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Theorem basellem4 26433

Description: Lemma for basel 26439. By basellem3 26432, 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 25668. (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 26430	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((𝑛 · π) / 𝑁) ∈ (0(,)(π / 2)))
3		tanrpcl 25861	. . . . . . . 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 12535	. . . . . . . 8 ⊢ 2 ∈ ℤ
6		znegcl 12538	. . . . . . . 8 ⊢ (2 ∈ ℤ → -2 ∈ ℤ)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ -2 ∈ ℤ
8		rpexpcl 13986	. . . . . . 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 12959	. . . . 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 26432	. . . . . . 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 13441	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℤ)
15	14	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑛 ∈ ℤ)
16	15	zred 12607	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑛 ∈ ℝ)
17		pire 25815	. . . . . . . . . . . 12 ⊢ π ∈ ℝ
18		remulcl 11136	. . . . . . . . . . . 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 11183	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑛 · π) ∈ ℂ)
21		2nn 12226	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
22		nnmulcl 12177	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (2 · 𝑀) ∈ ℕ)
23	21, 22	mpan 688	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) ∈ ℕ)
24	23	peano2nnd 12170	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → ((2 · 𝑀) + 1) ∈ ℕ)
25	1, 24	eqeltrid 2842	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ → 𝑁 ∈ ℕ)
26	25	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℕ)
27	26	nncnd 12169	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℂ)
28	26	nnne0d 12203	. . . . . . . . . 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 6846	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘(𝑁 · ((𝑛 · π) / 𝑁))) = (sin‘(𝑛 · π)))
31		sinkpi 25878	. . . . . . . . 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 2776	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘(𝑁 · ((𝑛 · π) / 𝑁))) = 0)
34	33	oveq1d 7372	. . . . . 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 12207	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((𝑛 · π) / 𝑁) ∈ ℝ)
36	35	resincld 16025	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ∈ ℝ)
37	36	recnd 11183	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ∈ ℂ)
38	26	nnnn0d 12473	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℕ₀)
39	37, 38	expcld 14051	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((sin‘((𝑛 · π) / 𝑁))↑𝑁) ∈ ℂ)
40		sincosq1sgn 25855	. . . . . . . . . . 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 495	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 0 < (sin‘((𝑛 · π) / 𝑁)))
43	42	gt0ne0d 11719	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (sin‘((𝑛 · π) / 𝑁)) ≠ 0)
44	26	nnzd 12526	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑁 ∈ ℤ)
45	37, 43, 44	expne0d 14057	. . . . . . 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 2780	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → (𝑃‘((tan‘((𝑛 · π) / 𝑁))↑-2)) = 0)
48	1, 11	basellem2 26431	. . . . . . . . 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 25559	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → 𝑃 Fn ℂ)
54		fniniseg 7010	. . . . . 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 711	. . . 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 7062	. . 3 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)⟶(^◡𝑃 “ {0}))
59		fveq2 6842	. . . . . 6 ⊢ (𝑘 = 𝑚 → (𝑇‘𝑘) = (𝑇‘𝑚))
60		fveq2 6842	. . . . . 6 ⊢ (𝑘 = 𝑥 → (𝑇‘𝑘) = (𝑇‘𝑥))
61		fveq2 6842	. . . . . 6 ⊢ (𝑘 = 𝑦 → (𝑇‘𝑘) = (𝑇‘𝑦))
62	14	zred 12607	. . . . . . 7 ⊢ (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℝ)
63	62	ssriv 3948	. . . . . 6 ⊢ (1...𝑀) ⊆ ℝ
64	9	rpred 12957	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ (1...𝑀)) → ((tan‘((𝑛 · π) / 𝑁))↑-2) ∈ ℝ)
65	64, 57	fmptd 7062	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑇:(1...𝑀)⟶ℝ)
66	65	ffvelcdmda 7035	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (1...𝑀)) → (𝑇‘𝑘) ∈ ℝ)
67		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑘 < 𝑚)
68	63	sseli 3940	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℝ)
69	68	ad2antrl 726	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑘 ∈ ℝ)
70	63	sseli 3940	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℝ)
71	70	ad2antll 727	. . . . . . . . . . . . . . 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 25817	. . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 11136	. . . . . . . . . . . . . . 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 11136	. . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑁 ∈ ℕ)
83	82	nnred 12168	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 𝑁 ∈ ℝ)
84	82	nngt0d 12202	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → 0 < 𝑁)
85		ltdiv1 12019	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 25823	. . . . . . . . . . . . . . 15 ⊢ -(π / 2) ∈ ℝ^*
89		pirp 25818	. . . . . . . . . . . . . . . . 17 ⊢ π ∈ ℝ⁺
90		rphalfcl 12942	. . . . . . . . . . . . . . . . 17 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
91		rpge0 12928	. . . . . . . . . . . . . . . . 17 ⊢ ((π / 2) ∈ ℝ⁺ → 0 ≤ (π / 2))
92	89, 90, 91	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ 0 ≤ (π / 2)
93		halfpire 25821	. . . . . . . . . . . . . . . . 17 ⊢ (π / 2) ∈ ℝ
94		le0neg2 11664	. . . . . . . . . . . . . . . . 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 13299	. . . . . . . . . . . . . . 15 ⊢ ((-(π / 2) ∈ ℝ^* ∧ -(π / 2) ≤ 0) → (0(,)(π / 2)) ⊆ (-(π / 2)(,)(π / 2)))
98	88, 96, 97	mp2an 690	. . . . . . . . . . . . . 14 ⊢ (0(,)(π / 2)) ⊆ (-(π / 2)(,)(π / 2))
99	1	basellem1 26430	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (1...𝑀)) → ((𝑘 · π) / 𝑁) ∈ (0(,)(π / 2)))
100	99	ad2ant2r 745	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑘 · π) / 𝑁) ∈ (0(,)(π / 2)))
101	98, 100	sselid 3942	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑘 · π) / 𝑁) ∈ (-(π / 2)(,)(π / 2)))
102	1	basellem1 26430	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑚 ∈ (1...𝑀)) → ((𝑚 · π) / 𝑁) ∈ (0(,)(π / 2)))
103	102	ad2ant2rl 747	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑚 · π) / 𝑁) ∈ (0(,)(π / 2)))
104	98, 103	sselid 3942	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((𝑚 · π) / 𝑁) ∈ (-(π / 2)(,)(π / 2)))
105		tanord 25894	. . . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑘 · π) / 𝑁)) < (tan‘((𝑚 · π) / 𝑁)))
108		tanrpcl 25861	. . . . . . . . . . . . 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 25861	. . . . . . . . . . . . 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 12930	. . . . . . . . . . . . 13 ⊢ ((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ⁺ → ((tan‘((𝑘 · π) / 𝑁)) ∈ ℝ ∧ 0 ≤ (tan‘((𝑘 · π) / 𝑁))))
113		rprege0 12930	. . . . . . . . . . . . 13 ⊢ ((tan‘((𝑚 · π) / 𝑁)) ∈ ℝ⁺ → ((tan‘((𝑚 · π) / 𝑁)) ∈ ℝ ∧ 0 ≤ (tan‘((𝑚 · π) / 𝑁))))
114		lt2sq 14038	. . . . . . . . . . . . 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 231	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → ((tan‘((𝑘 · π) / 𝑁))↑2) < ((tan‘((𝑚 · π) / 𝑁))↑2))
118		rpexpcl 13986	. . . . . . . . . . . 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 13986	. . . . . . . . . . . 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 12975	. . . . . . . . . 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 7364	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → (𝑛 · π) = (𝑚 · π))
125	124	fvoveq1d 7379	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (tan‘((𝑛 · π) / 𝑁)) = (tan‘((𝑚 · π) / 𝑁)))
126	125	oveq1d 7372	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((tan‘((𝑛 · π) / 𝑁))↑-2) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
127		ovex 7390	. . . . . . . . . . . 12 ⊢ ((tan‘((𝑚 · π) / 𝑁))↑-2) ∈ V
128	126, 57, 127	fvmpt 6948	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...𝑀) → (𝑇‘𝑚) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
129	128	ad2antll 727	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑚) = ((tan‘((𝑚 · π) / 𝑁))↑-2))
130	111	rpcnd 12959	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑚 · π) / 𝑁)) ∈ ℂ)
131		2nn0 12430	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
132		expneg 13975	. . . . . . . . . . 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 2776	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑚) = (1 / ((tan‘((𝑚 · π) / 𝑁))↑2)))
135		oveq1 7364	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑛 · π) = (𝑘 · π))
136	135	fvoveq1d 7379	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (tan‘((𝑛 · π) / 𝑁)) = (tan‘((𝑘 · π) / 𝑁)))
137	136	oveq1d 7372	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((tan‘((𝑛 · π) / 𝑁))↑-2) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
138		ovex 7390	. . . . . . . . . . . 12 ⊢ ((tan‘((𝑘 · π) / 𝑁))↑-2) ∈ V
139	137, 57, 138	fvmpt 6948	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑀) → (𝑇‘𝑘) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
140	139	ad2antrl 726	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑘) = ((tan‘((𝑘 · π) / 𝑁))↑-2))
141	109	rpcnd 12959	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (tan‘((𝑘 · π) / 𝑁)) ∈ ℂ)
142		expneg 13975	. . . . . . . . . . 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 2776	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑘) = (1 / ((tan‘((𝑘 · π) / 𝑁))↑2)))
145	123, 134, 144	3brtr4d 5137	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ ∧ 𝑘 < 𝑚) ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑇‘𝑚) < (𝑇‘𝑘))
146	145	an32s 650	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) ∧ 𝑘 < 𝑚) → (𝑇‘𝑚) < (𝑇‘𝑘))
147	146	ex 413	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ (𝑘 ∈ (1...𝑀) ∧ 𝑚 ∈ (1...𝑀))) → (𝑘 < 𝑚 → (𝑇‘𝑚) < (𝑇‘𝑘)))
148	59, 60, 61, 63, 66, 147	eqord2 11686	. . . . 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 3197	. . 3 ⊢ (𝑀 ∈ ℕ → ∀𝑥 ∈ (1...𝑀)∀𝑦 ∈ (1...𝑀)((𝑇‘𝑥) = (𝑇‘𝑦) → 𝑥 = 𝑦))
151		dff13 7202	. . 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 12187	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ≠ 0)
155	153, 154	eqnetrd 3011	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (deg‘𝑃) ≠ 0)
156		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑃 = 0_𝑝 → (deg‘𝑃) = (deg‘0_𝑝))
157		dgr0 25623	. . . . . . . . . 10 ⊢ (deg‘0_𝑝) = 0
158	156, 157	eqtrdi 2792	. . . . . . . . 9 ⊢ (𝑃 = 0_𝑝 → (deg‘𝑃) = 0)
159	158	necon3i 2976	. . . . . . . 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 25668	. . . . . . 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 495	. . . . 5 ⊢ (𝑀 ∈ ℕ → (^◡𝑃 “ {0}) ∈ Fin)
165		f1domg 8912	. . . . 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 496	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (♯‘(^◡𝑃 “ {0})) ≤ (deg‘𝑃))
168		nnnn0 12420	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
169		hashfz1 14246	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
170	168, 169	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (♯‘(1...𝑀)) = 𝑀)
171	153, 170	eqtr4d 2779	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (deg‘𝑃) = (♯‘(1...𝑀)))
172	167, 171	breqtrd 5131	. . . . 5 ⊢ (𝑀 ∈ ℕ → (♯‘(^◡𝑃 “ {0})) ≤ (♯‘(1...𝑀)))
173		fzfid 13878	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (1...𝑀) ∈ Fin)
174		hashdom 14279	. . . . . 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 231	. . . 4 ⊢ (𝑀 ∈ ℕ → (^◡𝑃 “ {0}) ≼ (1...𝑀))
177		sbth 9037	. . . 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 9215	. . 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 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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ⊆ wss 3910 {csn 4586 class class class wbr 5105 ↦ cmpt 5188 ^◡ccnv 5632 “ cima 5636 Fn wfn 6491 ⟶wf 6492 –1-1→wf1 6493 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7357 ≈ cen 8880 ≼ cdom 8881 Fincfn 8883 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 − cmin 11385 -cneg 11386 / cdiv 11812 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤcz 12499 ℝ⁺crp 12915 (,)cioo 13264 ...cfz 13424 ↑cexp 13967 Ccbc 14202 ♯chash 14230 Σcsu 15570 sincsin 15946 cosccos 15947 tanctan 15948 πcpi 15949 0_𝑝c0p 25033 Polycply 25545 coeffccoe 25547 degcdgr 25548

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-tan 15954 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-0p 25034 df-limc 25230 df-dv 25231 df-ply 25549 df-idp 25550 df-coe 25551 df-dgr 25552 df-quot 25651

This theorem is referenced by: basellem5 26434

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