rpabscxpbnd - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > rpabscxpbnd		GIF version

Theorem rpabscxpbnd 15965

Description: Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Jim Kingdon, 19-Jun-2024.)

**Hypotheses**
Ref	Expression
rpabscxpbnd.1	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
abscxpbnd.2	⊢ (𝜑 → 𝐵 ∈ ℂ)
rpabscxpbnd.3	⊢ (𝜑 → 0 < (ℜ‘𝐵))
abscxpbnd.4	⊢ (𝜑 → 𝑀 ∈ ℝ)
abscxpbnd.5	⊢ (𝜑 → (abs‘𝐴) ≤ 𝑀)

**Assertion**
Ref	Expression
rpabscxpbnd	⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))

**Proof of Theorem rpabscxpbnd**
Step	Hyp	Ref	Expression
1		rpabscxpbnd.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
2		abscxpbnd.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ)
3		rpcxpef 15919	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
4	1, 2, 3	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
5	4	fveq2d 5694	. . 3 ⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘(exp‘(𝐵 · (log‘𝐴)))))
6	1	relogcld 15906	. . . . . 6 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
7	6	recnd 8344	. . . . 5 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
8	2, 7	mulcld 8336	. . . 4 ⊢ (𝜑 → (𝐵 · (log‘𝐴)) ∈ ℂ)
9		absef 12515	. . . 4 ⊢ ((𝐵 · (log‘𝐴)) ∈ ℂ → (abs‘(exp‘(𝐵 · (log‘𝐴)))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
10	8, 9	syl 14	. . 3 ⊢ (𝜑 → (abs‘(exp‘(𝐵 · (log‘𝐴)))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
11	2	recld 11682	. . . . . . 7 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℝ)
12	7	recld 11682	. . . . . . 7 ⊢ (𝜑 → (ℜ‘(log‘𝐴)) ∈ ℝ)
13	11, 12	remulcld 8346	. . . . . 6 ⊢ (𝜑 → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℝ)
14	13	recnd 8344	. . . . 5 ⊢ (𝜑 → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℂ)
15	2	imcld 11683	. . . . . . 7 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℝ)
16	7	imcld 11683	. . . . . . . 8 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) ∈ ℝ)
17	16	renegcld 8697	. . . . . . 7 ⊢ (𝜑 → -(ℑ‘(log‘𝐴)) ∈ ℝ)
18	15, 17	remulcld 8346	. . . . . 6 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℝ)
19	18	recnd 8344	. . . . 5 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℂ)
20		efadd 12420	. . . . 5 ⊢ ((((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℂ ∧ ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℂ) → (exp‘(((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) = ((exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
21	14, 19, 20	syl2anc 415	. . . 4 ⊢ (𝜑 → (exp‘(((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) = ((exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
22	15, 16	remulcld 8346	. . . . . . . 8 ⊢ (𝜑 → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℝ)
23	22	recnd 8344	. . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℂ)
24	14, 23	negsubd 8633	. . . . . 6 ⊢ (𝜑 → (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + -((ℑ‘𝐵) · (ℑ‘(log‘𝐴)))) = (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) − ((ℑ‘𝐵) · (ℑ‘(log‘𝐴)))))
25	15	recnd 8344	. . . . . . . 8 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℂ)
26	16	recnd 8344	. . . . . . . 8 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) ∈ ℂ)
27	25, 26	mulneg2d 8729	. . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) = -((ℑ‘𝐵) · (ℑ‘(log‘𝐴))))
28	27	oveq2d 6091	. . . . . 6 ⊢ (𝜑 → (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + -((ℑ‘𝐵) · (ℑ‘(log‘𝐴)))))
29	2, 7	remuld 11707	. . . . . 6 ⊢ (𝜑 → (ℜ‘(𝐵 · (log‘𝐴))) = (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) − ((ℑ‘𝐵) · (ℑ‘(log‘𝐴)))))
30	24, 28, 29	3eqtr4d 2281	. . . . 5 ⊢ (𝜑 → (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = (ℜ‘(𝐵 · (log‘𝐴))))
31	30	fveq2d 5694	. . . 4 ⊢ (𝜑 → (exp‘(((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
32	6	rered 11713	. . . . . . . . 9 ⊢ (𝜑 → (ℜ‘(log‘𝐴)) = (log‘𝐴))
33	1	rpred 10076	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
34	1	rpge0d 10080	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 𝐴)
35	33, 34	absidd 11911	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘𝐴) = 𝐴)
36	35	fveq2d 5694	. . . . . . . . 9 ⊢ (𝜑 → (log‘(abs‘𝐴)) = (log‘𝐴))
37	32, 36	eqtr4d 2274	. . . . . . . 8 ⊢ (𝜑 → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
38	37	oveq2d 6091	. . . . . . 7 ⊢ (𝜑 → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) = ((ℜ‘𝐵) · (log‘(abs‘𝐴))))
39	38	fveq2d 5694	. . . . . 6 ⊢ (𝜑 → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
40	35, 1	eqeltrd 2315	. . . . . . 7 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ⁺)
41	11	recnd 8344	. . . . . . 7 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℂ)
42		rpcxpef 15919	. . . . . . 7 ⊢ (((abs‘𝐴) ∈ ℝ⁺ ∧ (ℜ‘𝐵) ∈ ℂ) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
43	40, 41, 42	syl2anc 415	. . . . . 6 ⊢ (𝜑 → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
44	39, 43	eqtr4d 2274	. . . . 5 ⊢ (𝜑 → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)))
45	44	oveq1d 6090	. . . 4 ⊢ (𝜑 → ((exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) = (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
46	21, 31, 45	3eqtr3d 2279	. . 3 ⊢ (𝜑 → (exp‘(ℜ‘(𝐵 · (log‘𝐴)))) = (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
47	5, 10, 46	3eqtrd 2275	. 2 ⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) = (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
48	40, 11	rpcxpcld 15958	. . . . 5 ⊢ (𝜑 → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ∈ ℝ⁺)
49	48	rpred 10076	. . . 4 ⊢ (𝜑 → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
50	18	reefcld 12414	. . . 4 ⊢ (𝜑 → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ)
51	49, 50	remulcld 8346	. . 3 ⊢ (𝜑 → (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
52		abscxpbnd.4	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ)
53		abscxpbnd.5	. . . . . . 7 ⊢ (𝜑 → (abs‘𝐴) ≤ 𝑀)
54	52, 40, 53	rpgecld 10116	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
55	54, 11	rpcxpcld 15958	. . . . 5 ⊢ (𝜑 → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ⁺)
56	55	rpred 10076	. . . 4 ⊢ (𝜑 → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
57	56, 50	remulcld 8346	. . 3 ⊢ (𝜑 → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
58	2	abscld 11925	. . . . . 6 ⊢ (𝜑 → (abs‘𝐵) ∈ ℝ)
59		pire 15810	. . . . . 6 ⊢ π ∈ ℝ
60		remulcl 8297	. . . . . 6 ⊢ (((abs‘𝐵) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘𝐵) · π) ∈ ℝ)
61	58, 59, 60	sylancl 417	. . . . 5 ⊢ (𝜑 → ((abs‘𝐵) · π) ∈ ℝ)
62	61	reefcld 12414	. . . 4 ⊢ (𝜑 → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
63	56, 62	remulcld 8346	. . 3 ⊢ (𝜑 → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) ∈ ℝ)
64	18	rpefcld 12431	. . . . 5 ⊢ (𝜑 → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ⁺)
65	64	rpge0d 10080	. . . 4 ⊢ (𝜑 → 0 ≤ (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
66	1	rpcnd 10078	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
67	1	rpap0d 10082	. . . . . . 7 ⊢ (𝜑 → 𝐴 # 0)
68	66, 67	absrpclapd 11932	. . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ⁺)
69	52, 68, 53	rpgecld 10116	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
70		rpabscxpbnd.3	. . . . . . 7 ⊢ (𝜑 → 0 < (ℜ‘𝐵))
71	11, 70	elrpd 10073	. . . . . 6 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℝ⁺)
72		rpcxple2 15943	. . . . . 6 ⊢ (((abs‘𝐴) ∈ ℝ⁺ ∧ 𝑀 ∈ ℝ⁺ ∧ (ℜ‘𝐵) ∈ ℝ⁺) → ((abs‘𝐴) ≤ 𝑀 ↔ ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ≤ (𝑀↑_𝑐(ℜ‘𝐵))))
73	68, 69, 71, 72	syl3anc 1278	. . . . 5 ⊢ (𝜑 → ((abs‘𝐴) ≤ 𝑀 ↔ ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ≤ (𝑀↑_𝑐(ℜ‘𝐵))))
74	53, 73	mpbid 147	. . . 4 ⊢ (𝜑 → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
75	49, 56, 50, 65, 74	lemul1ad 9259	. . 3 ⊢ (𝜑 → (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
76	55	rpge0d 10080	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
77	25	abscld 11925	. . . . . . 7 ⊢ (𝜑 → (abs‘(ℑ‘𝐵)) ∈ ℝ)
78	17	recnd 8344	. . . . . . . 8 ⊢ (𝜑 → -(ℑ‘(log‘𝐴)) ∈ ℂ)
79	78	abscld 11925	. . . . . . 7 ⊢ (𝜑 → (abs‘-(ℑ‘(log‘𝐴))) ∈ ℝ)
80	77, 79	remulcld 8346	. . . . . 6 ⊢ (𝜑 → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
81	18	leabsd 11905	. . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
82	25, 78	absmuld 11938	. . . . . . 7 ⊢ (𝜑 → (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
83	81, 82	breqtrd 4151	. . . . . 6 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
84	58, 79	remulcld 8346	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
85	78	absge0d 11928	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘-(ℑ‘(log‘𝐴))))
86		absimle 11828	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (abs‘(ℑ‘𝐵)) ≤ (abs‘𝐵))
87	2, 86	syl 14	. . . . . . . 8 ⊢ (𝜑 → (abs‘(ℑ‘𝐵)) ≤ (abs‘𝐵))
88	77, 58, 79, 85, 87	lemul1ad 9259	. . . . . . 7 ⊢ (𝜑 → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))))
89	59	a1i 9	. . . . . . . 8 ⊢ (𝜑 → π ∈ ℝ)
90	2	absge0d 11928	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘𝐵))
91	26	absnegd 11933	. . . . . . . . 9 ⊢ (𝜑 → (abs‘-(ℑ‘(log‘𝐴))) = (abs‘(ℑ‘(log‘𝐴))))
92	59	renegcli 8578	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ
93		0re 8316	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
94		pipos 15812	. . . . . . . . . . . . 13 ⊢ 0 < π
95		lt0neg2 8787	. . . . . . . . . . . . . 14 ⊢ (π ∈ ℝ → (0 < π ↔ -π < 0))
96	59, 95	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (0 < π ↔ -π < 0)
97	94, 96	mpbi 145	. . . . . . . . . . . 12 ⊢ -π < 0
98	92, 93, 97	ltleii 8418	. . . . . . . . . . 11 ⊢ -π ≤ 0
99	6	reim0d 11714	. . . . . . . . . . 11 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) = 0)
100	98, 99	breqtrrid 4163	. . . . . . . . . 10 ⊢ (𝜑 → -π ≤ (ℑ‘(log‘𝐴)))
101	93, 59, 94	ltleii 8418	. . . . . . . . . . 11 ⊢ 0 ≤ π
102	99, 101	eqbrtrdi 4164	. . . . . . . . . 10 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) ≤ π)
103		absle 11833	. . . . . . . . . . 11 ⊢ (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘(ℑ‘(log‘𝐴))) ≤ π ↔ (-π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π)))
104	16, 59, 103	sylancl 417	. . . . . . . . . 10 ⊢ (𝜑 → ((abs‘(ℑ‘(log‘𝐴))) ≤ π ↔ (-π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π)))
105	100, 102, 104	mpbir2and 957	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(ℑ‘(log‘𝐴))) ≤ π)
106	91, 105	eqbrtrd 4147	. . . . . . . 8 ⊢ (𝜑 → (abs‘-(ℑ‘(log‘𝐴))) ≤ π)
107	79, 89, 58, 90, 106	lemul2ad 9260	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
108	80, 84, 61, 88, 107	letrd 8440	. . . . . 6 ⊢ (𝜑 → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
109	18, 80, 61, 83, 108	letrd 8440	. . . . 5 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘𝐵) · π))
110		efle 15800	. . . . . 6 ⊢ ((((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℝ ∧ ((abs‘𝐵) · π) ∈ ℝ) → (((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘𝐵) · π) ↔ (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ≤ (exp‘((abs‘𝐵) · π))))
111	18, 61, 110	syl2anc 415	. . . . 5 ⊢ (𝜑 → (((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘𝐵) · π) ↔ (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ≤ (exp‘((abs‘𝐵) · π))))
112	109, 111	mpbid 147	. . . 4 ⊢ (𝜑 → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ≤ (exp‘((abs‘𝐵) · π)))
113	50, 62, 56, 76, 112	lemul2ad 9260	. . 3 ⊢ (𝜑 → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
114	51, 57, 63, 75, 113	letrd 8440	. 2 ⊢ (𝜑 → (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
115	47, 114	eqbrtrd 4147	1 ⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 class class class wbr 4125 ‘cfv 5372 (class class class)co 6075 ℂcc 8167 ℝcr 8168 0cc0 8169 + caddc 8172 · cmul 8174 < clt 8350 ≤ cle 8351 − cmin 8487 -cneg 8488 ℝ⁺crp 10033 ℜcre 11583 ℑcim 11584 abscabs 11741 expce 12387 πcpi 12392 logclog 15880 ↑_𝑐ccxp 15881

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289 ax-pre-suploc 8290 ax-addf 8291 ax-mulf 8292

This theorem depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-disj 4102 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-isom 5381 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-of 6292 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-frec 6652 df-1o 6677 df-oadd 6681 df-er 6797 df-map 6914 df-pm 6915 df-en 7013 df-dom 7014 df-fin 7015 df-sup 7314 df-inf 7315 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-5 9345 df-6 9346 df-7 9347 df-8 9348 df-9 9349 df-n0 9543 df-z 9624 df-uz 9901 df-q 9999 df-rp 10034 df-xneg 10153 df-xadd 10154 df-ioo 10273 df-ioc 10274 df-ico 10275 df-icc 10276 df-fz 10391 df-fzo 10528 df-seqfrec 10863 df-exp 10954 df-fac 11142 df-bc 11164 df-ihash 11193 df-shft 11558 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743 df-clim 12023 df-sumdc 12098 df-ef 12393 df-e 12394 df-sin 12395 df-cos 12396 df-pi 12398 df-rest 13572 df-topgen 13591 df-psmet 14852 df-xmet 14853 df-met 14854 df-bl 14855 df-mopn 14856 df-top 15022 df-topon 15035 df-bases 15067 df-ntr 15120 df-cn 15212 df-cnp 15213 df-tx 15277 df-cncf 15595 df-limced 15680 df-dvap 15681 df-relog 15882 df-rpcxp 15883

This theorem is referenced by: (None)

W3C validator