angpieqvd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > angpieqvd		Structured version Visualization version GIF version

Theorem angpieqvd 26333

Description: The angle ABC is π iff 𝐵 is a nontrivial convex combination of 𝐴 and 𝐶, i.e., iff 𝐵 is in the interior of the segment AC. (Contributed by David Moews, 28-Feb-2017.)

**Hypotheses**
Ref	Expression
angpieqvd.angdef	⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
angpieqvd.A	⊢ (𝜑 → 𝐴 ∈ ℂ)
angpieqvd.B	⊢ (𝜑 → 𝐵 ∈ ℂ)
angpieqvd.C	⊢ (𝜑 → 𝐶 ∈ ℂ)
angpieqvd.AneB	⊢ (𝜑 → 𝐴 ≠ 𝐵)
angpieqvd.BneC	⊢ (𝜑 → 𝐵 ≠ 𝐶)

**Assertion**
Ref	Expression
angpieqvd	⊢ (𝜑 → (((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π ↔ ∃𝑤 ∈ (0(,)1)𝐵 = ((𝑤 · 𝐴) + ((1 − 𝑤) · 𝐶))))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦 𝑥,𝐶,𝑦 𝑤,𝐹 𝜑,𝑤 𝑤,𝐴 𝑤,𝐵 𝑤,𝐶 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝐹(𝑥,𝑦)

**Proof of Theorem angpieqvd**
Step	Hyp	Ref	Expression
1		angpieqvd.angdef	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
2		angpieqvd.A	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		angpieqvd.B	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
4		angpieqvd.C	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5		angpieqvd.AneB	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
6		angpieqvd.BneC	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
7	1, 2, 3, 4, 5, 6	angpieqvdlem2 26331	. . . . . 6 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ ↔ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π))
8	7	biimpar 478	. . . . 5 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺)
9	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → 𝐴 ∈ ℂ)
10	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → 𝐵 ∈ ℂ)
11	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → 𝐶 ∈ ℂ)
12	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → 𝐴 ≠ 𝐵)
13	1, 2, 3, 4, 5, 6	angpined 26332	. . . . . . 7 ⊢ (𝜑 → (((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π → 𝐴 ≠ 𝐶))
14	13	imp 407	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → 𝐴 ≠ 𝐶)
15	9, 10, 11, 12, 14	angpieqvdlem 26330	. . . . 5 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ ↔ ((𝐶 − 𝐵) / (𝐶 − 𝐴)) ∈ (0(,)1)))
16	8, 15	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → ((𝐶 − 𝐵) / (𝐶 − 𝐴)) ∈ (0(,)1))
17	4, 3	subcld 11570	. . . . . . . 8 ⊢ (𝜑 → (𝐶 − 𝐵) ∈ ℂ)
18	17	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → (𝐶 − 𝐵) ∈ ℂ)
19	4, 2	subcld 11570	. . . . . . . 8 ⊢ (𝜑 → (𝐶 − 𝐴) ∈ ℂ)
20	19	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → (𝐶 − 𝐴) ∈ ℂ)
21	14	necomd 2996	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → 𝐶 ≠ 𝐴)
22	11, 9, 21	subne0d 11579	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → (𝐶 − 𝐴) ≠ 0)
23	18, 20, 22	divcan1d 11990	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → (((𝐶 − 𝐵) / (𝐶 − 𝐴)) · (𝐶 − 𝐴)) = (𝐶 − 𝐵))
24	23	eqcomd 2738	. . . . 5 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → (𝐶 − 𝐵) = (((𝐶 − 𝐵) / (𝐶 − 𝐴)) · (𝐶 − 𝐴)))
25	18, 20, 22	divcld 11989	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → ((𝐶 − 𝐵) / (𝐶 − 𝐴)) ∈ ℂ)
26	9, 10, 11, 25	affineequiv 26325	. . . . 5 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → (𝐵 = ((((𝐶 − 𝐵) / (𝐶 − 𝐴)) · 𝐴) + ((1 − ((𝐶 − 𝐵) / (𝐶 − 𝐴))) · 𝐶)) ↔ (𝐶 − 𝐵) = (((𝐶 − 𝐵) / (𝐶 − 𝐴)) · (𝐶 − 𝐴))))
27	24, 26	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → 𝐵 = ((((𝐶 − 𝐵) / (𝐶 − 𝐴)) · 𝐴) + ((1 − ((𝐶 − 𝐵) / (𝐶 − 𝐴))) · 𝐶)))
28		oveq1 7415	. . . . . 6 ⊢ (𝑤 = ((𝐶 − 𝐵) / (𝐶 − 𝐴)) → (𝑤 · 𝐴) = (((𝐶 − 𝐵) / (𝐶 − 𝐴)) · 𝐴))
29		oveq2 7416	. . . . . . 7 ⊢ (𝑤 = ((𝐶 − 𝐵) / (𝐶 − 𝐴)) → (1 − 𝑤) = (1 − ((𝐶 − 𝐵) / (𝐶 − 𝐴))))
30	29	oveq1d 7423	. . . . . 6 ⊢ (𝑤 = ((𝐶 − 𝐵) / (𝐶 − 𝐴)) → ((1 − 𝑤) · 𝐶) = ((1 − ((𝐶 − 𝐵) / (𝐶 − 𝐴))) · 𝐶))
31	28, 30	oveq12d 7426	. . . . 5 ⊢ (𝑤 = ((𝐶 − 𝐵) / (𝐶 − 𝐴)) → ((𝑤 · 𝐴) + ((1 − 𝑤) · 𝐶)) = ((((𝐶 − 𝐵) / (𝐶 − 𝐴)) · 𝐴) + ((1 − ((𝐶 − 𝐵) / (𝐶 − 𝐴))) · 𝐶)))
32	31	rspceeqv 3633	. . . 4 ⊢ ((((𝐶 − 𝐵) / (𝐶 − 𝐴)) ∈ (0(,)1) ∧ 𝐵 = ((((𝐶 − 𝐵) / (𝐶 − 𝐴)) · 𝐴) + ((1 − ((𝐶 − 𝐵) / (𝐶 − 𝐴))) · 𝐶))) → ∃𝑤 ∈ (0(,)1)𝐵 = ((𝑤 · 𝐴) + ((1 − 𝑤) · 𝐶)))
33	16, 27, 32	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π) → ∃𝑤 ∈ (0(,)1)𝐵 = ((𝑤 · 𝐴) + ((1 − 𝑤) · 𝐶)))
34	33	ex 413	. 2 ⊢ (𝜑 → (((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π → ∃𝑤 ∈ (0(,)1)𝐵 = ((𝑤 · 𝐴) + ((1 − 𝑤) · 𝐶))))
35	2	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1)) → 𝐴 ∈ ℂ)
36	3	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1)) → 𝐵 ∈ ℂ)
37	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1)) → 𝐶 ∈ ℂ)
38		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1)) → 𝑤 ∈ (0(,)1))
39		elioore 13353	. . . . . 6 ⊢ (𝑤 ∈ (0(,)1) → 𝑤 ∈ ℝ)
40		recn 11199	. . . . . 6 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
41	38, 39, 40	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1)) → 𝑤 ∈ ℂ)
42	35, 36, 37, 41	affineequiv 26325	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1)) → (𝐵 = ((𝑤 · 𝐴) + ((1 − 𝑤) · 𝐶)) ↔ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))))
43		simp3 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴)))
44	17	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → (𝐶 − 𝐵) ∈ ℂ)
45	41	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → 𝑤 ∈ ℂ)
46	19	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → (𝐶 − 𝐴) ∈ ℂ)
47	6	necomd 2996	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
48	4, 3, 47	subne0d 11579	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 − 𝐵) ≠ 0)
49	48	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → (𝐶 − 𝐵) ≠ 0)
50	43, 49	eqnetrrd 3009	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → (𝑤 · (𝐶 − 𝐴)) ≠ 0)
51	45, 46, 50	mulne0bbd 11869	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → (𝐶 − 𝐴) ≠ 0)
52	44, 45, 46, 51	divmul3d 12023	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → (((𝐶 − 𝐵) / (𝐶 − 𝐴)) = 𝑤 ↔ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))))
53	43, 52	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → ((𝐶 − 𝐵) / (𝐶 − 𝐴)) = 𝑤)
54		simp2 1137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → 𝑤 ∈ (0(,)1))
55	53, 54	eqeltrd 2833	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → ((𝐶 − 𝐵) / (𝐶 − 𝐴)) ∈ (0(,)1))
56	2	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → 𝐴 ∈ ℂ)
57	3	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → 𝐵 ∈ ℂ)
58	4	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → 𝐶 ∈ ℂ)
59	5	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → 𝐴 ≠ 𝐵)
60	58, 56, 51	subne0ad 11581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → 𝐶 ≠ 𝐴)
61	60	necomd 2996	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → 𝐴 ≠ 𝐶)
62	56, 57, 58, 59, 61	angpieqvdlem 26330	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ ↔ ((𝐶 − 𝐵) / (𝐶 − 𝐴)) ∈ (0(,)1)))
63	55, 62	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺)
64	6	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → 𝐵 ≠ 𝐶)
65	1, 56, 57, 58, 59, 64	angpieqvdlem2 26331	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ ↔ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π))
66	63, 65	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1) ∧ (𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴))) → ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π)
67	66	3expia 1121	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1)) → ((𝐶 − 𝐵) = (𝑤 · (𝐶 − 𝐴)) → ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π))
68	42, 67	sylbid 239	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ (0(,)1)) → (𝐵 = ((𝑤 · 𝐴) + ((1 − 𝑤) · 𝐶)) → ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π))
69	68	rexlimdva 3155	. 2 ⊢ (𝜑 → (∃𝑤 ∈ (0(,)1)𝐵 = ((𝑤 · 𝐴) + ((1 − 𝑤) · 𝐶)) → ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π))
70	34, 69	impbid 211	1 ⊢ (𝜑 → (((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π ↔ ∃𝑤 ∈ (0(,)1)𝐵 = ((𝑤 · 𝐴) + ((1 − 𝑤) · 𝐶))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 ∖ cdif 3945 {csn 4628 ‘cfv 6543 (class class class)co 7408 ∈ cmpo 7410 ℂcc 11107 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 − cmin 11443 -cneg 11444 / cdiv 11870 ℝ⁺crp 12973 (,)cioo 13323 ℑcim 15044 πcpi 16009 logclog 26062

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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ioc 13328 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-mod 13834 df-seq 13966 df-exp 14027 df-fac 14233 df-bc 14262 df-hash 14290 df-shft 15013 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-limsup 15414 df-clim 15431 df-rlim 15432 df-sum 15632 df-ef 16010 df-sin 16012 df-cos 16013 df-pi 16015 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-mulg 18950 df-cntz 19180 df-cmn 19649 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-lp 22639 df-perf 22640 df-cn 22730 df-cnp 22731 df-haus 22818 df-tx 23065 df-hmeo 23258 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-xms 23825 df-ms 23826 df-tms 23827 df-cncf 24393 df-limc 25382 df-dv 25383 df-log 26064

This theorem is referenced by: chordthm 26339 chordthmALT 43684

W3C validator