logcnlem3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > logcnlem3		Structured version Visualization version GIF version

Theorem logcnlem3 25504

Description: Lemma for logcn 25507. (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 25320	. . . . . 6 ⊢ π ∈ ℝ
2	1	renegcli 11122	. . . . 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 25499	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐷 ↔ (𝐵 ∈ ℂ ∧ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ⁺)))
7	6	simplbi 501	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ ℂ)
8	4, 7	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
9	5	logdmn0 25500	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ≠ 0)
10	4, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 0)
11	8, 10	logcld 25431	. . . . . 6 ⊢ (𝜑 → (log‘𝐵) ∈ ℂ)
12	11	imcld 14741	. . . . 5 ⊢ (𝜑 → (ℑ‘(log‘𝐵)) ∈ ℝ)
13	12	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐵)) ∈ ℝ)
14		logcnlem.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
15	5	ellogdm 25499	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ⁺)))
16	15	simplbi 501	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ ℂ)
17	14, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
18	5	logdmn0 25500	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ≠ 0)
19	14, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≠ 0)
20	17, 19	logcld 25431	. . . . . . 7 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
21	20	imcld 14741	. . . . . 6 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) ∈ ℝ)
22	12, 21	resubcld 11243	. . . . 5 ⊢ (𝜑 → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ∈ ℝ)
23	22	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ∈ ℝ)
24	8, 10	logimcld 25432	. . . . . 6 ⊢ (𝜑 → (-π < (ℑ‘(log‘𝐵)) ∧ (ℑ‘(log‘𝐵)) ≤ π))
25	24	simpld 498	. . . . 5 ⊢ (𝜑 → -π < (ℑ‘(log‘𝐵)))
26	25	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -π < (ℑ‘(log‘𝐵)))
27	12	recnd 10844	. . . . . . 7 ⊢ (𝜑 → (ℑ‘(log‘𝐵)) ∈ ℂ)
28	27	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐵)) ∈ ℂ)
29	28	subid1d 11161	. . . . 5 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐵)) − 0) = (ℑ‘(log‘𝐵)))
30	21	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐴)) ∈ ℝ)
31		0red 10819	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → 0 ∈ ℝ)
32		argimlt0 25473	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐴)) ∈ (-π(,)0))
33	17, 32	sylan 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐴)) ∈ (-π(,)0))
34		eliooord 12977	. . . . . . . 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 499	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐴)) < 0)
37	30, 31, 13, 36	ltsub2dd 11428	. . . . 5 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐵)) − 0) < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
38	29, 37	eqbrtrrd 5067	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐵)) < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
39	3, 13, 23, 26, 38	lttrd 10976	. . 3 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -π < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
40	25	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → -π < (ℑ‘(log‘𝐵)))
41		reim0b 14665	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0))
42	17, 41	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0))
43	15	simprbi 500	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐷 → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ⁺))
44	14, 43	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ⁺))
45	42, 44	sylbird 263	. . . . . . . . 9 ⊢ (𝜑 → ((ℑ‘𝐴) = 0 → 𝐴 ∈ ℝ⁺))
46	45	imp 410	. . . . . . . 8 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → 𝐴 ∈ ℝ⁺)
47	46	relogcld 25483	. . . . . . 7 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → (log‘𝐴) ∈ ℝ)
48	47	reim0d 14771	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → (ℑ‘(log‘𝐴)) = 0)
49	48	oveq2d 7218	. . . . 5 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) = ((ℑ‘(log‘𝐵)) − 0))
50	27	subid1d 11161	. . . . . 6 ⊢ (𝜑 → ((ℑ‘(log‘𝐵)) − 0) = (ℑ‘(log‘𝐵)))
51	50	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → ((ℑ‘(log‘𝐵)) − 0) = (ℑ‘(log‘𝐵)))
52	49, 51	eqtrd 2774	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) = (ℑ‘(log‘𝐵)))
53	40, 52	breqtrrd 5071	. . 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 11242	. . . . 5 ⊢ (𝜑 → -(ℑ‘(log‘𝐴)) ∈ ℝ)
56	55	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → -(ℑ‘(log‘𝐴)) ∈ ℝ)
57	22	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ∈ ℝ)
58		argimgt0 25472	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (0(,)π))
59	17, 58	sylan 583	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (0(,)π))
60		eliooord 12977	. . . . . . 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 499	. . . . 5 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐴)) < π)
63		ltneg 11315	. . . . . . 7 ⊢ (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ π ∈ ℝ) → ((ℑ‘(log‘𝐴)) < π ↔ -π < -(ℑ‘(log‘𝐴))))
64	21, 1, 63	sylancl 589	. . . . . 6 ⊢ (𝜑 → ((ℑ‘(log‘𝐴)) < π ↔ -π < -(ℑ‘(log‘𝐴))))
65	64	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐴)) < π ↔ -π < -(ℑ‘(log‘𝐴))))
66	62, 65	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → -π < -(ℑ‘(log‘𝐴)))
67		df-neg 11048	. . . . 5 ⊢ -(ℑ‘(log‘𝐴)) = (0 − (ℑ‘(log‘𝐴)))
68	8	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 𝐵 ∈ ℂ)
69	17, 8	imsubd 14763	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵)))
70	69	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵)))
71	17, 8	subcld 11172	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ)
72	71	imcld 14741	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℑ‘(𝐴 − 𝐵)) ∈ ℝ)
73	72	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(𝐴 − 𝐵)) ∈ ℝ)
74	71	abscld 14983	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ∈ ℝ)
75	74	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (abs‘(𝐴 − 𝐵)) ∈ ℝ)
76	17	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 𝐴 ∈ ℂ)
77	76	imcld 14741	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘𝐴) ∈ ℝ)
78		absimle 14856	. . . . . . . . . . . . . . . . 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 14977	. . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (-(abs‘(𝐴 − 𝐵)) ≤ (ℑ‘(𝐴 − 𝐵)) ∧ (ℑ‘(𝐴 − 𝐵)) ≤ (abs‘(𝐴 − 𝐵))))
82	81	simprd 499	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℑ‘(𝐴 − 𝐵)) ≤ (abs‘(𝐴 − 𝐵)))
83	82	adantr 484	. . . . . . . . . . . . 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 12577	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
86	85	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
87	17	imcld 14741	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ)
88	87	recnd 10844	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℂ)
89	88	abscld 14983	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ∈ ℝ)
90	89	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ⁺) → (abs‘(ℑ‘𝐴)) ∈ ℝ)
91	86, 90	ifclda 4464	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → if(𝐴 ∈ ℝ⁺, 𝐴, (abs‘(ℑ‘𝐴))) ∈ ℝ)
92	84, 91	eqeltrid 2838	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ∈ ℝ)
93		logcnlem.t	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅)))
94	17	abscld 14983	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
95		logcnlem.r	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
96	95	rpred 12611	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑅 ∈ ℝ)
97		1rp 12573	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ⁺
98		rpaddcl 12591	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℝ⁺ ∧ 𝑅 ∈ ℝ⁺) → (1 + 𝑅) ∈ ℝ⁺)
99	97, 95, 98	sylancr 590	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1 + 𝑅) ∈ ℝ⁺)
100	96, 99	rerpdivcld 12642	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑅 / (1 + 𝑅)) ∈ ℝ)
101	94, 100	remulcld 10846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((abs‘𝐴) · (𝑅 / (1 + 𝑅))) ∈ ℝ)
102	93, 101	eqeltrid 2838	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ ℝ)
103	92, 102	ifcld 4475	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑆 ≤ 𝑇, 𝑆, 𝑇) ∈ ℝ)
104		logcnlem.l	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) < if(𝑆 ≤ 𝑇, 𝑆, 𝑇))
105		min1 12762	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ ℝ ∧ 𝑇 ∈ ℝ) → if(𝑆 ≤ 𝑇, 𝑆, 𝑇) ≤ 𝑆)
106	92, 102, 105	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if(𝑆 ≤ 𝑇, 𝑆, 𝑇) ≤ 𝑆)
107	74, 103, 92, 104, 106	ltletrd 10975	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) < 𝑆)
108	107	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (abs‘(𝐴 − 𝐵)) < 𝑆)
109		gt0ne0 11280	. . . . . . . . . . . . . . . . 17 ⊢ (((ℑ‘𝐴) ∈ ℝ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘𝐴) ≠ 0)
110	87, 109	sylan 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘𝐴) ≠ 0)
111	85, 42	syl5ib 247	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐴 ∈ ℝ⁺ → (ℑ‘𝐴) = 0))
112	111	necon3ad 2948	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((ℑ‘𝐴) ≠ 0 → ¬ 𝐴 ∈ ℝ⁺))
113	112	imp 410	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (ℑ‘𝐴) ≠ 0) → ¬ 𝐴 ∈ ℝ⁺)
114		iffalse 4438	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝐴 ∈ ℝ⁺ → if(𝐴 ∈ ℝ⁺, 𝐴, (abs‘(ℑ‘𝐴))) = (abs‘(ℑ‘𝐴)))
115	84, 114	syl5eq 2786	. . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 𝑆 = (abs‘(ℑ‘𝐴)))
118		0re 10818	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
119		ltle 10904	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ (ℑ‘𝐴) ∈ ℝ) → (0 < (ℑ‘𝐴) → 0 ≤ (ℑ‘𝐴)))
120	118, 87, 119	sylancr 590	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0 < (ℑ‘𝐴) → 0 ≤ (ℑ‘𝐴)))
121	120	imp 410	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 0 ≤ (ℑ‘𝐴))
122	77, 121	absidd 14969	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (abs‘(ℑ‘𝐴)) = (ℑ‘𝐴))
123	117, 122	eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 𝑆 = (ℑ‘𝐴))
124	108, 123	breqtrd 5069	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (abs‘(𝐴 − 𝐵)) < (ℑ‘𝐴))
125	73, 75, 77, 83, 124	lelttrd 10973	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(𝐴 − 𝐵)) < (ℑ‘𝐴))
126	70, 125	eqbrtrrd 5067	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘𝐴) − (ℑ‘𝐵)) < (ℑ‘𝐴))
127	88	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘𝐴) ∈ ℂ)
128	127	subid1d 11161	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘𝐴) − 0) = (ℑ‘𝐴))
129	126, 128	breqtrrd 5071	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘𝐴) − (ℑ‘𝐵)) < ((ℑ‘𝐴) − 0))
130		0red 10819	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℝ)
131	8	imcld 14741	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℝ)
132	130, 131, 87	ltsub2d 11425	. . . . . . . . . . 11 ⊢ (𝜑 → (0 < (ℑ‘𝐵) ↔ ((ℑ‘𝐴) − (ℑ‘𝐵)) < ((ℑ‘𝐴) − 0)))
133	132	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (0 < (ℑ‘𝐵) ↔ ((ℑ‘𝐴) − (ℑ‘𝐵)) < ((ℑ‘𝐴) − 0)))
134	129, 133	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 0 < (ℑ‘𝐵))
135		argimgt0 25472	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 0 < (ℑ‘𝐵)) → (ℑ‘(log‘𝐵)) ∈ (0(,)π))
136	68, 134, 135	syl2anc 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐵)) ∈ (0(,)π))
137		eliooord 12977	. . . . . . . 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 498	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 0 < (ℑ‘(log‘𝐵)))
140	130, 12, 21	ltsub1d 11424	. . . . . . 7 ⊢ (𝜑 → (0 < (ℑ‘(log‘𝐵)) ↔ (0 − (ℑ‘(log‘𝐴))) < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴)))))
141	140	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (0 < (ℑ‘(log‘𝐵)) ↔ (0 − (ℑ‘(log‘𝐴))) < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴)))))
142	139, 141	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (0 − (ℑ‘(log‘𝐴))) < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
143	67, 142	eqbrtrid 5078	. . . 4 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → -(ℑ‘(log‘𝐴)) < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
144	54, 56, 57, 66, 143	lttrd 10976	. . 3 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → -π < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
145		lttri4 10900	. . . 4 ⊢ (((ℑ‘𝐴) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℑ‘𝐴) < 0 ∨ (ℑ‘𝐴) = 0 ∨ 0 < (ℑ‘𝐴)))
146	87, 118, 145	sylancl 589	. . 3 ⊢ (𝜑 → ((ℑ‘𝐴) < 0 ∨ (ℑ‘𝐴) = 0 ∨ 0 < (ℑ‘𝐴)))
147	39, 53, 144, 146	mpjao3dan 1433	. 2 ⊢ (𝜑 → -π < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))))
148	1	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → π ∈ ℝ)
149	30	renegcld 11242	. . . . 5 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -(ℑ‘(log‘𝐴)) ∈ ℝ)
150	8	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → 𝐵 ∈ ℂ)
151	88	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) ∈ ℂ)
152	151	subid1d 11161	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘𝐴) − 0) = (ℑ‘𝐴))
153	87	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) ∈ ℝ)
154	74	renegcld 11242	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → -(abs‘(𝐴 − 𝐵)) ∈ ℝ)
155	154	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -(abs‘(𝐴 − 𝐵)) ∈ ℝ)
156	72	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(𝐴 − 𝐵)) ∈ ℝ)
157	74	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (abs‘(𝐴 − 𝐵)) ∈ ℝ)
158	107	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (abs‘(𝐴 − 𝐵)) < 𝑆)
159	118	ltnri 10924	. . . . . . . . . . . . . . . . . . . 20 ⊢ ¬ 0 < 0
160		breq1 5046	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℑ‘𝐴) = 0 → ((ℑ‘𝐴) < 0 ↔ 0 < 0))
161	159, 160	mtbiri 330	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℑ‘𝐴) = 0 → ¬ (ℑ‘𝐴) < 0)
162	161	necon2ai 2964	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℑ‘𝐴) < 0 → (ℑ‘𝐴) ≠ 0)
163	162, 116	sylan2 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → 𝑆 = (abs‘(ℑ‘𝐴)))
164		ltle 10904	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ℑ‘𝐴) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℑ‘𝐴) < 0 → (ℑ‘𝐴) ≤ 0))
165	87, 118, 164	sylancl 589	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((ℑ‘𝐴) < 0 → (ℑ‘𝐴) ≤ 0))
166	165	imp 410	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) ≤ 0)
167	153, 166	absnidd 14960	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (abs‘(ℑ‘𝐴)) = -(ℑ‘𝐴))
168	163, 167	eqtrd 2774	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → 𝑆 = -(ℑ‘𝐴))
169	158, 168	breqtrd 5069	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (abs‘(𝐴 − 𝐵)) < -(ℑ‘𝐴))
170	157, 153, 169	ltnegcon2d 11396	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) < -(abs‘(𝐴 − 𝐵)))
171	81	simpld 498	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → -(abs‘(𝐴 − 𝐵)) ≤ (ℑ‘(𝐴 − 𝐵)))
172	171	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -(abs‘(𝐴 − 𝐵)) ≤ (ℑ‘(𝐴 − 𝐵)))
173	153, 155, 156, 170, 172	ltletrd 10975	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) < (ℑ‘(𝐴 − 𝐵)))
174	69	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵)))
175	173, 174	breqtrd 5069	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) < ((ℑ‘𝐴) − (ℑ‘𝐵)))
176	152, 175	eqbrtrd 5065	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘𝐴) − 0) < ((ℑ‘𝐴) − (ℑ‘𝐵)))
177	150	imcld 14741	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐵) ∈ ℝ)
178	177, 31, 153	ltsub2d 11425	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘𝐵) < 0 ↔ ((ℑ‘𝐴) − 0) < ((ℑ‘𝐴) − (ℑ‘𝐵))))
179	176, 178	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐵) < 0)
180		argimlt0 25473	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℂ ∧ (ℑ‘𝐵) < 0) → (ℑ‘(log‘𝐵)) ∈ (-π(,)0))
181	150, 179, 180	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐵)) ∈ (-π(,)0))
182		eliooord 12977	. . . . . . . . 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 499	. . . . . . 7 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐵)) < 0)
185	13, 31, 30, 184	ltsub1dd 11427	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) < (0 − (ℑ‘(log‘𝐴))))
186	185, 67	breqtrrdi 5085	. . . . 5 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) < -(ℑ‘(log‘𝐴)))
187	35	simpld 498	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -π < (ℑ‘(log‘𝐴)))
188		ltnegcon1 11316	. . . . . . 7 ⊢ ((π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (-π < (ℑ‘(log‘𝐴)) ↔ -(ℑ‘(log‘𝐴)) < π))
189	1, 30, 188	sylancr 590	. . . . . 6 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → (-π < (ℑ‘(log‘𝐴)) ↔ -(ℑ‘(log‘𝐴)) < π))
190	187, 189	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → -(ℑ‘(log‘𝐴)) < π)
191	23, 149, 148, 186, 190	lttrd 10976	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) < π)
192	23, 148, 191	ltled 10963	. . 3 ⊢ ((𝜑 ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ≤ π)
193	24	simprd 499	. . . . 5 ⊢ (𝜑 → (ℑ‘(log‘𝐵)) ≤ π)
194	193	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (ℑ‘𝐴) = 0) → (ℑ‘(log‘𝐵)) ≤ π)
195	52, 194	eqbrtrd 5065	. . 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 484	. . . . 5 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐵)) ∈ ℝ)
198		0red 10819	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 0 ∈ ℝ)
199	21	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ ℝ)
200	61	simpld 498	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → 0 < (ℑ‘(log‘𝐴)))
201	198, 199, 197, 200	ltsub2dd 11428	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) < ((ℑ‘(log‘𝐵)) − 0))
202	27	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐵)) ∈ ℂ)
203	202	subid1d 11161	. . . . . 6 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐵)) − 0) = (ℑ‘(log‘𝐵)))
204	201, 203	breqtrd 5069	. . . . 5 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) < (ℑ‘(log‘𝐵)))
205	138	simprd 499	. . . . 5 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐵)) < π)
206	57, 197, 196, 204, 205	lttrd 10976	. . . 4 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) < π)
207	57, 196, 206	ltled 10963	. . 3 ⊢ ((𝜑 ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ≤ π)
208	192, 195, 207, 146	mpjao3dan 1433	. 2 ⊢ (𝜑 → ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ≤ π)
209	147, 208	jca 515	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 209 ∧ wa 399 ∨ w3o 1088 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∖ cdif 3854 ifcif 4429 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 ℂcc 10710 ℝcr 10711 0cc0 10712 1c1 10713 + caddc 10715 · cmul 10717 -∞cmnf 10848 < clt 10850 ≤ cle 10851 − cmin 11045 -cneg 11046 / cdiv 11472 ℝ⁺crp 12569 (,)cioo 12918 (,]cioc 12919 ℑcim 14644 abscabs 14780 πcpi 15609 logclog 25415

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-fi 9016 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ioc 12923 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-mod 13426 df-seq 13558 df-exp 13619 df-fac 13823 df-bc 13852 df-hash 13880 df-shft 14613 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-limsup 15015 df-clim 15032 df-rlim 15033 df-sum 15233 df-ef 15610 df-sin 15612 df-cos 15613 df-pi 15615 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-pt 16921 df-prds 16924 df-xrs 16979 df-qtop 16984 df-imas 16985 df-xps 16987 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-mulg 18461 df-cntz 18683 df-cmn 19144 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-fbas 20332 df-fg 20333 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cld 21888 df-ntr 21889 df-cls 21890 df-nei 21967 df-lp 22005 df-perf 22006 df-cn 22096 df-cnp 22097 df-haus 22184 df-tx 22431 df-hmeo 22624 df-fil 22715 df-fm 22807 df-flim 22808 df-flf 22809 df-xms 23190 df-ms 23191 df-tms 23192 df-cncf 23747 df-limc 24735 df-dv 24736 df-log 25417

This theorem is referenced by: logcnlem4 25505

W3C validator