . . 4 ⊢ (𝑀 ∈ ℕ → (^◡𝑃 “ {0}) ≼ (1...𝑀))
179		sbth 8080	. . . 4 ⊢ (((1...𝑀) ≼ (^◡𝑃 “ {0}) ∧ (^◡𝑃 “ {0}) ≼ (1...𝑀)) → (1...𝑀) ≈ (^◡𝑃 “ {0}))
180	168, 178, 179	syl2anc 693	. . 3 ⊢ (𝑀 ∈ ℕ → (1...𝑀) ≈ (^◡𝑃 “ {0}))
181		f1finf1o 8187	. . 3 ⊢ (((1...𝑀) ≈ (^◡𝑃 “ {0}) ∧ (^◡𝑃 “ {0}) ∈ Fin) → (𝑇:(1...𝑀)–1-1→(^◡𝑃 “ {0}) ↔ 𝑇:(1...𝑀)–1-1-onto→(^◡𝑃 “ {0})))
182	180, 166, 181	syl2anc 693	. 2 ⊢ (𝑀 ∈ ℕ → (𝑇:(1...𝑀)–1-1→(^◡𝑃 “ {0}) ↔ 𝑇:(1...𝑀)–1-1-onto→(^◡𝑃 “ {0})))
183	154, 182	mpbid 222	1 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)–1-1-onto→(^◡𝑃 “ {0}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ⊆ wss 3574 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 ^◡ccnv 5113 “ cima 5117 Fn wfn 5883 ⟶wf 5884 –1-1→wf1 5885 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ≈ cen 7952 ≼ cdom 7953 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℝ⁺crp 11832 (,)cioo 12175 ...cfz 12326 ↑cexp 12860 Ccbc 13089 #chash 13117 Σcsu 14416 sincsin 14794 cosccos 14795 tanctan 14796 πcpi 14797 0_𝑝c0p 23436 Polycply 23940 coeffccoe 23942 degcdgr 23943

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-tan 14802 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-0p 23437 df-limc 23630 df-dv 23631 df-ply 23944 df-idp 23945 df-coe 23946 df-dgr 23947 df-quot 24046

This theorem is referenced by: basellem5 24811

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