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Theorem rpabscxpbnd 15608

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 15562	. . . . 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 5630	. . 3 ⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘(exp‘(𝐵 · (log‘𝐴)))))
6	1	relogcld 15550	. . . . . 6 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
7	6	recnd 8171	. . . . 5 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
8	2, 7	mulcld 8163	. . . 4 ⊢ (𝜑 → (𝐵 · (log‘𝐴)) ∈ ℂ)
9		absef 12276	. . . 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 11444	. . . . . . 7 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℝ)
12	7	recld 11444	. . . . . . 7 ⊢ (𝜑 → (ℜ‘(log‘𝐴)) ∈ ℝ)
13	11, 12	remulcld 8173	. . . . . 6 ⊢ (𝜑 → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℝ)
14	13	recnd 8171	. . . . 5 ⊢ (𝜑 → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℂ)
15	2	imcld 11445	. . . . . . 7 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℝ)
16	7	imcld 11445	. . . . . . . 8 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) ∈ ℝ)
17	16	renegcld 8522	. . . . . . 7 ⊢ (𝜑 → -(ℑ‘(log‘𝐴)) ∈ ℝ)
18	15, 17	remulcld 8173	. . . . . 6 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℝ)
19	18	recnd 8171	. . . . 5 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℂ)
20		efadd 12181	. . . . 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 8173	. . . . . . . 8 ⊢ (𝜑 → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℝ)
23	22	recnd 8171	. . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℂ)
24	14, 23	negsubd 8459	. . . . . 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 8171	. . . . . . . 8 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℂ)
26	16	recnd 8171	. . . . . . . 8 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) ∈ ℂ)
27	25, 26	mulneg2d 8554	. . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) = -((ℑ‘𝐵) · (ℑ‘(log‘𝐴))))
28	27	oveq2d 6016	. . . . . 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 11469	. . . . . 6 ⊢ (𝜑 → (ℜ‘(𝐵 · (log‘𝐴))) = (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) − ((ℑ‘𝐵) · (ℑ‘(log‘𝐴)))))
30	24, 28, 29	3eqtr4d 2272	. . . . 5 ⊢ (𝜑 → (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = (ℜ‘(𝐵 · (log‘𝐴))))
31	30	fveq2d 5630	. . . 4 ⊢ (𝜑 → (exp‘(((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
32	6	rered 11475	. . . . . . . . 9 ⊢ (𝜑 → (ℜ‘(log‘𝐴)) = (log‘𝐴))
33	1	rpred 9888	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
34	1	rpge0d 9892	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 𝐴)
35	33, 34	absidd 11673	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘𝐴) = 𝐴)
36	35	fveq2d 5630	. . . . . . . . 9 ⊢ (𝜑 → (log‘(abs‘𝐴)) = (log‘𝐴))
37	32, 36	eqtr4d 2265	. . . . . . . 8 ⊢ (𝜑 → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
38	37	oveq2d 6016	. . . . . . 7 ⊢ (𝜑 → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) = ((ℜ‘𝐵) · (log‘(abs‘𝐴))))
39	38	fveq2d 5630	. . . . . 6 ⊢ (𝜑 → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
40	35, 1	eqeltrd 2306	. . . . . . 7 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ⁺)
41	11	recnd 8171	. . . . . . 7 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℂ)
42		rpcxpef 15562	. . . . . . 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 2265	. . . . 5 ⊢ (𝜑 → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)))
45	44	oveq1d 6015	. . . 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 2270	. . 3 ⊢ (𝜑 → (exp‘(ℜ‘(𝐵 · (log‘𝐴)))) = (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
47	5, 10, 46	3eqtrd 2266	. 2 ⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) = (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
48	40, 11	rpcxpcld 15601	. . . . 5 ⊢ (𝜑 → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ∈ ℝ⁺)
49	48	rpred 9888	. . . 4 ⊢ (𝜑 → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
50	18	reefcld 12175	. . . 4 ⊢ (𝜑 → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ)
51	49, 50	remulcld 8173	. . 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 9928	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
55	54, 11	rpcxpcld 15601	. . . . 5 ⊢ (𝜑 → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ⁺)
56	55	rpred 9888	. . . 4 ⊢ (𝜑 → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
57	56, 50	remulcld 8173	. . 3 ⊢ (𝜑 → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
58	2	abscld 11687	. . . . . 6 ⊢ (𝜑 → (abs‘𝐵) ∈ ℝ)
59		pire 15454	. . . . . 6 ⊢ π ∈ ℝ
60		remulcl 8123	. . . . . 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 12175	. . . 4 ⊢ (𝜑 → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
63	56, 62	remulcld 8173	. . 3 ⊢ (𝜑 → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) ∈ ℝ)
64	18	rpefcld 12192	. . . . 5 ⊢ (𝜑 → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ⁺)
65	64	rpge0d 9892	. . . 4 ⊢ (𝜑 → 0 ≤ (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
66	1	rpcnd 9890	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
67	1	rpap0d 9894	. . . . . . 7 ⊢ (𝜑 → 𝐴 # 0)
68	66, 67	absrpclapd 11694	. . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ⁺)
69	52, 68, 53	rpgecld 9928	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
70		rpabscxpbnd.3	. . . . . . 7 ⊢ (𝜑 → 0 < (ℜ‘𝐵))
71	11, 70	elrpd 9885	. . . . . 6 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℝ⁺)
72		rpcxple2 15586	. . . . . 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 1271	. . . . 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 9082	. . 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 9892	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
77	25	abscld 11687	. . . . . . 7 ⊢ (𝜑 → (abs‘(ℑ‘𝐵)) ∈ ℝ)
78	17	recnd 8171	. . . . . . . 8 ⊢ (𝜑 → -(ℑ‘(log‘𝐴)) ∈ ℂ)
79	78	abscld 11687	. . . . . . 7 ⊢ (𝜑 → (abs‘-(ℑ‘(log‘𝐴))) ∈ ℝ)
80	77, 79	remulcld 8173	. . . . . 6 ⊢ (𝜑 → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
81	18	leabsd 11667	. . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
82	25, 78	absmuld 11700	. . . . . . 7 ⊢ (𝜑 → (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
83	81, 82	breqtrd 4108	. . . . . 6 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
84	58, 79	remulcld 8173	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
85	78	absge0d 11690	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘-(ℑ‘(log‘𝐴))))
86		absimle 11590	. . . . . . . . 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 9082	. . . . . . 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 11690	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘𝐵))
91	26	absnegd 11695	. . . . . . . . 9 ⊢ (𝜑 → (abs‘-(ℑ‘(log‘𝐴))) = (abs‘(ℑ‘(log‘𝐴))))
92	59	renegcli 8404	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ
93		0re 8142	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
94		pipos 15456	. . . . . . . . . . . . 13 ⊢ 0 < π
95		lt0neg2 8612	. . . . . . . . . . . . . 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 8245	. . . . . . . . . . 11 ⊢ -π ≤ 0
99	6	reim0d 11476	. . . . . . . . . . 11 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) = 0)
100	98, 99	breqtrrid 4120	. . . . . . . . . 10 ⊢ (𝜑 → -π ≤ (ℑ‘(log‘𝐴)))
101	93, 59, 94	ltleii 8245	. . . . . . . . . . 11 ⊢ 0 ≤ π
102	99, 101	eqbrtrdi 4121	. . . . . . . . . 10 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) ≤ π)
103		absle 11595	. . . . . . . . . . 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 950	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(ℑ‘(log‘𝐴))) ≤ π)
106	91, 105	eqbrtrd 4104	. . . . . . . 8 ⊢ (𝜑 → (abs‘-(ℑ‘(log‘𝐴))) ≤ π)
107	79, 89, 58, 90, 106	lemul2ad 9083	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
108	80, 84, 61, 88, 107	letrd 8266	. . . . . 6 ⊢ (𝜑 → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
109	18, 80, 61, 83, 108	letrd 8266	. . . . 5 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘𝐵) · π))
110		efle 15444	. . . . . 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 9083	. . 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 8266	. 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 4104	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 1395 ∈ wcel 2200 class class class wbr 4082 ‘cfv 5317 (class class class)co 6000 ℂcc 7993 ℝcr 7994 0cc0 7995 + caddc 7998 · cmul 8000 < clt 8177 ≤ cle 8178 − cmin 8313 -cneg 8314 ℝ⁺crp 9845 ℜcre 11346 ℑcim 11347 abscabs 11503 expce 12148 πcpi 12153 logclog 15524 ↑_𝑐ccxp 15525

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113 ax-arch 8114 ax-caucvg 8115 ax-pre-suploc 8116 ax-addf 8117 ax-mulf 8118

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-disj 4059 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-ilim 4459 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-isom 5326 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-of 6216 df-1st 6284 df-2nd 6285 df-recs 6449 df-irdg 6514 df-frec 6535 df-1o 6560 df-oadd 6564 df-er 6678 df-map 6795 df-pm 6796 df-en 6886 df-dom 6887 df-fin 6888 df-sup 7147 df-inf 7148 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-2 9165 df-3 9166 df-4 9167 df-5 9168 df-6 9169 df-7 9170 df-8 9171 df-9 9172 df-n0 9366 df-z 9443 df-uz 9719 df-q 9811 df-rp 9846 df-xneg 9964 df-xadd 9965 df-ioo 10084 df-ioc 10085 df-ico 10086 df-icc 10087 df-fz 10201 df-fzo 10335 df-seqfrec 10665 df-exp 10756 df-fac 10943 df-bc 10965 df-ihash 10993 df-shft 11321 df-cj 11348 df-re 11349 df-im 11350 df-rsqrt 11504 df-abs 11505 df-clim 11785 df-sumdc 11860 df-ef 12154 df-e 12155 df-sin 12156 df-cos 12157 df-pi 12159 df-rest 13269 df-topgen 13288 df-psmet 14501 df-xmet 14502 df-met 14503 df-bl 14504 df-mopn 14505 df-top 14666 df-topon 14679 df-bases 14711 df-ntr 14764 df-cn 14856 df-cnp 14857 df-tx 14921 df-cncf 15239 df-limced 15324 df-dvap 15325 df-relog 15526 df-rpcxp 15527

This theorem is referenced by: (None)

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