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Theorem logcnlem3 26633

Description: Lemma for logcn 26636. (Contributed by Mario Carneiro, 25-Feb-2015.)

**Hypotheses**
Ref	Expression
logcn.d	⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
logcnlem.s	⊢ 𝑆 = if(𝐴 ∈ ℝ⁺, 𝐴, (abs‘(ℑ‘𝐴)))
logcnlem.t	⊢ 𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅)))
logcnlem.a	⊢ (𝜑 → 𝐴 ∈ 𝐷)
logcnlem.r	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
logcnlem.b	⊢ (𝜑 → 𝐵 ∈ 𝐷)
logcnlem.l	⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) < if(𝑆 ≤ 𝑇, 𝑆, 𝑇))

**Assertion**
Ref	Expression
logcnlem3	⊢ (𝜑 → (-π < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ∧ ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ≤ π))

**Proof of Theorem logcnlem3**
Step	Hyp	Ref	Expression
1		pire 26446	. . . . . 6 ⊢ π ∈ ℝ
2	1	renegcli 11453	. . . . 5 ⊢ -π ∈ ℝ
3	2	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -π ∈ ℝ)
4		logcnlem.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
5		logcn.d	. . . . . . . . . 10 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
6	5	ellogdm 26628	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐷 ↔ (𝐵 ∈ ℂ ∧ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ⁺)))
7	6	simplbi 497	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ ℂ)
8	4, 7	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
9	5	logdmn0 26629	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ≠ 0)
10	4, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 0)
11	8, 10	logcld 26559	. . . . . 6 ⊢ (𝜑 → (log‘𝐵) ∈ ℂ)
12	11	imcld 15155	. . . . 5 ⊢ (𝜑 → (ℑ‘(log‘𝐵)) ∈ ℝ)
13	12	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐵)) ∈ ℝ)
14		logcnlem.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
15	5	ellogdm 26628	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ⁺)))
16	15	simplbi 497	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ ℂ)
17	14, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
18	5	logdmn0 26629	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ≠ 0)
19	14, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≠ 0)
20	17, 19	logcld 26559	. . . . . . 7 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
21	20	imcld 15155	. . . . . 6 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) ∈ ℝ)
22	12, 21	resubcld 11576	. . . . 5 ⊢ (𝜑 → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ∈ ℝ)
23	22	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ∈ ℝ)
24	8, 10	logimcld 26560	. . . . . 6 ⊢ (𝜑 → (-π < (ℑ‘(log‘𝐵)) ∧ (ℑ‘(log‘𝐵)) ≤ π))
25	24	simpld 495	. . . . 5 ⊢ (𝜑 → -π < (ℑ‘(log‘𝐵)))
26	25	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -π < (ℑ‘(log‘𝐵)))
27	12	recnd 11171	. . . . . . 7 ⊢ (𝜑 → (ℑ‘(log‘𝐵)) ∈ ℂ)
28	27	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐵)) ∈ ℂ)
29	28	subid1d 11492	. . . . 5 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐵)) − 0) = (ℑ‘(log‘𝐵)))
30	21	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐴)) ∈ ℝ)
31		0red 11145	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → 0 ∈ ℝ)
32		argimlt0 26602	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐴)) ∈ (-π(,)0))
33	17, 32	sylan 586	. . . . . . . 8 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐴)) ∈ (-π(,)0))
34		eliooord 13356	. . . . . . . 8 ⊢ ((ℑ‘(log‘𝐴)) ∈ (-π(,)0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) < 0))
35	33, 34	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) < 0))
36	35	simprd 496	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐴)) < 0)
37	30, 31, 13, 36	ltsub2dd 11761	. . . . 5 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐵)) − 0) < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
38	29, 37	eqbrtrrd 5103	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐵)) < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
39	3, 13, 23, 26, 38	lttrd 11305	. . 3 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -π < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
40	25	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → -π < (ℑ‘(log‘𝐵)))
41		reim0b 15079	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0))
42	17, 41	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0))
43	15	simprbi 498	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐷 → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ⁺))
44	14, 43	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ⁺))
45	42, 44	sylbird 261	. . . . . . . . 9 ⊢ (𝜑 → ((ℑ‘𝐴) = 0 → 𝐴 ∈ ℝ⁺))
46	45	imp 407	. . . . . . . 8 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → 𝐴 ∈ ℝ⁺)
47	46	relogcld 26612	. . . . . . 7 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → (log‘𝐴) ∈ ℝ)
48	47	reim0d 15185	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → (ℑ‘(log‘𝐴)) = 0)
49	48	oveq2d 7379	. . . . 5 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) = ((ℑ‘(log‘𝐵)) − 0))
50	27	subid1d 11492	. . . . . 6 ⊢ (𝜑 → ((ℑ‘(log‘𝐵)) − 0) = (ℑ‘(log‘𝐵)))
51	50	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → ((ℑ‘(log‘𝐵)) − 0) = (ℑ‘(log‘𝐵)))
52	49, 51	eqtrd 2775	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) = (ℑ‘(log‘𝐵)))
53	40, 52	breqtrrd 5107	. . 3 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → -π < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
54	2	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → -π ∈ ℝ)
55	21	renegcld 11575	. . . . 5 ⊢ (𝜑 → -(ℑ‘(log‘𝐴)) ∈ ℝ)
56	55	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → -(ℑ‘(log‘𝐴)) ∈ ℝ)
57	22	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ∈ ℝ)
58		argimgt0 26601	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (0(,)π))
59	17, 58	sylan 586	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (0(,)π))
60		eliooord 13356	. . . . . . 7 ⊢ ((ℑ‘(log‘𝐴)) ∈ (0(,)π) → (0 < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) < π))
61	59, 60	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (0 < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) < π))
62	61	simprd 496	. . . . 5 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐴)) < π)
63		ltneg 11648	. . . . . . 7 ⊢ (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ π ∈ ℝ) → ((ℑ‘(log‘𝐴)) < π ↔ -π < -(ℑ‘(log‘𝐴))))
64	21, 1, 63	sylancl 592	. . . . . 6 ⊢ (𝜑 → ((ℑ‘(log‘𝐴)) < π ↔ -π < -(ℑ‘(log‘𝐴))))
65	64	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐴)) < π ↔ -π < -(ℑ‘(log‘𝐴))))
66	62, 65	mpbid 233	. . . 4 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → -π < -(ℑ‘(log‘𝐴)))
67		df-neg 11378	. . . . 5 ⊢ -(ℑ‘(log‘𝐴)) = (0 − (ℑ‘(log‘𝐴)))
68	8	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 𝐵 ∈ ℂ)
69	17, 8	imsubd 15177	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵)))
70	69	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵)))
71	17, 8	subcld 11503	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ)
72	71	imcld 15155	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℑ‘(𝐴 − 𝐵)) ∈ ℝ)
73	72	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(𝐴 − 𝐵)) ∈ ℝ)
74	71	abscld 15399	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ∈ ℝ)
75	74	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (abs‘(𝐴 − 𝐵)) ∈ ℝ)
76	17	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 𝐴 ∈ ℂ)
77	76	imcld 15155	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘𝐴) ∈ ℝ)
78		absimle 15269	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 − 𝐵) ∈ ℂ → (abs‘(ℑ‘(𝐴 − 𝐵))) ≤ (abs‘(𝐴 − 𝐵)))
79	71, 78	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (abs‘(ℑ‘(𝐴 − 𝐵))) ≤ (abs‘(𝐴 − 𝐵)))
80	72, 74	absled 15393	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((abs‘(ℑ‘(𝐴 − 𝐵))) ≤ (abs‘(𝐴 − 𝐵)) ↔ (-(abs‘(𝐴 − 𝐵)) ≤ (ℑ‘(𝐴 − 𝐵)) ∧ (ℑ‘(𝐴 − 𝐵)) ≤ (abs‘(𝐴 − 𝐵)))))
81	79, 80	mpbid 233	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (-(abs‘(𝐴 − 𝐵)) ≤ (ℑ‘(𝐴 − 𝐵)) ∧ (ℑ‘(𝐴 − 𝐵)) ≤ (abs‘(𝐴 − 𝐵))))
82	81	simprd 496	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℑ‘(𝐴 − 𝐵)) ≤ (abs‘(𝐴 − 𝐵)))
83	82	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(𝐴 − 𝐵)) ≤ (abs‘(𝐴 − 𝐵)))
84		logcnlem.s	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑆 = if(𝐴 ∈ ℝ⁺, 𝐴, (abs‘(ℑ‘𝐴)))
85		rpre 12949	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
86	85	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
87	17	imcld 15155	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ)
88	87	recnd 11171	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℂ)
89	88	abscld 15399	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ∈ ℝ)
90	89	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ⁺) → (abs‘(ℑ‘𝐴)) ∈ ℝ)
91	86, 90	ifclda 4497	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → if(𝐴 ∈ ℝ⁺, 𝐴, (abs‘(ℑ‘𝐴))) ∈ ℝ)
92	84, 91	eqeltrid 2844	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ∈ ℝ)
93		logcnlem.t	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅)))
94	17	abscld 15399	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
95		logcnlem.r	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
96	95	rpred 12984	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑅 ∈ ℝ)
97		1rp 12944	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ⁺
98		rpaddcl 12964	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑅 ∈ ℝ⁺) → (1 + 𝑅) ∈ ℝ⁺)
99	97, 95, 98	sylancr 593	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1 + 𝑅) ∈ ℝ⁺)
100	96, 99	rerpdivcld 13015	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑅 / (1 + 𝑅)) ∈ ℝ)
101	94, 100	remulcld 11173	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((abs‘𝐴) · (𝑅 / (1 + 𝑅))) ∈ ℝ)
102	93, 101	eqeltrid 2844	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ ℝ)
103	92, 102	ifcld 4508	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑆 ≤ 𝑇, 𝑆, 𝑇) ∈ ℝ)
104		logcnlem.l	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) < if(𝑆 ≤ 𝑇, 𝑆, 𝑇))
105		min1 13139	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ ℝ ∧ 𝑇 ∈ ℝ) → if(𝑆 ≤ 𝑇, 𝑆, 𝑇) ≤ 𝑆)
106	92, 102, 105	syl2anc 590	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑆 ≤ 𝑇, 𝑆, 𝑇) ≤ 𝑆)
107	74, 103, 92, 104, 106	ltletrd 11304	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) < 𝑆)
108	107	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (abs‘(𝐴 − 𝐵)) < 𝑆)
109		gt0ne0 11613	. . . . . . . . . . . . . . . . 17 ⊢ (((ℑ‘𝐴) ∈ ℝ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘𝐴) ≠ 0)
110	87, 109	sylan 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘𝐴) ≠ 0)
111	85, 42	imbitrid 245	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐴 ∈ ℝ⁺ → (ℑ‘𝐴) = 0))
112	111	necon3ad 2948	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((ℑ‘𝐴) ≠ 0 → ¬ 𝐴 ∈ ℝ⁺))
113	112	imp 407	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (ℑ‘𝐴) ≠ 0) → ¬ 𝐴 ∈ ℝ⁺)
114		iffalse 4470	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝐴 ∈ ℝ⁺ → if(𝐴 ∈ ℝ⁺, 𝐴, (abs‘(ℑ‘𝐴))) = (abs‘(ℑ‘𝐴)))
115	84, 114	eqtrid 2787	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝐴 ∈ ℝ⁺ → 𝑆 = (abs‘(ℑ‘𝐴)))
116	113, 115	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (ℑ‘𝐴) ≠ 0) → 𝑆 = (abs‘(ℑ‘𝐴)))
117	110, 116	syldan 597	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 𝑆 = (abs‘(ℑ‘𝐴)))
118		0re 11144	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
119		ltle 11232	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ (ℑ‘𝐴) ∈ ℝ) → (0 < (ℑ‘𝐴) → 0 ≤ (ℑ‘𝐴)))
120	118, 87, 119	sylancr 593	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0 < (ℑ‘𝐴) → 0 ≤ (ℑ‘𝐴)))
121	120	imp 407	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 0 ≤ (ℑ‘𝐴))
122	77, 121	absidd 15383	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (abs‘(ℑ‘𝐴)) = (ℑ‘𝐴))
123	117, 122	eqtrd 2775	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 𝑆 = (ℑ‘𝐴))
124	108, 123	breqtrd 5105	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (abs‘(𝐴 − 𝐵)) < (ℑ‘𝐴))
125	73, 75, 77, 83, 124	lelttrd 11302	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(𝐴 − 𝐵)) < (ℑ‘𝐴))
126	70, 125	eqbrtrrd 5103	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘𝐴) − (ℑ‘𝐵)) < (ℑ‘𝐴))
127	88	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘𝐴) ∈ ℂ)
128	127	subid1d 11492	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘𝐴) − 0) = (ℑ‘𝐴))
129	126, 128	breqtrrd 5107	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘𝐴) − (ℑ‘𝐵)) < ((ℑ‘𝐴) − 0))
130		0red 11145	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℝ)
131	8	imcld 15155	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℝ)
132	130, 131, 87	ltsub2d 11758	. . . . . . . . . . 11 ⊢ (𝜑 → (0 < (ℑ‘𝐵) ↔ ((ℑ‘𝐴) − (ℑ‘𝐵)) < ((ℑ‘𝐴) − 0)))
133	132	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (0 < (ℑ‘𝐵) ↔ ((ℑ‘𝐴) − (ℑ‘𝐵)) < ((ℑ‘𝐴) − 0)))
134	129, 133	mpbird 258	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 0 < (ℑ‘𝐵))
135		argimgt0 26601	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 0 < (ℑ‘𝐵)) → (ℑ‘(log‘𝐵)) ∈ (0(,)π))
136	68, 134, 135	syl2anc 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐵)) ∈ (0(,)π))
137		eliooord 13356	. . . . . . . 8 ⊢ ((ℑ‘(log‘𝐵)) ∈ (0(,)π) → (0 < (ℑ‘(log‘𝐵)) ∧ (ℑ‘(log‘𝐵)) < π))
138	136, 137	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (0 < (ℑ‘(log‘𝐵)) ∧ (ℑ‘(log‘𝐵)) < π))
139	138	simpld 495	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 0 < (ℑ‘(log‘𝐵)))
140	130, 12, 21	ltsub1d 11757	. . . . . . 7 ⊢ (𝜑 → (0 < (ℑ‘(log‘𝐵)) ↔ (0 − (ℑ‘(log‘𝐴))) < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴)))))
141	140	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (0 < (ℑ‘(log‘𝐵)) ↔ (0 − (ℑ‘(log‘𝐴))) < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴)))))
142	139, 141	mpbid 233	. . . . 5 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (0 − (ℑ‘(log‘𝐴))) < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
143	67, 142	eqbrtrid 5114	. . . 4 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → -(ℑ‘(log‘𝐴)) < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
144	54, 56, 57, 66, 143	lttrd 11305	. . 3 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → -π < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
145		lttri4 11228	. . . 4 ⊢ (((ℑ‘𝐴) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℑ‘𝐴) < 0 ∨ (ℑ‘𝐴) = 0 ∨ 0 < (ℑ‘𝐴)))
146	87, 118, 145	sylancl 592	. . 3 ⊢ (𝜑 → ((ℑ‘𝐴) < 0 ∨ (ℑ‘𝐴) = 0 ∨ 0 < (ℑ‘𝐴)))
147	39, 53, 144, 146	mpjao3dan 1440	. 2 ⊢ (𝜑 → -π < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
148	1	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → π ∈ ℝ)
149	30	renegcld 11575	. . . . 5 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -(ℑ‘(log‘𝐴)) ∈ ℝ)
150	8	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → 𝐵 ∈ ℂ)
151	88	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) ∈ ℂ)
152	151	subid1d 11492	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘𝐴) − 0) = (ℑ‘𝐴))
153	87	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) ∈ ℝ)
154	74	renegcld 11575	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → -(abs‘(𝐴 − 𝐵)) ∈ ℝ)
155	154	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -(abs‘(𝐴 − 𝐵)) ∈ ℝ)
156	72	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(𝐴 − 𝐵)) ∈ ℝ)
157	74	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (abs‘(𝐴 − 𝐵)) ∈ ℝ)
158	107	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (abs‘(𝐴 − 𝐵)) < 𝑆)
159	118	ltnri 11253	. . . . . . . . . . . . . . . . . . . 20 ⊢ ¬ 0 < 0
160		breq1 5082	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℑ‘𝐴) = 0 → ((ℑ‘𝐴) < 0 ↔ 0 < 0))
161	159, 160	mtbiri 328	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℑ‘𝐴) = 0 → ¬ (ℑ‘𝐴) < 0)
162	161	necon2ai 2964	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℑ‘𝐴) < 0 → (ℑ‘𝐴) ≠ 0)
163	162, 116	sylan2 599	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → 𝑆 = (abs‘(ℑ‘𝐴)))
164		ltle 11232	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ℑ‘𝐴) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℑ‘𝐴) < 0 → (ℑ‘𝐴) ≤ 0))
165	87, 118, 164	sylancl 592	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((ℑ‘𝐴) < 0 → (ℑ‘𝐴) ≤ 0))
166	165	imp 407	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) ≤ 0)
167	153, 166	absnidd 15374	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (abs‘(ℑ‘𝐴)) = -(ℑ‘𝐴))
168	163, 167	eqtrd 2775	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → 𝑆 = -(ℑ‘𝐴))
169	158, 168	breqtrd 5105	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (abs‘(𝐴 − 𝐵)) < -(ℑ‘𝐴))
170	157, 153, 169	ltnegcon2d 11729	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) < -(abs‘(𝐴 − 𝐵)))
171	81	simpld 495	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → -(abs‘(𝐴 − 𝐵)) ≤ (ℑ‘(𝐴 − 𝐵)))
172	171	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -(abs‘(𝐴 − 𝐵)) ≤ (ℑ‘(𝐴 − 𝐵)))
173	153, 155, 156, 170, 172	ltletrd 11304	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) < (ℑ‘(𝐴 − 𝐵)))
174	69	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵)))
175	173, 174	breqtrd 5105	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) < ((ℑ‘𝐴) − (ℑ‘𝐵)))
176	152, 175	eqbrtrd 5101	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘𝐴) − 0) < ((ℑ‘𝐴) − (ℑ‘𝐵)))
177	150	imcld 15155	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐵) ∈ ℝ)
178	177, 31, 153	ltsub2d 11758	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘𝐵) < 0 ↔ ((ℑ‘𝐴) − 0) < ((ℑ‘𝐴) − (ℑ‘𝐵))))
179	176, 178	mpbird 258	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐵) < 0)
180		argimlt0 26602	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℂ ∧ (ℑ‘𝐵) < 0) → (ℑ‘(log‘𝐵)) ∈ (-π(,)0))
181	150, 179, 180	syl2anc 590	. . . . . . . . 9 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐵)) ∈ (-π(,)0))
182		eliooord 13356	. . . . . . . . 9 ⊢ ((ℑ‘(log‘𝐵)) ∈ (-π(,)0) → (-π < (ℑ‘(log‘𝐵)) ∧ (ℑ‘(log‘𝐵)) < 0))
183	181, 182	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (-π < (ℑ‘(log‘𝐵)) ∧ (ℑ‘(log‘𝐵)) < 0))
184	183	simprd 496	. . . . . . 7 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐵)) < 0)
185	13, 31, 30, 184	ltsub1dd 11760	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) < (0 − (ℑ‘(log‘𝐴))))
186	185, 67	breqtrrdi 5121	. . . . 5 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) < -(ℑ‘(log‘𝐴)))
187	35	simpld 495	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -π < (ℑ‘(log‘𝐴)))
188		ltnegcon1 11649	. . . . . . 7 ⊢ ((π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (-π < (ℑ‘(log‘𝐴)) ↔ -(ℑ‘(log‘𝐴)) < π))
189	1, 30, 188	sylancr 593	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (-π < (ℑ‘(log‘𝐴)) ↔ -(ℑ‘(log‘𝐴)) < π))
190	187, 189	mpbid 233	. . . . 5 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -(ℑ‘(log‘𝐴)) < π)
191	23, 149, 148, 186, 190	lttrd 11305	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) < π)
192	23, 148, 191	ltled 11292	. . 3 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ≤ π)
193	24	simprd 496	. . . . 5 ⊢ (𝜑 → (ℑ‘(log‘𝐵)) ≤ π)
194	193	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → (ℑ‘(log‘𝐵)) ≤ π)
195	52, 194	eqbrtrd 5101	. . 3 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ≤ π)
196	1	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → π ∈ ℝ)
197	12	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐵)) ∈ ℝ)
198		0red 11145	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 0 ∈ ℝ)
199	21	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ ℝ)
200	61	simpld 495	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 0 < (ℑ‘(log‘𝐴)))
201	198, 199, 197, 200	ltsub2dd 11761	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) < ((ℑ‘(log‘𝐵)) − 0))
202	27	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐵)) ∈ ℂ)
203	202	subid1d 11492	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐵)) − 0) = (ℑ‘(log‘𝐵)))
204	201, 203	breqtrd 5105	. . . . 5 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) < (ℑ‘(log‘𝐵)))
205	138	simprd 496	. . . . 5 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐵)) < π)
206	57, 197, 196, 204, 205	lttrd 11305	. . . 4 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) < π)
207	57, 196, 206	ltled 11292	. . 3 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ≤ π)
208	192, 195, 207, 146	mpjao3dan 1440	. 2 ⊢ (𝜑 → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ≤ π)
209	147, 208	jca 516	1 ⊢ (𝜑 → (-π < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ∧ ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ≤ π))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ w3o 1091 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 ∖ cdif 3887 ifcif 4461 class class class wbr 5079 ‘cfv 6492 (class class class)co 7363 ℂcc 11034 ℝcr 11035 0cc0 11036 1c1 11037 + caddc 11039 · cmul 11041 -∞cmnf 11175 < clt 11177 ≤ cle 11178 − cmin 11375 -cneg 11376 / cdiv 11805 ℝ⁺crp 12940 (,)cioo 13296 (,]cioc 13297 ℑcim 15058 abscabs 15194 πcpi 16029 logclog 26543

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-ioc 13301 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-mod 13827 df-seq 13962 df-exp 14022 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15027 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-limsup 15431 df-clim 15448 df-rlim 15449 df-sum 15647 df-ef 16030 df-sin 16032 df-cos 16033 df-pi 16035 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-fbas 21351 df-fg 21352 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-haus 23305 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24310 df-ms 24311 df-tms 24312 df-cncf 24870 df-limc 25858 df-dv 25859 df-log 26545

This theorem is referenced by: logcnlem4 26634

W3C validator