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Theorem rpabscxpbnd 16042

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 15996	. . . . 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 5699	. . 3 ⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘(exp‘(𝐵 · (log‘𝐴)))))
6	1	relogcld 15983	. . . . . 6 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
7	6	recnd 8354	. . . . 5 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
8	2, 7	mulcld 8346	. . . 4 ⊢ (𝜑 → (𝐵 · (log‘𝐴)) ∈ ℂ)
9		absef 12537	. . . 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 11704	. . . . . . 7 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℝ)
12	7	recld 11704	. . . . . . 7 ⊢ (𝜑 → (ℜ‘(log‘𝐴)) ∈ ℝ)
13	11, 12	remulcld 8356	. . . . . 6 ⊢ (𝜑 → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℝ)
14	13	recnd 8354	. . . . 5 ⊢ (𝜑 → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℂ)
15	2	imcld 11705	. . . . . . 7 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℝ)
16	7	imcld 11705	. . . . . . . 8 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) ∈ ℝ)
17	16	renegcld 8707	. . . . . . 7 ⊢ (𝜑 → -(ℑ‘(log‘𝐴)) ∈ ℝ)
18	15, 17	remulcld 8356	. . . . . 6 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℝ)
19	18	recnd 8354	. . . . 5 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℂ)
20		efadd 12442	. . . . 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 8356	. . . . . . . 8 ⊢ (𝜑 → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℝ)
23	22	recnd 8354	. . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℂ)
24	14, 23	negsubd 8643	. . . . . 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 8354	. . . . . . . 8 ⊢ (𝜑 → (ℑ‘𝐵) ∈ ℂ)
26	16	recnd 8354	. . . . . . . 8 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) ∈ ℂ)
27	25, 26	mulneg2d 8739	. . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) = -((ℑ‘𝐵) · (ℑ‘(log‘𝐴))))
28	27	oveq2d 6101	. . . . . 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 11729	. . . . . 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 5699	. . . 4 ⊢ (𝜑 → (exp‘(((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
32	6	rered 11735	. . . . . . . . 9 ⊢ (𝜑 → (ℜ‘(log‘𝐴)) = (log‘𝐴))
33	1	rpred 10097	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
34	1	rpge0d 10101	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 𝐴)
35	33, 34	absidd 11933	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘𝐴) = 𝐴)
36	35	fveq2d 5699	. . . . . . . . 9 ⊢ (𝜑 → (log‘(abs‘𝐴)) = (log‘𝐴))
37	32, 36	eqtr4d 2274	. . . . . . . 8 ⊢ (𝜑 → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
38	37	oveq2d 6101	. . . . . . 7 ⊢ (𝜑 → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) = ((ℜ‘𝐵) · (log‘(abs‘𝐴))))
39	38	fveq2d 5699	. . . . . 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 8354	. . . . . . 7 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℂ)
42		rpcxpef 15996	. . . . . . 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 6100	. . . 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 16035	. . . . 5 ⊢ (𝜑 → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ∈ ℝ⁺)
49	48	rpred 10097	. . . 4 ⊢ (𝜑 → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
50	18	reefcld 12436	. . . 4 ⊢ (𝜑 → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ)
51	49, 50	remulcld 8356	. . 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 10137	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
55	54, 11	rpcxpcld 16035	. . . . 5 ⊢ (𝜑 → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ⁺)
56	55	rpred 10097	. . . 4 ⊢ (𝜑 → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
57	56, 50	remulcld 8356	. . 3 ⊢ (𝜑 → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
58	2	abscld 11947	. . . . . 6 ⊢ (𝜑 → (abs‘𝐵) ∈ ℝ)
59		pire 15887	. . . . . 6 ⊢ π ∈ ℝ
60		remulcl 8307	. . . . . 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 12436	. . . 4 ⊢ (𝜑 → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
63	56, 62	remulcld 8356	. . 3 ⊢ (𝜑 → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) ∈ ℝ)
64	18	rpefcld 12453	. . . . 5 ⊢ (𝜑 → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ⁺)
65	64	rpge0d 10101	. . . 4 ⊢ (𝜑 → 0 ≤ (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
66	1	rpcnd 10099	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
67	1	rpap0d 10103	. . . . . . 7 ⊢ (𝜑 → 𝐴 # 0)
68	66, 67	absrpclapd 11954	. . . . . 6 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ⁺)
69	52, 68, 53	rpgecld 10137	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
70		rpabscxpbnd.3	. . . . . . 7 ⊢ (𝜑 → 0 < (ℜ‘𝐵))
71	11, 70	elrpd 10094	. . . . . 6 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℝ⁺)
72		rpcxple2 16020	. . . . . 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 9269	. . 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 10101	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
77	25	abscld 11947	. . . . . . 7 ⊢ (𝜑 → (abs‘(ℑ‘𝐵)) ∈ ℝ)
78	17	recnd 8354	. . . . . . . 8 ⊢ (𝜑 → -(ℑ‘(log‘𝐴)) ∈ ℂ)
79	78	abscld 11947	. . . . . . 7 ⊢ (𝜑 → (abs‘-(ℑ‘(log‘𝐴))) ∈ ℝ)
80	77, 79	remulcld 8356	. . . . . 6 ⊢ (𝜑 → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
81	18	leabsd 11927	. . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
82	25, 78	absmuld 11960	. . . . . . 7 ⊢ (𝜑 → (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
83	81, 82	breqtrd 4156	. . . . . 6 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
84	58, 79	remulcld 8356	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
85	78	absge0d 11950	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘-(ℑ‘(log‘𝐴))))
86		absimle 11850	. . . . . . . . 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 9269	. . . . . . 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 11950	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘𝐵))
91	26	absnegd 11955	. . . . . . . . 9 ⊢ (𝜑 → (abs‘-(ℑ‘(log‘𝐴))) = (abs‘(ℑ‘(log‘𝐴))))
92	59	renegcli 8588	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ
93		0re 8326	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
94		pipos 15889	. . . . . . . . . . . . 13 ⊢ 0 < π
95		lt0neg2 8797	. . . . . . . . . . . . . 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 8428	. . . . . . . . . . 11 ⊢ -π ≤ 0
99	6	reim0d 11736	. . . . . . . . . . 11 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) = 0)
100	98, 99	breqtrrid 4168	. . . . . . . . . 10 ⊢ (𝜑 → -π ≤ (ℑ‘(log‘𝐴)))
101	93, 59, 94	ltleii 8428	. . . . . . . . . . 11 ⊢ 0 ≤ π
102	99, 101	eqbrtrdi 4169	. . . . . . . . . 10 ⊢ (𝜑 → (ℑ‘(log‘𝐴)) ≤ π)
103		absle 11855	. . . . . . . . . . 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 4152	. . . . . . . 8 ⊢ (𝜑 → (abs‘-(ℑ‘(log‘𝐴))) ≤ π)
107	79, 89, 58, 90, 106	lemul2ad 9270	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
108	80, 84, 61, 88, 107	letrd 8450	. . . . . 6 ⊢ (𝜑 → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
109	18, 80, 61, 83, 108	letrd 8450	. . . . 5 ⊢ (𝜑 → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘𝐵) · π))
110		efle 15877	. . . . . 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 9270	. . 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 8450	. 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 4152	1 ⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 class class class wbr 4130 ‘cfv 5377 (class class class)co 6085 ℂcc 8177 ℝcr 8178 0cc0 8179 + caddc 8182 · cmul 8184 < clt 8360 ≤ cle 8361 − cmin 8497 -cneg 8498 ℝ⁺crp 10054 ℜcre 11605 ℑcim 11606 abscabs 11763 expce 12409 πcpi 12414 logclog 15957 ↑_𝑐ccxp 15958

This proof depends on 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 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299 ax-pre-suploc 8300 ax-addf 8301 ax-mulf 8302

This proof 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 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-disj 4107 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-of 6302 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-map 6924 df-pm 6925 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7324 df-inf 7325 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-2 9363 df-3 9364 df-4 9365 df-5 9366 df-6 9367 df-7 9368 df-8 9369 df-9 9370 df-n0 9564 df-z 9645 df-uz 9922 df-q 10020 df-rp 10055 df-xneg 10174 df-xadd 10175 df-ioo 10294 df-ioc 10295 df-ico 10296 df-icc 10297 df-fz 10412 df-fzo 10550 df-seqfrec 10885 df-exp 10976 df-fac 11164 df-bc 11186 df-ihash 11215 df-shft 11580 df-cj 11607 df-re 11608 df-im 11609 df-rsqrt 11764 df-abs 11765 df-clim 12045 df-sumdc 12120 df-ef 12415 df-e 12416 df-sin 12417 df-cos 12418 df-pi 12420 df-rest 13595 df-topgen 13614 df-psmet 14880 df-xmet 14881 df-met 14882 df-bl 14883 df-mopn 14884 df-top 15099 df-topon 15112 df-bases 15144 df-ntr 15197 df-cn 15289 df-cnp 15290 df-tx 15354 df-cncf 15672 df-limced 15757 df-dvap 15758 df-relog 15959 df-rpcxp 15960

This theorem is used by: (None)

W3C validator