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Theorem basellem4 26931

Description: Lemma for basel 26937. By basellem3 26930, 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 26161. (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 26928	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((𝑛 · π) / 𝑁) ∈ (0(,)(π / 2)))
3		tanrpcl 26355	. . . . . . . 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 12590	. . . . . . . 8 ⊢ 2 ∈ ℤ
6		znegcl 12593	. . . . . . . 8 ⊢ (2 ∈ ℤ → -2 ∈ ℤ)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ -2 ∈ ℤ
8		rpexpcl 14042	. . . . . . 7 ⊢ (((tan‘((𝑛 · π) / 𝑁)) ∈ ℝ⁺ ∧ -2 ∈ ℤ) → ((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℝ⁺)
9	4, 7, 8	sylancl 585	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℝ⁺)
10	9	rpcnd 13014	. . . . 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 26930	. . . . . . 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 590	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑃‘((tan‘((𝑛 · π) / 𝑁))↑-2)) = ((sin‘(𝑁 · ((𝑛 · π) / 𝑁))) / ((sin‘((𝑛 · π) / 𝑁))↑𝑁)))
14		elfzelz 13497	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℤ)
15	14	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑛 ∈ ℤ)
16	15	zred 12662	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑛 ∈ ℝ)
17		pire 26309	. . . . . . . . . . . 12 ⊢ π ∈ ℝ
18		remulcl 11190	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℝ ∧ π ∈ ℝ) → (𝑛 · π) ∈ ℝ)
19	16, 17, 18	sylancl 585	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑛 · π) ∈ ℝ)
20	19	recnd 11238	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑛 · π) ∈ ℂ)
21		2nn 12281	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
22		nnmulcl 12232	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (2 · 𝑀) ∈ ℕ)
23	21, 22	mpan 687	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ ℕ)
24	23	peano2nnd 12225	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → ((2 · 𝑀) + 1) ∈ ℕ)
25	1, 24	eqeltrid 2829	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ → 𝑁 ∈ ℕ)
26	25	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℕ)
27	26	nncnd 12224	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℂ)
28	26	nnne0d 12258	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ≠ 0)
29	20, 27, 28	divcan2d 11988	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑁 · ((𝑛 · π) / 𝑁)) = (𝑛 · π))
30	29	fveq2d 6885	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘(𝑁 · ((𝑛 · π) / 𝑁))) = (sin‘(𝑛 · π)))
31		sinkpi 26372	. . . . . . . . 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 2764	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘(𝑁 · ((𝑛 · π) / 𝑁))) = 0)
34	33	oveq1d 7416	. . . . . 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 12262	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((𝑛 · π) / 𝑁) ∈ ℝ)
36	35	resincld 16082	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ∈ ℝ)
37	36	recnd 11238	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ∈ ℂ)
38	26	nnnn0d 12528	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℕ₀)
39	37, 38	expcld 14107	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((sin‘((𝑛 · π) / 𝑁))↑𝑁) ∈ ℂ)
40		sincosq1sgn 26349	. . . . . . . . . . 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 494	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 0 < (sin‘((𝑛 · π) / 𝑁)))
43	42	gt0ne0d 11774	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ≠ 0)
44	26	nnzd 12581	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℤ)
45	37, 43, 44	expne0d 14113	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((sin‘((𝑛 · π) / 𝑁))↑𝑁) ≠ 0)
46	39, 45	div0d 11985	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (0 / ((sin‘((𝑛 · π) / 𝑁))↑𝑁)) = 0)
47	13, 34, 46	3eqtrd 2768	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑃‘((tan‘((𝑛 · π) / 𝑁))↑-2)) = 0)
48	1, 11	basellem2 26929	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (Poly‘ℂ) ∧ (deg‘𝑃) = 𝑀 ∧ (coeff‘𝑃) = (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))))
49	48	simp1d 1139	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑃 ∈ (Poly‘ℂ))
50		plyf 26051	. . . . . . . 8 ⊢ (𝑃 ∈ (Poly‘ℂ) → 𝑃:ℂ⟶ℂ)
51		ffn 6707	. . . . . . . 8 ⊢ (𝑃:ℂ⟶ℂ → 𝑃 Fn ℂ)
52	49, 50, 51	3syl 18	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑃 Fn ℂ)
53	52	adantr 480	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑃 Fn ℂ)
54		fniniseg 7051	. . . . . 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 710	. . . 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 7105	. . 3 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)⟶(^◡𝑃 “ {0}))
59		fveq2 6881	. . . . . 6 ⊢ (𝑘 = 𝑚 → (𝑇‘𝑘) = (𝑇‘𝑚))
60		fveq2 6881	. . . . . 6 ⊢ (𝑘 = 𝑥 → (𝑇‘𝑘) = (𝑇‘𝑥))
61		fveq2 6881	. . . . . 6 ⊢ (𝑘 = 𝑦 → (𝑇‘𝑘) = (𝑇‘𝑦))
62	14	zred 12662	. . . . . . 7 ⊢ (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℝ)
63	62	ssriv 3978	. . . . . 6 ⊢ (1...𝑀) ⊆ ℝ
64	9	rpred 13012	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℝ)
65	64, 57	fmptd 7105	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)⟶ℝ)
66	65	ffvelcdmda 7076	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (1...𝑀)) → (𝑇‘𝑘) ∈ ℝ)
67		simplr 766	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑘 < 𝑚)
68	63	sseli 3970	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℝ)
69	68	ad2antrl 725	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑘 ∈ ℝ)
70	63	sseli 3970	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℝ)
71	70	ad2antll 726	. . . . . . . . . . . . . . 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 26311	. . . . . . . . . . . . . . . 16 ⊢ 0 < π
74	73	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 0 < π)
75		ltmul1 12060	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ ∧ (π ∈ ℝ ∧ 0 < π)) → (𝑘 < 𝑚 ↔ (𝑘 · π) < (𝑚 · π)))
76	69, 71, 72, 74, 75	syl112anc 1371	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑘 < 𝑚 ↔ (𝑘 · π) < (𝑚 · π)))
77	67, 76	mpbid 231	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑘 · π) < (𝑚 · π))
78		remulcl 11190	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ π ∈ ℝ) → (𝑘 · π) ∈ ℝ)
79	69, 17, 78	sylancl 585	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑘 · π) ∈ ℝ)
80		remulcl 11190	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℝ ∧ π ∈ ℝ) → (𝑚 · π) ∈ ℝ)
81	71, 17, 80	sylancl 585	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑚 · π) ∈ ℝ)
82	25	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑁 ∈ ℕ)
83	82	nnred 12223	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑁 ∈ ℝ)
84	82	nngt0d 12257	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 0 < 𝑁)
85		ltdiv1 12074	. . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑘 · π) / 𝑁) < ((𝑚 · π) / 𝑁))
88		neghalfpirx 26317	. . . . . . . . . . . . . . 15 ⊢ -(π / 2) ∈ ℝ^*
89		pirp 26312	. . . . . . . . . . . . . . . . 17 ⊢ π ∈ ℝ⁺
90		rphalfcl 12997	. . . . . . . . . . . . . . . . 17 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
91		rpge0 12983	. . . . . . . . . . . . . . . . 17 ⊢ ((π / 2) ∈ ℝ⁺ → 0 ≤ (π / 2))
92	89, 90, 91	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ 0 ≤ (π / 2)
93		halfpire 26315	. . . . . . . . . . . . . . . . 17 ⊢ (π / 2) ∈ ℝ
94		le0neg2 11719	. . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . 15 ⊢ -(π / 2) ≤ 0
97		iooss1 13355	. . . . . . . . . . . . . . 15 ⊢ ((-(π / 2) ∈ ℝ^* ∧ -(π / 2) ≤ 0) → (0(,)(π / 2)) ⊆ (-(π / 2)(,)(π / 2)))
98	88, 96, 97	mp2an 689	. . . . . . . . . . . . . 14 ⊢ (0(,)(π / 2)) ⊆ (-(π / 2)(,)(π / 2))
99	1	basellem1 26928	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (1...𝑀)) → ((𝑘 · π) / 𝑁) ∈ (0(,)(π / 2)))
100	99	ad2ant2r 744	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑘 · π) / 𝑁) ∈ (0(,)(π / 2)))
101	98, 100	sselid 3972	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑘 · π) / 𝑁) ∈ (-(π / 2)(,)(π / 2)))
102	1	basellem1 26928	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑚 ∈ (1...𝑀)) → ((𝑚 · π) / 𝑁) ∈ (0(,)(π / 2)))
103	102	ad2ant2rl 746	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑚 · π) / 𝑁) ∈ (0(,)(π / 2)))
104	98, 103	sselid 3972	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑚 · π) / 𝑁) ∈ (-(π / 2)(,)(π / 2)))
105		tanord 26388	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑘 · π) / 𝑁)) < (tan‘((𝑚 · π) / 𝑁)))
108		tanrpcl 26355	. . . . . . . . . . . . 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 26355	. . . . . . . . . . . . 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 12985	. . . . . . . . . . . . 13 ⊢ ((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ⁺ → ((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ ∧ 0 ≤ (tan‘((𝑘 · π) / 𝑁))))
113		rprege0 12985	. . . . . . . . . . . . 13 ⊢ ((tan‘((𝑚 · π) / 𝑁)) ∈ ℝ⁺ → ((tan‘((𝑚 · π) / 𝑁)) ∈ ℝ ∧ 0 ≤ (tan‘((𝑚 · π) / 𝑁))))
114		lt2sq 14094	. . . . . . . . . . . . 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 595	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 231	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑘 · π) / 𝑁))↑2) < ((tan‘((𝑚 · π) / 𝑁))↑2))
118		rpexpcl 14042	. . . . . . . . . . . 12 ⊢ (((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ⁺ ∧ 2 ∈ ℤ) → ((tan‘((𝑘 · π) / 𝑁))↑2) ∈ ℝ⁺)
119	109, 5, 118	sylancl 585	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑘 · π) / 𝑁))↑2) ∈ ℝ⁺)
120		rpexpcl 14042	. . . . . . . . . . . 12 ⊢ (((tan‘((𝑚 · π) / 𝑁)) ∈ ℝ⁺ ∧ 2 ∈ ℤ) → ((tan‘((𝑚 · π) / 𝑁))↑2) ∈ ℝ⁺)
121	111, 5, 120	sylancl 585	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑚 · π) / 𝑁))↑2) ∈ ℝ⁺)
122	119, 121	ltrecd 13030	. . . . . . . . . 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 231	. . . . . . . . 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 7408	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → (𝑛 · π) = (𝑚 · π))
125	124	fvoveq1d 7423	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (tan‘((𝑛 · π) / 𝑁)) = (tan‘((𝑚 · π) / 𝑁)))
126	125	oveq1d 7416	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((tan‘((𝑛 · π) / 𝑁))↑-2) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
127		ovex 7434	. . . . . . . . . . . 12 ⊢ ((tan‘((𝑚 · π) / 𝑁))↑-2) ∈ V
128	126, 57, 127	fvmpt 6988	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...𝑀) → (𝑇‘𝑚) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
129	128	ad2antll 726	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑚) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
130	111	rpcnd 13014	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑚 · π) / 𝑁)) ∈ ℂ)
131		2nn0 12485	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
132		expneg 14031	. . . . . . . . . . 11 ⊢ (((tan‘((𝑚 · π) / 𝑁)) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((tan‘((𝑚 · π) / 𝑁))↑-2) = (1 / ((tan‘((𝑚 · π) / 𝑁))↑2)))
133	130, 131, 132	sylancl 585	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑚 · π) / 𝑁))↑-2) = (1 / ((tan‘((𝑚 · π) / 𝑁))↑2)))
134	129, 133	eqtrd 2764	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑚) = (1 / ((tan‘((𝑚 · π) / 𝑁))↑2)))
135		oveq1 7408	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑛 · π) = (𝑘 · π))
136	135	fvoveq1d 7423	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (tan‘((𝑛 · π) / 𝑁)) = (tan‘((𝑘 · π) / 𝑁)))
137	136	oveq1d 7416	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((tan‘((𝑛 · π) / 𝑁))↑-2) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
138		ovex 7434	. . . . . . . . . . . 12 ⊢ ((tan‘((𝑘 · π) / 𝑁))↑-2) ∈ V
139	137, 57, 138	fvmpt 6988	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑀) → (𝑇‘𝑘) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
140	139	ad2antrl 725	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑘) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
141	109	rpcnd 13014	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑘 · π) / 𝑁)) ∈ ℂ)
142		expneg 14031	. . . . . . . . . . 11 ⊢ (((tan‘((𝑘 · π) / 𝑁)) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((tan‘((𝑘 · π) / 𝑁))↑-2) = (1 / ((tan‘((𝑘 · π) / 𝑁))↑2)))
143	141, 131, 142	sylancl 585	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑘 · π) / 𝑁))↑-2) = (1 / ((tan‘((𝑘 · π) / 𝑁))↑2)))
144	140, 143	eqtrd 2764	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑘) = (1 / ((tan‘((𝑘 · π) / 𝑁))↑2)))
145	123, 134, 144	3brtr4d 5170	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑚) < (𝑇‘𝑘))
146	145	an32s 649	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) ∧ 𝑘 < 𝑚) → (𝑇‘𝑚) < (𝑇‘𝑘))
147	146	ex 412	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑘 < 𝑚 → (𝑇‘𝑚) < (𝑇‘𝑘)))
148	59, 60, 61, 63, 66, 147	eqord2 11741	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑀))) → (𝑥 = 𝑦 ↔ (𝑇‘𝑥) = (𝑇‘𝑦)))
149	148	biimprd 247	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑀))) → ((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
150	149	ralrimivva 3192	. . 3 ⊢ (𝑀 ∈ ℕ → ∀𝑥 ∈ (1...𝑀)∀𝑦 ∈ (1...𝑀)((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
151		dff13 7246	. . 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)))
152	58, 150, 151	sylanbrc 582	. 2 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)–1-1→(^◡𝑃 “ {0}))
153	48	simp2d 1140	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (deg‘𝑃) = 𝑀)
154		nnne0 12242	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ≠ 0)
155	153, 154	eqnetrd 3000	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (deg‘𝑃) ≠ 0)
156		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑃 = 0_𝑝 → (deg‘𝑃) = (deg‘0_𝑝))
157		dgr0 26116	. . . . . . . . . 10 ⊢ (deg‘0_𝑝) = 0
158	156, 157	eqtrdi 2780	. . . . . . . . 9 ⊢ (𝑃 = 0_𝑝 → (deg‘𝑃) = 0)
159	158	necon3i 2965	. . . . . . . 8 ⊢ ((deg‘𝑃) ≠ 0 → 𝑃 ≠ 0_𝑝)
160	155, 159	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑃 ≠ 0_𝑝)
161		eqid 2724	. . . . . . . 8 ⊢ (^◡𝑃 “ {0}) = (^◡𝑃 “ {0})
162	161	fta1 26161	. . . . . . 7 ⊢ ((𝑃 ∈ (Poly‘ℂ) ∧ 𝑃 ≠ 0_𝑝) → ((^◡𝑃 “ {0}) ∈ Fin ∧ (♯‘(^◡𝑃 “ {0})) ≤ (deg‘𝑃)))
163	49, 160, 162	syl2anc 583	. . . . . 6 ⊢ (𝑀 ∈ ℕ → ((^◡𝑃 “ {0}) ∈ Fin ∧ (♯‘(^◡𝑃 “ {0})) ≤ (deg‘𝑃)))
164	163	simpld 494	. . . . 5 ⊢ (𝑀 ∈ ℕ → (^◡𝑃 “ {0}) ∈ Fin)
165		f1domg 8963	. . . . 5 ⊢ ((^◡𝑃 “ {0}) ∈ Fin → (𝑇:(1...𝑀)–1-1→(^◡𝑃 “ {0}) → (1...𝑀) ≼ (^◡𝑃 “ {0})))
166	164, 152, 165	sylc 65	. . . 4 ⊢ (𝑀 ∈ ℕ → (1...𝑀) ≼ (^◡𝑃 “ {0}))
167	163	simprd 495	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (♯‘(^◡𝑃 “ {0})) ≤ (deg‘𝑃))
168		nnnn0 12475	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
169		hashfz1 14302	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
170	168, 169	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (♯‘(1...𝑀)) = 𝑀)
171	153, 170	eqtr4d 2767	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (deg‘𝑃) = (♯‘(1...𝑀)))
172	167, 171	breqtrd 5164	. . . . 5 ⊢ (𝑀 ∈ ℕ → (♯‘(^◡𝑃 “ {0})) ≤ (♯‘(1...𝑀)))
173		fzfid 13934	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (1...𝑀) ∈ Fin)
174		hashdom 14335	. . . . . 6 ⊢ (((^◡𝑃 “ {0}) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(^◡𝑃 “ {0})) ≤ (♯‘(1...𝑀)) ↔ (^◡𝑃 “ {0}) ≼ (1...𝑀)))
175	164, 173, 174	syl2anc 583	. . . . 5 ⊢ (𝑀 ∈ ℕ → ((♯‘(^◡𝑃 “ {0})) ≤ (♯‘(1...𝑀)) ↔ (^◡𝑃 “ {0}) ≼ (1...𝑀)))
176	172, 175	mpbid 231	. . . 4 ⊢ (𝑀 ∈ ℕ → (^◡𝑃 “ {0}) ≼ (1...𝑀))
177		sbth 9088	. . . 4 ⊢ (((1...𝑀) ≼ (^◡𝑃 “ {0}) ∧ (^◡𝑃 “ {0}) ≼ (1...𝑀)) → (1...𝑀) ≈ (^◡𝑃 “ {0}))
178	166, 176, 177	syl2anc 583	. . 3 ⊢ (𝑀 ∈ ℕ → (1...𝑀) ≈ (^◡𝑃 “ {0}))
179		f1finf1o 9266	. . 3 ⊢ (((1...𝑀) ≈ (^◡𝑃 “ {0}) ∧ (^◡𝑃 “ {0}) ∈ Fin) → (𝑇:(1...𝑀)–1-1→(^◡𝑃 “ {0}) ↔ 𝑇:(1...𝑀)–1-1-onto→(^◡𝑃 “ {0})))
180	178, 164, 179	syl2anc 583	. 2 ⊢ (𝑀 ∈ ℕ → (𝑇:(1...𝑀)–1-1→(^◡𝑃 “ {0}) ↔ 𝑇:(1...𝑀)–1-1-onto→(^◡𝑃 “ {0})))
181	152, 180	mpbid 231	1 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)–1-1-onto→(^◡𝑃 “ {0}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ⊆ wss 3940 {csn 4620 class class class wbr 5138 ↦ cmpt 5221 ^◡ccnv 5665 “ cima 5669 Fn wfn 6528 ⟶wf 6529 –1-1→wf1 6530 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7401 ≈ cen 8931 ≼ cdom 8932 Fincfn 8934 ℂcc 11103 ℝcr 11104 0cc0 11105 1c1 11106 + caddc 11108 · cmul 11110 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11440 -cneg 11441 / cdiv 11867 ℕcn 12208 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ℝ⁺crp 12970 (,)cioo 13320 ...cfz 13480 ↑cexp 14023 Ccbc 14258 ♯chash 14286 Σcsu 15628 sincsin 16003 cosccos 16004 tanctan 16005 πcpi 16006 0_𝑝c0p 25519 Polycply 26037 coeffccoe 26039 degcdgr 26040

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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-dju 9891 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-tan 16011 df-pi 16012 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-starv 17210 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-unif 17218 df-hom 17219 df-cco 17220 df-rest 17366 df-topn 17367 df-0g 17385 df-gsum 17386 df-topgen 17387 df-pt 17388 df-prds 17391 df-xrs 17446 df-qtop 17451 df-imas 17452 df-xps 17454 df-mre 17528 df-mrc 17529 df-acs 17531 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-submnd 18703 df-mulg 18985 df-cntz 19222 df-cmn 19691 df-psmet 21219 df-xmet 21220 df-met 21221 df-bl 21222 df-mopn 21223 df-fbas 21224 df-fg 21225 df-cnfld 21228 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cld 22844 df-ntr 22845 df-cls 22846 df-nei 22923 df-lp 22961 df-perf 22962 df-cn 23052 df-cnp 23053 df-haus 23140 df-tx 23387 df-hmeo 23580 df-fil 23671 df-fm 23763 df-flim 23764 df-flf 23765 df-xms 24147 df-ms 24148 df-tms 24149 df-cncf 24719 df-0p 25520 df-limc 25716 df-dv 25717 df-ply 26041 df-idp 26042 df-coe 26043 df-dgr 26044 df-quot 26144

This theorem is referenced by: basellem5 26932

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