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Theorem rpabscxpbnd 15284

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 15238	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
4	1, 2, 3	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
5	4	fveq2d 5565	. . 3 ⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘(exp‘(𝐵 · (log‘𝐴)))))
6	1	relogcld 15226	. . . . . 6 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
7	6	recnd 8074	. . . . 5 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
8	2, 7	mulcld 8066	. . . 4 ⊢ (𝜑 → (𝐵 · (log‘𝐴)) ∈ ℂ)
9		absef 11954	. . . 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 11122	. . . . . . 7 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℝ)
12	7	recld 11122	. . . . . . 7 ⊢ (𝜑 → (ℜ‘(log‘𝐴)) ∈ ℝ)
13	11, 12	remulcld 8076	. . . . . 6 ⊢ (𝜑 → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℝ)
14	13	recnd 8074	. . . . 5 ⊢ (𝜑 → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℂ)
15	2	imcld 11123	. . . . . . 7 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℝ)
16	7	imcld 11123	. . . . . . . 8 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) ∈ ℝ)
17	16	renegcld 8425	. . . . . . 7 ⊢ (𝜑 → -(ℑ‘(log‘𝐴)) ∈ ℝ)
18	15, 17	remulcld 8076	. . . . . 6 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℝ)
19	18	recnd 8074	. . . . 5 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℂ)
20		efadd 11859	. . . . 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 411	. . . 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 8076	. . . . . . . 8 ⊢ (𝜑 → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℝ)
23	22	recnd 8074	. . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℂ)
24	14, 23	negsubd 8362	. . . . . 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 8074	. . . . . . . 8 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℂ)
26	16	recnd 8074	. . . . . . . 8 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) ∈ ℂ)
27	25, 26	mulneg2d 8457	. . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) = -((ℑ‘𝐵) · (ℑ‘(log‘𝐴))))
28	27	oveq2d 5941	. . . . . 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 11147	. . . . . 6 ⊢ (𝜑 → (ℜ‘(𝐵 · (log‘𝐴))) = (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) − ((ℑ‘𝐵) · (ℑ‘(log‘𝐴)))))
30	24, 28, 29	3eqtr4d 2239	. . . . 5 ⊢ (𝜑 → (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = (ℜ‘(𝐵 · (log‘𝐴))))
31	30	fveq2d 5565	. . . 4 ⊢ (𝜑 → (exp‘(((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
32	6	rered 11153	. . . . . . . . 9 ⊢ (𝜑 → (ℜ‘(log‘𝐴)) = (log‘𝐴))
33	1	rpred 9790	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
34	1	rpge0d 9794	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 𝐴)
35	33, 34	absidd 11351	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘𝐴) = 𝐴)
36	35	fveq2d 5565	. . . . . . . . 9 ⊢ (𝜑 → (log‘(abs‘𝐴)) = (log‘𝐴))
37	32, 36	eqtr4d 2232	. . . . . . . 8 ⊢ (𝜑 → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
38	37	oveq2d 5941	. . . . . . 7 ⊢ (𝜑 → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) = ((ℜ‘𝐵) · (log‘(abs‘𝐴))))
39	38	fveq2d 5565	. . . . . 6 ⊢ (𝜑 → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
40	35, 1	eqeltrd 2273	. . . . . . 7 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ⁺)
41	11	recnd 8074	. . . . . . 7 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℂ)
42		rpcxpef 15238	. . . . . . 7 ⊢ (((abs‘𝐴) ∈ ℝ⁺ ∧ (ℜ‘𝐵) ∈ ℂ) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
43	40, 41, 42	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
44	39, 43	eqtr4d 2232	. . . . 5 ⊢ (𝜑 → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)))
45	44	oveq1d 5940	. . . 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 2237	. . 3 ⊢ (𝜑 → (exp‘(ℜ‘(𝐵 · (log‘𝐴)))) = (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
47	5, 10, 46	3eqtrd 2233	. 2 ⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) = (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
48	40, 11	rpcxpcld 15277	. . . . 5 ⊢ (𝜑 → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ∈ ℝ⁺)
49	48	rpred 9790	. . . 4 ⊢ (𝜑 → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
50	18	reefcld 11853	. . . 4 ⊢ (𝜑 → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ)
51	49, 50	remulcld 8076	. . 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 9830	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
55	54, 11	rpcxpcld 15277	. . . . 5 ⊢ (𝜑 → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ⁺)
56	55	rpred 9790	. . . 4 ⊢ (𝜑 → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
57	56, 50	remulcld 8076	. . 3 ⊢ (𝜑 → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
58	2	abscld 11365	. . . . . 6 ⊢ (𝜑 → (abs‘𝐵) ∈ ℝ)
59		pire 15130	. . . . . 6 ⊢ π ∈ ℝ
60		remulcl 8026	. . . . . 6 ⊢ (((abs‘𝐵) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘𝐵) · π) ∈ ℝ)
61	58, 59, 60	sylancl 413	. . . . 5 ⊢ (𝜑 → ((abs‘𝐵) · π) ∈ ℝ)
62	61	reefcld 11853	. . . 4 ⊢ (𝜑 → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
63	56, 62	remulcld 8076	. . 3 ⊢ (𝜑 → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) ∈ ℝ)
64	18	rpefcld 11870	. . . . 5 ⊢ (𝜑 → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ⁺)
65	64	rpge0d 9794	. . . 4 ⊢ (𝜑 → 0 ≤ (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
66	1	rpcnd 9792	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
67	1	rpap0d 9796	. . . . . . 7 ⊢ (𝜑 → 𝐴 # 0)
68	66, 67	absrpclapd 11372	. . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ⁺)
69	52, 68, 53	rpgecld 9830	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
70		rpabscxpbnd.3	. . . . . . 7 ⊢ (𝜑 → 0 < (ℜ‘𝐵))
71	11, 70	elrpd 9787	. . . . . 6 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℝ⁺)
72		rpcxple2 15262	. . . . . 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 1249	. . . . 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 8985	. . 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 9794	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
77	25	abscld 11365	. . . . . . 7 ⊢ (𝜑 → (abs‘(ℑ‘𝐵)) ∈ ℝ)
78	17	recnd 8074	. . . . . . . 8 ⊢ (𝜑 → -(ℑ‘(log‘𝐴)) ∈ ℂ)
79	78	abscld 11365	. . . . . . 7 ⊢ (𝜑 → (abs‘-(ℑ‘(log‘𝐴))) ∈ ℝ)
80	77, 79	remulcld 8076	. . . . . 6 ⊢ (𝜑 → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
81	18	leabsd 11345	. . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
82	25, 78	absmuld 11378	. . . . . . 7 ⊢ (𝜑 → (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
83	81, 82	breqtrd 4060	. . . . . 6 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
84	58, 79	remulcld 8076	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
85	78	absge0d 11368	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘-(ℑ‘(log‘𝐴))))
86		absimle 11268	. . . . . . . . 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 8985	. . . . . . 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 11368	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘𝐵))
91	26	absnegd 11373	. . . . . . . . 9 ⊢ (𝜑 → (abs‘-(ℑ‘(log‘𝐴))) = (abs‘(ℑ‘(log‘𝐴))))
92	59	renegcli 8307	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ
93		0re 8045	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
94		pipos 15132	. . . . . . . . . . . . 13 ⊢ 0 < π
95		lt0neg2 8515	. . . . . . . . . . . . . 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 8148	. . . . . . . . . . 11 ⊢ -π ≤ 0
99	6	reim0d 11154	. . . . . . . . . . 11 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) = 0)
100	98, 99	breqtrrid 4072	. . . . . . . . . 10 ⊢ (𝜑 → -π ≤ (ℑ‘(log‘𝐴)))
101	93, 59, 94	ltleii 8148	. . . . . . . . . . 11 ⊢ 0 ≤ π
102	99, 101	eqbrtrdi 4073	. . . . . . . . . 10 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) ≤ π)
103		absle 11273	. . . . . . . . . . 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 413	. . . . . . . . . 10 ⊢ (𝜑 → ((abs‘(ℑ‘(log‘𝐴))) ≤ π ↔ (-π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π)))
105	100, 102, 104	mpbir2and 946	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(ℑ‘(log‘𝐴))) ≤ π)
106	91, 105	eqbrtrd 4056	. . . . . . . 8 ⊢ (𝜑 → (abs‘-(ℑ‘(log‘𝐴))) ≤ π)
107	79, 89, 58, 90, 106	lemul2ad 8986	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
108	80, 84, 61, 88, 107	letrd 8169	. . . . . 6 ⊢ (𝜑 → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
109	18, 80, 61, 83, 108	letrd 8169	. . . . 5 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘𝐵) · π))
110		efle 15120	. . . . . 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 411	. . . . 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 8986	. . 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 8169	. 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 4056	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 1364 ∈ wcel 2167 class class class wbr 4034 ‘cfv 5259 (class class class)co 5925 ℂcc 7896 ℝcr 7897 0cc0 7898 + caddc 7901 · cmul 7903 < clt 8080 ≤ cle 8081 − cmin 8216 -cneg 8217 ℝ⁺crp 9747 ℜcre 11024 ℑcim 11025 abscabs 11181 expce 11826 πcpi 11831 logclog 15200 ↑_𝑐ccxp 15201

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7989 ax-resscn 7990 ax-1cn 7991 ax-1re 7992 ax-icn 7993 ax-addcl 7994 ax-addrcl 7995 ax-mulcl 7996 ax-mulrcl 7997 ax-addcom 7998 ax-mulcom 7999 ax-addass 8000 ax-mulass 8001 ax-distr 8002 ax-i2m1 8003 ax-0lt1 8004 ax-1rid 8005 ax-0id 8006 ax-rnegex 8007 ax-precex 8008 ax-cnre 8009 ax-pre-ltirr 8010 ax-pre-ltwlin 8011 ax-pre-lttrn 8012 ax-pre-apti 8013 ax-pre-ltadd 8014 ax-pre-mulgt0 8015 ax-pre-mulext 8016 ax-arch 8017 ax-caucvg 8018 ax-pre-suploc 8019 ax-addf 8020 ax-mulf 8021

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-disj 4012 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-of 6139 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-frec 6458 df-1o 6483 df-oadd 6487 df-er 6601 df-map 6718 df-pm 6719 df-en 6809 df-dom 6810 df-fin 6811 df-sup 7059 df-inf 7060 df-pnf 8082 df-mnf 8083 df-xr 8084 df-ltxr 8085 df-le 8086 df-sub 8218 df-neg 8219 df-reap 8621 df-ap 8628 df-div 8719 df-inn 9010 df-2 9068 df-3 9069 df-4 9070 df-5 9071 df-6 9072 df-7 9073 df-8 9074 df-9 9075 df-n0 9269 df-z 9346 df-uz 9621 df-q 9713 df-rp 9748 df-xneg 9866 df-xadd 9867 df-ioo 9986 df-ioc 9987 df-ico 9988 df-icc 9989 df-fz 10103 df-fzo 10237 df-seqfrec 10559 df-exp 10650 df-fac 10837 df-bc 10859 df-ihash 10887 df-shft 10999 df-cj 11026 df-re 11027 df-im 11028 df-rsqrt 11182 df-abs 11183 df-clim 11463 df-sumdc 11538 df-ef 11832 df-e 11833 df-sin 11834 df-cos 11835 df-pi 11837 df-rest 12945 df-topgen 12964 df-psmet 14177 df-xmet 14178 df-met 14179 df-bl 14180 df-mopn 14181 df-top 14342 df-topon 14355 df-bases 14387 df-ntr 14440 df-cn 14532 df-cnp 14533 df-tx 14597 df-cncf 14915 df-limced 15000 df-dvap 15001 df-relog 15202 df-rpcxp 15203

This theorem is referenced by: (None)

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