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Theorem abscxpbnd 25050

Description: Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.)

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

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

**Proof of Theorem abscxpbnd**
Step	Hyp	Ref	Expression
1		1le1 11075	. . . . 5 ⊢ 1 ≤ 1
2	1	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → 1 ≤ 1)
3		oveq12 6991	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴↑_𝑐𝐵) = (0↑_𝑐0))
4	3	adantll 702	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐴↑_𝑐𝐵) = (0↑_𝑐0))
5		0cn 10437	. . . . . . . 8 ⊢ 0 ∈ ℂ
6		cxp0 24969	. . . . . . . 8 ⊢ (0 ∈ ℂ → (0↑_𝑐0) = 1)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (0↑_𝑐0) = 1
8	4, 7	syl6eq 2832	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐴↑_𝑐𝐵) = 1)
9	8	fveq2d 6508	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘1))
10		abs1 14524	. . . . 5 ⊢ (abs‘1) = 1
11	9, 10	syl6eq 2832	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘(𝐴↑_𝑐𝐵)) = 1)
12		fveq2 6504	. . . . . . . . 9 ⊢ (𝐵 = 0 → (ℜ‘𝐵) = (ℜ‘0))
13		re0 14378	. . . . . . . . 9 ⊢ (ℜ‘0) = 0
14	12, 13	syl6eq 2832	. . . . . . . 8 ⊢ (𝐵 = 0 → (ℜ‘𝐵) = 0)
15	14	oveq2d 6998	. . . . . . 7 ⊢ (𝐵 = 0 → (𝑀↑_𝑐(ℜ‘𝐵)) = (𝑀↑_𝑐0))
16		abscxpbnd.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
17	16	recnd 10474	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
18	17	cxp0d 25004	. . . . . . . 8 ⊢ (𝜑 → (𝑀↑_𝑐0) = 1)
19	18	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 0) → (𝑀↑_𝑐0) = 1)
20	15, 19	sylan9eqr 2838	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝑀↑_𝑐(ℜ‘𝐵)) = 1)
21		simpr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → 𝐵 = 0)
22	21	abs00bd 14518	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘𝐵) = 0)
23	22	oveq1d 6997	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((abs‘𝐵) · π) = (0 · π))
24		picn 24763	. . . . . . . . . 10 ⊢ π ∈ ℂ
25	24	mul02i 10635	. . . . . . . . 9 ⊢ (0 · π) = 0
26	23, 25	syl6eq 2832	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((abs‘𝐵) · π) = 0)
27	26	fveq2d 6508	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (exp‘((abs‘𝐵) · π)) = (exp‘0))
28		ef0 15310	. . . . . . 7 ⊢ (exp‘0) = 1
29	27, 28	syl6eq 2832	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (exp‘((abs‘𝐵) · π)) = 1)
30	20, 29	oveq12d 7000	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) = (1 · 1))
31		1t1e1 11615	. . . . 5 ⊢ (1 · 1) = 1
32	30, 31	syl6eq 2832	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) = 1)
33	2, 11, 32	3brtr4d 4966	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
34		simplr 757	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 𝐴 = 0)
35	34	oveq1d 6997	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑_𝑐𝐵) = (0↑_𝑐𝐵))
36		abscxpbnd.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
37	36	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
38		0cxp 24965	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (0↑_𝑐𝐵) = 0)
39	37, 38	sylan 572	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (0↑_𝑐𝐵) = 0)
40	35, 39	eqtrd 2816	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑_𝑐𝐵) = 0)
41	40	abs00bd 14518	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) = 0)
42		0red 10449	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
43		abscxpbnd.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
44	43	abscld 14663	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
45	43	absge0d 14671	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘𝐴))
46		abscxpbnd.5	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐴) ≤ 𝑀)
47	42, 44, 16, 45, 46	letrd 10603	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑀)
48	36	recld 14420	. . . . . . 7 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℝ)
49	16, 47, 48	recxpcld 25022	. . . . . 6 ⊢ (𝜑 → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
50	49	ad2antrr 714	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
51	36	abscld 14663	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐵) ∈ ℝ)
52	51	ad2antrr 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘𝐵) ∈ ℝ)
53		pire 24762	. . . . . . 7 ⊢ π ∈ ℝ
54		remulcl 10426	. . . . . . 7 ⊢ (((abs‘𝐵) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘𝐵) · π) ∈ ℝ)
55	52, 53, 54	sylancl 578	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → ((abs‘𝐵) · π) ∈ ℝ)
56	55	reefcld 15307	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
57	16, 47, 48	cxpge0d 25023	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
58	57	ad2antrr 714	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
59	55	rpefcld 15324	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ⁺)
60	59	rpge0d 12258	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 0 ≤ (exp‘((abs‘𝐵) · π)))
61	50, 56, 58, 60	mulge0d 11024	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 0 ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
62	41, 61	eqbrtrd 4956	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
63	33, 62	pm2.61dane 3057	. 2 ⊢ ((𝜑 ∧ 𝐴 = 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
64	43	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
65		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
66	36	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐵 ∈ ℂ)
67	64, 65, 66	cxpefd 25011	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
68	67	fveq2d 6508	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘(exp‘(𝐵 · (log‘𝐴)))))
69		logcl 24868	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
70	43, 69	sylan 572	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
71	66, 70	mulcld 10466	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐵 · (log‘𝐴)) ∈ ℂ)
72		absef 15416	. . . . 5 ⊢ ((𝐵 · (log‘𝐴)) ∈ ℂ → (abs‘(exp‘(𝐵 · (log‘𝐴)))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
73	71, 72	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(exp‘(𝐵 · (log‘𝐴)))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
74	66	recld 14420	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘𝐵) ∈ ℝ)
75	70	recld 14420	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) ∈ ℝ)
76	74, 75	remulcld 10476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℝ)
77	76	recnd 10474	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℂ)
78	66	imcld 14421	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘𝐵) ∈ ℝ)
79	70	imcld 14421	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℝ)
80	79	renegcld 10874	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -(ℑ‘(log‘𝐴)) ∈ ℝ)
81	78, 80	remulcld 10476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℝ)
82	81	recnd 10474	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℂ)
83		efadd 15313	. . . . . 6 ⊢ ((((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))))))
84	77, 82, 83	syl2anc 576	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘(((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) = ((exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
85	78, 79	remulcld 10476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℝ)
86	85	recnd 10474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℂ)
87	77, 86	negsubd 10810	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + -((ℑ‘𝐵) · (ℑ‘(log‘𝐴)))) = (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) − ((ℑ‘𝐵) · (ℑ‘(log‘𝐴)))))
88	78	recnd 10474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘𝐵) ∈ ℂ)
89	79	recnd 10474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℂ)
90	88, 89	mulneg2d 10901	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) = -((ℑ‘𝐵) · (ℑ‘(log‘𝐴))))
91	90	oveq2d 6998	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + -((ℑ‘𝐵) · (ℑ‘(log‘𝐴)))))
92	66, 70	remuld 14444	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘(𝐵 · (log‘𝐴))) = (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) − ((ℑ‘𝐵) · (ℑ‘(log‘𝐴)))))
93	87, 91, 92	3eqtr4d 2826	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = (ℜ‘(𝐵 · (log‘𝐴))))
94	93	fveq2d 6508	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘(((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
95		relog 24896	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
96	43, 95	sylan 572	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
97	96	oveq2d 6998	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) = ((ℜ‘𝐵) · (log‘(abs‘𝐴))))
98	97	fveq2d 6508	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
99	44	recnd 10474	. . . . . . . . 9 ⊢ (𝜑 → (abs‘𝐴) ∈ ℂ)
100	99	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℂ)
101	43	abs00ad 14517	. . . . . . . . . 10 ⊢ (𝜑 → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
102	101	necon3bid 3013	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
103	102	biimpar 470	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≠ 0)
104	74	recnd 10474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘𝐵) ∈ ℂ)
105	100, 103, 104	cxpefd 25011	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
106	98, 105	eqtr4d 2819	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)))
107	106	oveq1d 6997	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) = (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
108	84, 94, 107	3eqtr3d 2824	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘(𝐵 · (log‘𝐴)))) = (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
109	68, 73, 108	3eqtrd 2820	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) = (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
110	64	abscld 14663	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ)
111	64	absge0d 14671	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘𝐴))
112	110, 111, 74	recxpcld 25022	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
113	81	reefcld 15307	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ)
114	112, 113	remulcld 10476	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
115	49	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
116	115, 113	remulcld 10476	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
117	51, 53, 54	sylancl 578	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐵) · π) ∈ ℝ)
118	117	reefcld 15307	. . . . . 6 ⊢ (𝜑 → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
119	118	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
120	115, 119	remulcld 10476	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) ∈ ℝ)
121	81	rpefcld 15324	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ⁺)
122	121	rpge0d 12258	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
123	16	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝑀 ∈ ℝ)
124		abscxpbnd.3	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (ℜ‘𝐵))
125	124	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (ℜ‘𝐵))
126	46	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≤ 𝑀)
127	110, 111, 123, 74, 125, 126	cxple2ad 25024	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
128	112, 115, 113, 122, 127	lemul1ad 11386	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
129	57	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
130	88	abscld 14663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘𝐵)) ∈ ℝ)
131	80	recnd 10474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -(ℑ‘(log‘𝐴)) ∈ ℂ)
132	131	abscld 14663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) ∈ ℝ)
133	130, 132	remulcld 10476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
134	117	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · π) ∈ ℝ)
135	81	leabsd 14641	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
136	88, 131	absmuld 14681	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
137	135, 136	breqtrd 4960	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
138	66	abscld 14663	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐵) ∈ ℝ)
139	138, 132	remulcld 10476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
140	131	absge0d 14671	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘-(ℑ‘(log‘𝐴))))
141		absimle 14536	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℂ → (abs‘(ℑ‘𝐵)) ≤ (abs‘𝐵))
142	66, 141	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘𝐵)) ≤ (abs‘𝐵))
143	130, 138, 132, 140, 142	lemul1ad 11386	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))))
144	53	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → π ∈ ℝ)
145	66	absge0d 14671	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘𝐵))
146	89	absnegd 14676	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) = (abs‘(ℑ‘(log‘𝐴))))
147		logimcl 24869	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
148	43, 147	sylan 572	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
149	148	simpld 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -π < (ℑ‘(log‘𝐴)))
150	53	renegcli 10754	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ
151		ltle 10535	. . . . . . . . . . . . 13 ⊢ ((-π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
152	150, 79, 151	sylancr 579	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
153	149, 152	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -π ≤ (ℑ‘(log‘𝐴)))
154	148	simprd 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ≤ π)
155		absle 14542	. . . . . . . . . . . 12 ⊢ (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘(ℑ‘(log‘𝐴))) ≤ π ↔ (-π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π)))
156	79, 53, 155	sylancl 578	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘(log‘𝐴))) ≤ π ↔ (-π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π)))
157	153, 154, 156	mpbir2and 701	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘(log‘𝐴))) ≤ π)
158	146, 157	eqbrtrd 4956	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) ≤ π)
159	132, 144, 138, 145, 158	lemul2ad 11387	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
160	133, 139, 134, 143, 159	letrd 10603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
161	81, 133, 134, 137, 160	letrd 10603	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘𝐵) · π))
162		efle 15337	. . . . . . 7 ⊢ ((((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))))
163	81, 134, 162	syl2anc 576	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘𝐵) · π) ↔ (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ≤ (exp‘((abs‘𝐵) · π))))
164	161, 163	mpbid 224	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ≤ (exp‘((abs‘𝐵) · π)))
165	113, 119, 115, 129, 164	lemul2ad 11387	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
166	114, 116, 120, 128, 165	letrd 10603	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
167	109, 166	eqbrtrd 4956	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
168	63, 167	pm2.61dane 3057	1 ⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ≠ wne 2969 class class class wbr 4934 ‘cfv 6193 (class class class)co 6982 ℂcc 10339 ℝcr 10340 0cc0 10341 1c1 10342 + caddc 10344 · cmul 10346 < clt 10480 ≤ cle 10481 − cmin 10676 -cneg 10677 ℜcre 14323 ℑcim 14324 abscabs 14460 expce 15281 πcpi 15286 logclog 24854 ↑_𝑐ccxp 24855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-inf2 8904 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 ax-pre-sup 10419 ax-addf 10420 ax-mulf 10421

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-int 4755 df-iun 4799 df-iin 4800 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-se 5371 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-isom 6202 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-of 7233 df-om 7403 df-1st 7507 df-2nd 7508 df-supp 7640 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-1o 7911 df-2o 7912 df-oadd 7915 df-er 8095 df-map 8214 df-pm 8215 df-ixp 8266 df-en 8313 df-dom 8314 df-sdom 8315 df-fin 8316 df-fsupp 8635 df-fi 8676 df-sup 8707 df-inf 8708 df-oi 8775 df-card 9168 df-cda 9394 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-div 11105 df-nn 11446 df-2 11509 df-3 11510 df-4 11511 df-5 11512 df-6 11513 df-7 11514 df-8 11515 df-9 11516 df-n0 11714 df-z 11800 df-dec 11918 df-uz 12065 df-q 12169 df-rp 12211 df-xneg 12330 df-xadd 12331 df-xmul 12332 df-ioo 12564 df-ioc 12565 df-ico 12566 df-icc 12567 df-fz 12715 df-fzo 12856 df-fl 12983 df-mod 13059 df-seq 13191 df-exp 13251 df-fac 13455 df-bc 13484 df-hash 13512 df-shft 14293 df-cj 14325 df-re 14326 df-im 14327 df-sqrt 14461 df-abs 14462 df-limsup 14695 df-clim 14712 df-rlim 14713 df-sum 14910 df-ef 15287 df-sin 15289 df-cos 15290 df-pi 15292 df-struct 16347 df-ndx 16348 df-slot 16349 df-base 16351 df-sets 16352 df-ress 16353 df-plusg 16440 df-mulr 16441 df-starv 16442 df-sca 16443 df-vsca 16444 df-ip 16445 df-tset 16446 df-ple 16447 df-ds 16449 df-unif 16450 df-hom 16451 df-cco 16452 df-rest 16558 df-topn 16559 df-0g 16577 df-gsum 16578 df-topgen 16579 df-pt 16580 df-prds 16583 df-xrs 16637 df-qtop 16642 df-imas 16643 df-xps 16645 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-mulg 18024 df-cntz 18230 df-cmn 18680 df-psmet 20254 df-xmet 20255 df-met 20256 df-bl 20257 df-mopn 20258 df-fbas 20259 df-fg 20260 df-cnfld 20263 df-top 21221 df-topon 21238 df-topsp 21260 df-bases 21273 df-cld 21346 df-ntr 21347 df-cls 21348 df-nei 21425 df-lp 21463 df-perf 21464 df-cn 21554 df-cnp 21555 df-haus 21642 df-tx 21889 df-hmeo 22082 df-fil 22173 df-fm 22265 df-flim 22266 df-flf 22267 df-xms 22648 df-ms 22649 df-tms 22650 df-cncf 23204 df-limc 24182 df-dv 24183 df-log 24856 df-cxp 24857

This theorem is referenced by: o1cxp 25269

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