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Theorem basellem4 27051

Description: Lemma for basel 27057. By basellem3 27050, 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 26273. (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 27048	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((𝑛 · π) / 𝑁) ∈ (0(,)(π / 2)))
3		tanrpcl 26470	. . . . . . . 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 12629	. . . . . . . 8 ⊢ 2 ∈ ℤ
6		znegcl 12632	. . . . . . . 8 ⊢ (2 ∈ ℤ → -2 ∈ ℤ)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ -2 ∈ ℤ
8		rpexpcl 14103	. . . . . . 7 ⊢ (((tan‘((𝑛 · π) / 𝑁)) ∈ ℝ⁺ ∧ -2 ∈ ℤ) → ((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℝ⁺)
9	4, 7, 8	sylancl 586	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℝ⁺)
10	9	rpcnd 13058	. . . . 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 27050	. . . . . . 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 591	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑃‘((tan‘((𝑛 · π) / 𝑁))↑-2)) = ((sin‘(𝑁 · ((𝑛 · π) / 𝑁))) / ((sin‘((𝑛 · π) / 𝑁))↑𝑁)))
14		elfzelz 13546	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℤ)
15	14	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑛 ∈ ℤ)
16	15	zred 12702	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑛 ∈ ℝ)
17		pire 26423	. . . . . . . . . . . 12 ⊢ π ∈ ℝ
18		remulcl 11219	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℝ ∧ π ∈ ℝ) → (𝑛 · π) ∈ ℝ)
19	16, 17, 18	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑛 · π) ∈ ℝ)
20	19	recnd 11268	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑛 · π) ∈ ℂ)
21		2nn 12318	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
22		nnmulcl 12269	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (2 · 𝑀) ∈ ℕ)
23	21, 22	mpan 690	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ ℕ)
24	23	peano2nnd 12262	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → ((2 · 𝑀) + 1) ∈ ℕ)
25	1, 24	eqeltrid 2839	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ → 𝑁 ∈ ℕ)
26	25	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℕ)
27	26	nncnd 12261	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℂ)
28	26	nnne0d 12295	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ≠ 0)
29	20, 27, 28	divcan2d 12024	. . . . . . . . 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 26488	. . . . . . . . 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 2771	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘(𝑁 · ((𝑛 · π) / 𝑁))) = 0)
34	33	oveq1d 7425	. . . . . 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 12299	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((𝑛 · π) / 𝑁) ∈ ℝ)
36	35	resincld 16166	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ∈ ℝ)
37	36	recnd 11268	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ∈ ℂ)
38	26	nnnn0d 12567	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℕ₀)
39	37, 38	expcld 14169	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((sin‘((𝑛 · π) / 𝑁))↑𝑁) ∈ ℂ)
40		sincosq1sgn 26464	. . . . . . . . . . 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 11806	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ≠ 0)
44	26	nnzd 12620	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℤ)
45	37, 43, 44	expne0d 14175	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((sin‘((𝑛 · π) / 𝑁))↑𝑁) ≠ 0)
46	39, 45	div0d 12021	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (0 / ((sin‘((𝑛 · π) / 𝑁))↑𝑁)) = 0)
47	13, 34, 46	3eqtrd 2775	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑃‘((tan‘((𝑛 · π) / 𝑁))↑-2)) = 0)
48	1, 11	basellem2 27049	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (Poly‘ℂ) ∧ (deg‘𝑃) = 𝑀 ∧ (coeff‘𝑃) = (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))))
49	48	simp1d 1142	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑃 ∈ (Poly‘ℂ))
50		plyf 26160	. . . . . . . 8 ⊢ (𝑃 ∈ (Poly‘ℂ) → 𝑃:ℂ⟶ℂ)
51		ffn 6711	. . . . . . . 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 7055	. . . . . 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 713	. . . 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 7109	. . 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 12702	. . . . . . 7 ⊢ (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℝ)
63	62	ssriv 3967	. . . . . 6 ⊢ (1...𝑀) ⊆ ℝ
64	9	rpred 13056	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℝ)
65	64, 57	fmptd 7109	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)⟶ℝ)
66	65	ffvelcdmda 7079	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (1...𝑀)) → (𝑇‘𝑘) ∈ ℝ)
67		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑘 < 𝑚)
68	63	sseli 3959	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℝ)
69	68	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑘 ∈ ℝ)
70	63	sseli 3959	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℝ)
71	70	ad2antll 729	. . . . . . . . . . . . . . 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 26425	. . . . . . . . . . . . . . . 16 ⊢ 0 < π
74	73	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 0 < π)
75		ltmul1 12096	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ ∧ (π ∈ ℝ ∧ 0 < π)) → (𝑘 < 𝑚 ↔ (𝑘 · π) < (𝑚 · π)))
76	69, 71, 72, 74, 75	syl112anc 1376	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑘 < 𝑚 ↔ (𝑘 · π) < (𝑚 · π)))
77	67, 76	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑘 · π) < (𝑚 · π))
78		remulcl 11219	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ π ∈ ℝ) → (𝑘 · π) ∈ ℝ)
79	69, 17, 78	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑘 · π) ∈ ℝ)
80		remulcl 11219	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℝ ∧ π ∈ ℝ) → (𝑚 · π) ∈ ℝ)
81	71, 17, 80	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑚 · π) ∈ ℝ)
82	25	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑁 ∈ ℕ)
83	82	nnred 12260	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑁 ∈ ℝ)
84	82	nngt0d 12294	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 0 < 𝑁)
85		ltdiv1 12111	. . . . . . . . . . . . . 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 1376	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑘 · π) / 𝑁) < ((𝑚 · π) / 𝑁))
88		neghalfpirx 26432	. . . . . . . . . . . . . . 15 ⊢ -(π / 2) ∈ ℝ^*
89		pirp 26427	. . . . . . . . . . . . . . . . 17 ⊢ π ∈ ℝ⁺
90		rphalfcl 13041	. . . . . . . . . . . . . . . . 17 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
91		rpge0 13027	. . . . . . . . . . . . . . . . 17 ⊢ ((π / 2) ∈ ℝ⁺ → 0 ≤ (π / 2))
92	89, 90, 91	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ 0 ≤ (π / 2)
93		halfpire 26430	. . . . . . . . . . . . . . . . 17 ⊢ (π / 2) ∈ ℝ
94		le0neg2 11751	. . . . . . . . . . . . . . . . 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 230	. . . . . . . . . . . . . . 15 ⊢ -(π / 2) ≤ 0
97		iooss1 13402	. . . . . . . . . . . . . . 15 ⊢ ((-(π / 2) ∈ ℝ^* ∧ -(π / 2) ≤ 0) → (0(,)(π / 2)) ⊆ (-(π / 2)(,)(π / 2)))
98	88, 96, 97	mp2an 692	. . . . . . . . . . . . . 14 ⊢ (0(,)(π / 2)) ⊆ (-(π / 2)(,)(π / 2))
99	1	basellem1 27048	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (1...𝑀)) → ((𝑘 · π) / 𝑁) ∈ (0(,)(π / 2)))
100	99	ad2ant2r 747	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑘 · π) / 𝑁) ∈ (0(,)(π / 2)))
101	98, 100	sselid 3961	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑘 · π) / 𝑁) ∈ (-(π / 2)(,)(π / 2)))
102	1	basellem1 27048	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑚 ∈ (1...𝑀)) → ((𝑚 · π) / 𝑁) ∈ (0(,)(π / 2)))
103	102	ad2ant2rl 749	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑚 · π) / 𝑁) ∈ (0(,)(π / 2)))
104	98, 103	sselid 3961	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑚 · π) / 𝑁) ∈ (-(π / 2)(,)(π / 2)))
105		tanord 26504	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 232	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑘 · π) / 𝑁)) < (tan‘((𝑚 · π) / 𝑁)))
108		tanrpcl 26470	. . . . . . . . . . . . 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 26470	. . . . . . . . . . . . 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 13029	. . . . . . . . . . . . 13 ⊢ ((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ⁺ → ((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ ∧ 0 ≤ (tan‘((𝑘 · π) / 𝑁))))
113		rprege0 13029	. . . . . . . . . . . . 13 ⊢ ((tan‘((𝑚 · π) / 𝑁)) ∈ ℝ⁺ → ((tan‘((𝑚 · π) / 𝑁)) ∈ ℝ ∧ 0 ≤ (tan‘((𝑚 · π) / 𝑁))))
114		lt2sq 14156	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑘 · π) / 𝑁))↑2) < ((tan‘((𝑚 · π) / 𝑁))↑2))
118		rpexpcl 14103	. . . . . . . . . . . 12 ⊢ (((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ⁺ ∧ 2 ∈ ℤ) → ((tan‘((𝑘 · π) / 𝑁))↑2) ∈ ℝ⁺)
119	109, 5, 118	sylancl 586	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑘 · π) / 𝑁))↑2) ∈ ℝ⁺)
120		rpexpcl 14103	. . . . . . . . . . . 12 ⊢ (((tan‘((𝑚 · π) / 𝑁)) ∈ ℝ⁺ ∧ 2 ∈ ℤ) → ((tan‘((𝑚 · π) / 𝑁))↑2) ∈ ℝ⁺)
121	111, 5, 120	sylancl 586	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑚 · π) / 𝑁))↑2) ∈ ℝ⁺)
122	119, 121	ltrecd 13074	. . . . . . . . . 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 232	. . . . . . . . 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 7417	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → (𝑛 · π) = (𝑚 · π))
125	124	fvoveq1d 7432	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (tan‘((𝑛 · π) / 𝑁)) = (tan‘((𝑚 · π) / 𝑁)))
126	125	oveq1d 7425	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((tan‘((𝑛 · π) / 𝑁))↑-2) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
127		ovex 7443	. . . . . . . . . . . 12 ⊢ ((tan‘((𝑚 · π) / 𝑁))↑-2) ∈ V
128	126, 57, 127	fvmpt 6991	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...𝑀) → (𝑇‘𝑚) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
129	128	ad2antll 729	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑚) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
130	111	rpcnd 13058	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑚 · π) / 𝑁)) ∈ ℂ)
131		2nn0 12523	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
132		expneg 14092	. . . . . . . . . . 11 ⊢ (((tan‘((𝑚 · π) / 𝑁)) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((tan‘((𝑚 · π) / 𝑁))↑-2) = (1 / ((tan‘((𝑚 · π) / 𝑁))↑2)))
133	130, 131, 132	sylancl 586	. . . . . . . . . 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 2771	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑚) = (1 / ((tan‘((𝑚 · π) / 𝑁))↑2)))
135		oveq1 7417	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑛 · π) = (𝑘 · π))
136	135	fvoveq1d 7432	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (tan‘((𝑛 · π) / 𝑁)) = (tan‘((𝑘 · π) / 𝑁)))
137	136	oveq1d 7425	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((tan‘((𝑛 · π) / 𝑁))↑-2) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
138		ovex 7443	. . . . . . . . . . . 12 ⊢ ((tan‘((𝑘 · π) / 𝑁))↑-2) ∈ V
139	137, 57, 138	fvmpt 6991	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑀) → (𝑇‘𝑘) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
140	139	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑘) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
141	109	rpcnd 13058	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑘 · π) / 𝑁)) ∈ ℂ)
142		expneg 14092	. . . . . . . . . . 11 ⊢ (((tan‘((𝑘 · π) / 𝑁)) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((tan‘((𝑘 · π) / 𝑁))↑-2) = (1 / ((tan‘((𝑘 · π) / 𝑁))↑2)))
143	141, 131, 142	sylancl 586	. . . . . . . . . 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 2771	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑘) = (1 / ((tan‘((𝑘 · π) / 𝑁))↑2)))
145	123, 134, 144	3brtr4d 5156	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑚) < (𝑇‘𝑘))
146	145	an32s 652	. . . . . . 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 11773	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑀))) → (𝑥 = 𝑦 ↔ (𝑇‘𝑥) = (𝑇‘𝑦)))
149	148	biimprd 248	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑀))) → ((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
150	149	ralrimivva 3188	. . 3 ⊢ (𝑀 ∈ ℕ → ∀𝑥 ∈ (1...𝑀)∀𝑦 ∈ (1...𝑀)((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
151		dff13 7252	. . 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 583	. 2 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)–1-1→(^◡𝑃 “ {0}))
153	48	simp2d 1143	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (deg‘𝑃) = 𝑀)
154		nnne0 12279	. . . . . . . . 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 26225	. . . . . . . . . 10 ⊢ (deg‘0_𝑝) = 0
158	156, 157	eqtrdi 2787	. . . . . . . . 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 2736	. . . . . . . 8 ⊢ (^◡𝑃 “ {0}) = (^◡𝑃 “ {0})
162	161	fta1 26273	. . . . . . 7 ⊢ ((𝑃 ∈ (Poly‘ℂ) ∧ 𝑃 ≠ 0_𝑝) → ((^◡𝑃 “ {0}) ∈ Fin ∧ (♯‘(^◡𝑃 “ {0})) ≤ (deg‘𝑃)))
163	49, 160, 162	syl2anc 584	. . . . . 6 ⊢ (𝑀 ∈ ℕ → ((^◡𝑃 “ {0}) ∈ Fin ∧ (♯‘(^◡𝑃 “ {0})) ≤ (deg‘𝑃)))
164	163	simpld 494	. . . . 5 ⊢ (𝑀 ∈ ℕ → (^◡𝑃 “ {0}) ∈ Fin)
165		f1domg 8991	. . . . 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 12513	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
169		hashfz1 14369	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
170	168, 169	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (♯‘(1...𝑀)) = 𝑀)
171	153, 170	eqtr4d 2774	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (deg‘𝑃) = (♯‘(1...𝑀)))
172	167, 171	breqtrd 5150	. . . . 5 ⊢ (𝑀 ∈ ℕ → (♯‘(^◡𝑃 “ {0})) ≤ (♯‘(1...𝑀)))
173		fzfid 13996	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (1...𝑀) ∈ Fin)
174		hashdom 14402	. . . . . 6 ⊢ (((^◡𝑃 “ {0}) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(^◡𝑃 “ {0})) ≤ (♯‘(1...𝑀)) ↔ (^◡𝑃 “ {0}) ≼ (1...𝑀)))
175	164, 173, 174	syl2anc 584	. . . . 5 ⊢ (𝑀 ∈ ℕ → ((♯‘(^◡𝑃 “ {0})) ≤ (♯‘(1...𝑀)) ↔ (^◡𝑃 “ {0}) ≼ (1...𝑀)))
176	172, 175	mpbid 232	. . . 4 ⊢ (𝑀 ∈ ℕ → (^◡𝑃 “ {0}) ≼ (1...𝑀))
177		sbth 9112	. . . 4 ⊢ (((1...𝑀) ≼ (^◡𝑃 “ {0}) ∧ (^◡𝑃 “ {0}) ≼ (1...𝑀)) → (1...𝑀) ≈ (^◡𝑃 “ {0}))
178	166, 176, 177	syl2anc 584	. . 3 ⊢ (𝑀 ∈ ℕ → (1...𝑀) ≈ (^◡𝑃 “ {0}))
179		f1finf1o 9282	. . 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 584	. 2 ⊢ (𝑀 ∈ ℕ → (𝑇:(1...𝑀)–1-1→(^◡𝑃 “ {0}) ↔ 𝑇:(1...𝑀)–1-1-onto→(^◡𝑃 “ {0})))
181	152, 180	mpbid 232	1 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)–1-1-onto→(^◡𝑃 “ {0}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 ⊆ wss 3931 {csn 4606 class class class wbr 5124 ↦ cmpt 5206 ^◡ccnv 5658 “ cima 5662 Fn wfn 6531 ⟶wf 6532 –1-1→wf1 6533 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7410 ≈ cen 8961 ≼ cdom 8962 Fincfn 8964 ℂcc 11132 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 · cmul 11139 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275 − cmin 11471 -cneg 11472 / cdiv 11899 ℕcn 12245 2c2 12300 ℕ₀cn0 12506 ℤcz 12593 ℝ⁺crp 13013 (,)cioo 13367 ...cfz 13529 ↑cexp 14084 Ccbc 14325 ♯chash 14353 Σcsu 15707 sincsin 16084 cosccos 16085 tanctan 16086 πcpi 16087 0_𝑝c0p 25627 Polycply 26146 coeffccoe 26148 degcdgr 26149

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 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 12507 df-xnn0 12580 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-shft 15091 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-limsup 15492 df-clim 15509 df-rlim 15510 df-sum 15708 df-ef 16088 df-sin 16090 df-cos 16091 df-tan 16092 df-pi 16093 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19768 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-0p 25628 df-limc 25824 df-dv 25825 df-ply 26150 df-idp 26151 df-coe 26152 df-dgr 26153 df-quot 26256

This theorem is referenced by: basellem5 27052

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