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Theorem basellem4 27047

Description: Lemma for basel 27053. By basellem3 27046, 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 26274. (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 27044	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((𝑛 · π) / 𝑁) ∈ (0(,)(π / 2)))
3		tanrpcl 26468	. . . . . . . 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 12559	. . . . . . . 8 ⊢ 2 ∈ ℤ
6		znegcl 12562	. . . . . . . 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 587	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℝ⁺)
10	9	rpcnd 12988	. . . . 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 27046	. . . . . . 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 592	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑃‘((tan‘((𝑛 · π) / 𝑁))↑-2)) = ((sin‘(𝑁 · ((𝑛 · π) / 𝑁))) / ((sin‘((𝑛 · π) / 𝑁))↑𝑁)))
14		elfzelz 13478	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℤ)
15	14	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑛 ∈ ℤ)
16	15	zred 12633	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑛 ∈ ℝ)
17		pire 26421	. . . . . . . . . . . 12 ⊢ π ∈ ℝ
18		remulcl 11123	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℝ ∧ π ∈ ℝ) → (𝑛 · π) ∈ ℝ)
19	16, 17, 18	sylancl 587	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑛 · π) ∈ ℝ)
20	19	recnd 11173	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑛 · π) ∈ ℂ)
21		2nn 12254	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
22		nnmulcl 12198	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (2 · 𝑀) ∈ ℕ)
23	21, 22	mpan 691	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ ℕ)
24	23	peano2nnd 12191	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → ((2 · 𝑀) + 1) ∈ ℕ)
25	1, 24	eqeltrid 2840	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ → 𝑁 ∈ ℕ)
26	25	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℕ)
27	26	nncnd 12190	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℂ)
28	26	nnne0d 12227	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ≠ 0)
29	20, 27, 28	divcan2d 11933	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑁 · ((𝑛 · π) / 𝑁)) = (𝑛 · π))
30	29	fveq2d 6844	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘(𝑁 · ((𝑛 · π) / 𝑁))) = (sin‘(𝑛 · π)))
31		sinkpi 26486	. . . . . . . . 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 7382	. . . . . 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 12231	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((𝑛 · π) / 𝑁) ∈ ℝ)
36	35	resincld 16110	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ∈ ℝ)
37	36	recnd 11173	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ∈ ℂ)
38	26	nnnn0d 12498	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℕ₀)
39	37, 38	expcld 14108	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((sin‘((𝑛 · π) / 𝑁))↑𝑁) ∈ ℂ)
40		sincosq1sgn 26462	. . . . . . . . . . 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 11714	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ≠ 0)
44	26	nnzd 12550	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℤ)
45	37, 43, 44	expne0d 14114	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((sin‘((𝑛 · π) / 𝑁))↑𝑁) ≠ 0)
46	39, 45	div0d 11930	. . . . . 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 27045	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (Poly‘ℂ) ∧ (deg‘𝑃) = 𝑀 ∧ (coeff‘𝑃) = (𝑛 ∈ ℕ₀ ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))))
49	48	simp1d 1143	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑃 ∈ (Poly‘ℂ))
50		plyf 26163	. . . . . . . 8 ⊢ (𝑃 ∈ (Poly‘ℂ) → 𝑃:ℂ⟶ℂ)
51		ffn 6668	. . . . . . . 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 7012	. . . . . 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 714	. . . 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 7066	. . 3 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)⟶(^◡𝑃 “ {0}))
59		fveq2 6840	. . . . . 6 ⊢ (𝑘 = 𝑚 → (𝑇‘𝑘) = (𝑇‘𝑚))
60		fveq2 6840	. . . . . 6 ⊢ (𝑘 = 𝑥 → (𝑇‘𝑘) = (𝑇‘𝑥))
61		fveq2 6840	. . . . . 6 ⊢ (𝑘 = 𝑦 → (𝑇‘𝑘) = (𝑇‘𝑦))
62	14	zred 12633	. . . . . . 7 ⊢ (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℝ)
63	62	ssriv 3925	. . . . . 6 ⊢ (1...𝑀) ⊆ ℝ
64	9	rpred 12986	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℝ)
65	64, 57	fmptd 7066	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)⟶ℝ)
66	65	ffvelcdmda 7036	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (1...𝑀)) → (𝑇‘𝑘) ∈ ℝ)
67		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑘 < 𝑚)
68	63	sseli 3917	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℝ)
69	68	ad2antrl 729	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑘 ∈ ℝ)
70	63	sseli 3917	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℝ)
71	70	ad2antll 730	. . . . . . . . . . . . . . 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 26423	. . . . . . . . . . . . . . . 16 ⊢ 0 < π
74	73	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 0 < π)
75		ltmul1 12005	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ ∧ (π ∈ ℝ ∧ 0 < π)) → (𝑘 < 𝑚 ↔ (𝑘 · π) < (𝑚 · π)))
76	69, 71, 72, 74, 75	syl112anc 1377	. . . . . . . . . . . . . 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 11123	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ π ∈ ℝ) → (𝑘 · π) ∈ ℝ)
79	69, 17, 78	sylancl 587	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑘 · π) ∈ ℝ)
80		remulcl 11123	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℝ ∧ π ∈ ℝ) → (𝑚 · π) ∈ ℝ)
81	71, 17, 80	sylancl 587	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑚 · π) ∈ ℝ)
82	25	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑁 ∈ ℕ)
83	82	nnred 12189	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑁 ∈ ℝ)
84	82	nngt0d 12226	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 0 < 𝑁)
85		ltdiv1 12020	. . . . . . . . . . . . . 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 1377	. . . . . . . . . . . . 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 26430	. . . . . . . . . . . . . . 15 ⊢ -(π / 2) ∈ ℝ^*
89		pirp 26425	. . . . . . . . . . . . . . . . 17 ⊢ π ∈ ℝ⁺
90		rphalfcl 12971	. . . . . . . . . . . . . . . . 17 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
91		rpge0 12956	. . . . . . . . . . . . . . . . 17 ⊢ ((π / 2) ∈ ℝ⁺ → 0 ≤ (π / 2))
92	89, 90, 91	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ 0 ≤ (π / 2)
93		halfpire 26428	. . . . . . . . . . . . . . . . 17 ⊢ (π / 2) ∈ ℝ
94		le0neg2 11659	. . . . . . . . . . . . . . . . 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 13333	. . . . . . . . . . . . . . 15 ⊢ ((-(π / 2) ∈ ℝ^* ∧ -(π / 2) ≤ 0) → (0(,)(π / 2)) ⊆ (-(π / 2)(,)(π / 2)))
98	88, 96, 97	mp2an 693	. . . . . . . . . . . . . 14 ⊢ (0(,)(π / 2)) ⊆ (-(π / 2)(,)(π / 2))
99	1	basellem1 27044	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (1...𝑀)) → ((𝑘 · π) / 𝑁) ∈ (0(,)(π / 2)))
100	99	ad2ant2r 748	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑘 · π) / 𝑁) ∈ (0(,)(π / 2)))
101	98, 100	sselid 3919	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑘 · π) / 𝑁) ∈ (-(π / 2)(,)(π / 2)))
102	1	basellem1 27044	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑚 ∈ (1...𝑀)) → ((𝑚 · π) / 𝑁) ∈ (0(,)(π / 2)))
103	102	ad2ant2rl 750	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑚 · π) / 𝑁) ∈ (0(,)(π / 2)))
104	98, 103	sselid 3919	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑚 · π) / 𝑁) ∈ (-(π / 2)(,)(π / 2)))
105		tanord 26502	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 26468	. . . . . . . . . . . . 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 26468	. . . . . . . . . . . . 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 12958	. . . . . . . . . . . . 13 ⊢ ((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ⁺ → ((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ ∧ 0 ≤ (tan‘((𝑘 · π) / 𝑁))))
113		rprege0 12958	. . . . . . . . . . . . 13 ⊢ ((tan‘((𝑚 · π) / 𝑁)) ∈ ℝ⁺ → ((tan‘((𝑚 · π) / 𝑁)) ∈ ℝ ∧ 0 ≤ (tan‘((𝑚 · π) / 𝑁))))
114		lt2sq 14095	. . . . . . . . . . . . 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 597	. . . . . . . . . . . 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 585	. . . . . . . . . . 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 14042	. . . . . . . . . . . 12 ⊢ (((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ⁺ ∧ 2 ∈ ℤ) → ((tan‘((𝑘 · π) / 𝑁))↑2) ∈ ℝ⁺)
119	109, 5, 118	sylancl 587	. . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑚 · π) / 𝑁))↑2) ∈ ℝ⁺)
122	119, 121	ltrecd 13004	. . . . . . . . . 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 7374	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → (𝑛 · π) = (𝑚 · π))
125	124	fvoveq1d 7389	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (tan‘((𝑛 · π) / 𝑁)) = (tan‘((𝑚 · π) / 𝑁)))
126	125	oveq1d 7382	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((tan‘((𝑛 · π) / 𝑁))↑-2) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
127		ovex 7400	. . . . . . . . . . . 12 ⊢ ((tan‘((𝑚 · π) / 𝑁))↑-2) ∈ V
128	126, 57, 127	fvmpt 6947	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...𝑀) → (𝑇‘𝑚) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
129	128	ad2antll 730	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑚) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
130	111	rpcnd 12988	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑚 · π) / 𝑁)) ∈ ℂ)
131		2nn0 12454	. . . . . . . . . . 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 587	. . . . . . . . . 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 7374	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑛 · π) = (𝑘 · π))
136	135	fvoveq1d 7389	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (tan‘((𝑛 · π) / 𝑁)) = (tan‘((𝑘 · π) / 𝑁)))
137	136	oveq1d 7382	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((tan‘((𝑛 · π) / 𝑁))↑-2) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
138		ovex 7400	. . . . . . . . . . . 12 ⊢ ((tan‘((𝑘 · π) / 𝑁))↑-2) ∈ V
139	137, 57, 138	fvmpt 6947	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑀) → (𝑇‘𝑘) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
140	139	ad2antrl 729	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑘) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
141	109	rpcnd 12988	. . . . . . . . . . 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 587	. . . . . . . . . 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 5117	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑚) < (𝑇‘𝑘))
146	145	an32s 653	. . . . . . 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 11681	. . . . 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 3180	. . 3 ⊢ (𝑀 ∈ ℕ → ∀𝑥 ∈ (1...𝑀)∀𝑦 ∈ (1...𝑀)((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
151		dff13 7209	. . 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 584	. 2 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)–1-1→(^◡𝑃 “ {0}))
153	48	simp2d 1144	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (deg‘𝑃) = 𝑀)
154		nnne0 12211	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ≠ 0)
155	153, 154	eqnetrd 2999	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (deg‘𝑃) ≠ 0)
156		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑃 = 0_𝑝 → (deg‘𝑃) = (deg‘0_𝑝))
157		dgr0 26227	. . . . . . . . . 10 ⊢ (deg‘0_𝑝) = 0
158	156, 157	eqtrdi 2787	. . . . . . . . 9 ⊢ (𝑃 = 0_𝑝 → (deg‘𝑃) = 0)
159	158	necon3i 2964	. . . . . . . 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 26274	. . . . . . 7 ⊢ ((𝑃 ∈ (Poly‘ℂ) ∧ 𝑃 ≠ 0_𝑝) → ((^◡𝑃 “ {0}) ∈ Fin ∧ (♯‘(^◡𝑃 “ {0})) ≤ (deg‘𝑃)))
163	49, 160, 162	syl2anc 585	. . . . . 6 ⊢ (𝑀 ∈ ℕ → ((^◡𝑃 “ {0}) ∈ Fin ∧ (♯‘(^◡𝑃 “ {0})) ≤ (deg‘𝑃)))
164	163	simpld 494	. . . . 5 ⊢ (𝑀 ∈ ℕ → (^◡𝑃 “ {0}) ∈ Fin)
165		f1domg 8918	. . . . 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 12444	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
169		hashfz1 14308	. . . . . . . 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 5111	. . . . 5 ⊢ (𝑀 ∈ ℕ → (♯‘(^◡𝑃 “ {0})) ≤ (♯‘(1...𝑀)))
173		fzfid 13935	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (1...𝑀) ∈ Fin)
174		hashdom 14341	. . . . . 6 ⊢ (((^◡𝑃 “ {0}) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(^◡𝑃 “ {0})) ≤ (♯‘(1...𝑀)) ↔ (^◡𝑃 “ {0}) ≼ (1...𝑀)))
175	164, 173, 174	syl2anc 585	. . . . 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 9035	. . . 4 ⊢ (((1...𝑀) ≼ (^◡𝑃 “ {0}) ∧ (^◡𝑃 “ {0}) ≼ (1...𝑀)) → (1...𝑀) ≈ (^◡𝑃 “ {0}))
178	166, 176, 177	syl2anc 585	. . 3 ⊢ (𝑀 ∈ ℕ → (1...𝑀) ≈ (^◡𝑃 “ {0}))
179		f1finf1o 9183	. . 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 585	. 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 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ⊆ wss 3889 {csn 4567 class class class wbr 5085 ↦ cmpt 5166 ^◡ccnv 5630 “ cima 5634 Fn wfn 6493 ⟶wf 6494 –1-1→wf1 6495 –1-1-onto→wf1o 6497 ‘cfv 6498 (class class class)co 7367 ≈ cen 8890 ≼ cdom 8891 Fincfn 8893 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 − cmin 11377 -cneg 11378 / cdiv 11807 ℕcn 12174 2c2 12236 ℕ₀cn0 12437 ℤcz 12524 ℝ⁺crp 12942 (,)cioo 13298 ...cfz 13461 ↑cexp 14023 Ccbc 14264 ♯chash 14292 Σcsu 15648 sincsin 16028 cosccos 16029 tanctan 16030 πcpi 16031 0_𝑝c0p 25636 Polycply 26149 coeffccoe 26151 degcdgr 26152

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-tan 16036 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-0p 25637 df-limc 25833 df-dv 25834 df-ply 26153 df-idp 26154 df-coe 26155 df-dgr 26156 df-quot 26257

This theorem is referenced by: basellem5 27048

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