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Theorem abscxpbnd 25334

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 11268	. . . . 5 ⊢ 1 ≤ 1
2	1	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → 1 ≤ 1)
3		oveq12 7165	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴↑_𝑐𝐵) = (0↑_𝑐0))
4	3	adantll 712	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐴↑_𝑐𝐵) = (0↑_𝑐0))
5		0cn 10633	. . . . . . . 8 ⊢ 0 ∈ ℂ
6		cxp0 25253	. . . . . . . 8 ⊢ (0 ∈ ℂ → (0↑_𝑐0) = 1)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (0↑_𝑐0) = 1
8	4, 7	syl6eq 2872	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐴↑_𝑐𝐵) = 1)
9	8	fveq2d 6674	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘1))
10		abs1 14657	. . . . 5 ⊢ (abs‘1) = 1
11	9, 10	syl6eq 2872	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘(𝐴↑_𝑐𝐵)) = 1)
12		fveq2 6670	. . . . . . . . 9 ⊢ (𝐵 = 0 → (ℜ‘𝐵) = (ℜ‘0))
13		re0 14511	. . . . . . . . 9 ⊢ (ℜ‘0) = 0
14	12, 13	syl6eq 2872	. . . . . . . 8 ⊢ (𝐵 = 0 → (ℜ‘𝐵) = 0)
15	14	oveq2d 7172	. . . . . . 7 ⊢ (𝐵 = 0 → (𝑀↑_𝑐(ℜ‘𝐵)) = (𝑀↑_𝑐0))
16		abscxpbnd.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
17	16	recnd 10669	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
18	17	cxp0d 25288	. . . . . . . 8 ⊢ (𝜑 → (𝑀↑_𝑐0) = 1)
19	18	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 0) → (𝑀↑_𝑐0) = 1)
20	15, 19	sylan9eqr 2878	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝑀↑_𝑐(ℜ‘𝐵)) = 1)
21		simpr 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → 𝐵 = 0)
22	21	abs00bd 14651	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘𝐵) = 0)
23	22	oveq1d 7171	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((abs‘𝐵) · π) = (0 · π))
24		picn 25045	. . . . . . . . . 10 ⊢ π ∈ ℂ
25	24	mul02i 10829	. . . . . . . . 9 ⊢ (0 · π) = 0
26	23, 25	syl6eq 2872	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((abs‘𝐵) · π) = 0)
27	26	fveq2d 6674	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (exp‘((abs‘𝐵) · π)) = (exp‘0))
28		ef0 15444	. . . . . . 7 ⊢ (exp‘0) = 1
29	27, 28	syl6eq 2872	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (exp‘((abs‘𝐵) · π)) = 1)
30	20, 29	oveq12d 7174	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) = (1 · 1))
31		1t1e1 11800	. . . . 5 ⊢ (1 · 1) = 1
32	30, 31	syl6eq 2872	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) = 1)
33	2, 11, 32	3brtr4d 5098	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
34		simplr 767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 𝐴 = 0)
35	34	oveq1d 7171	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑_𝑐𝐵) = (0↑_𝑐𝐵))
36		abscxpbnd.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
37	36	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
38		0cxp 25249	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (0↑_𝑐𝐵) = 0)
39	37, 38	sylan 582	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (0↑_𝑐𝐵) = 0)
40	35, 39	eqtrd 2856	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑_𝑐𝐵) = 0)
41	40	abs00bd 14651	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) = 0)
42		0red 10644	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
43		abscxpbnd.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
44	43	abscld 14796	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
45	43	absge0d 14804	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘𝐴))
46		abscxpbnd.5	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐴) ≤ 𝑀)
47	42, 44, 16, 45, 46	letrd 10797	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑀)
48	36	recld 14553	. . . . . . 7 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℝ)
49	16, 47, 48	recxpcld 25306	. . . . . 6 ⊢ (𝜑 → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
50	49	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
51	36	abscld 14796	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐵) ∈ ℝ)
52	51	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘𝐵) ∈ ℝ)
53		pire 25044	. . . . . . 7 ⊢ π ∈ ℝ
54		remulcl 10622	. . . . . . 7 ⊢ (((abs‘𝐵) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘𝐵) · π) ∈ ℝ)
55	52, 53, 54	sylancl 588	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → ((abs‘𝐵) · π) ∈ ℝ)
56	55	reefcld 15441	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
57	16, 47, 48	cxpge0d 25307	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
58	57	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
59	55	rpefcld 15458	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ⁺)
60	59	rpge0d 12436	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 0 ≤ (exp‘((abs‘𝐵) · π)))
61	50, 56, 58, 60	mulge0d 11217	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 0 ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
62	41, 61	eqbrtrd 5088	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
63	33, 62	pm2.61dane 3104	. 2 ⊢ ((𝜑 ∧ 𝐴 = 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
64	43	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
65		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
66	36	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐵 ∈ ℂ)
67	64, 65, 66	cxpefd 25295	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
68	67	fveq2d 6674	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘(exp‘(𝐵 · (log‘𝐴)))))
69		logcl 25152	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
70	43, 69	sylan 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
71	66, 70	mulcld 10661	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐵 · (log‘𝐴)) ∈ ℂ)
72		absef 15550	. . . . 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 14553	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘𝐵) ∈ ℝ)
75	70	recld 14553	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) ∈ ℝ)
76	74, 75	remulcld 10671	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℝ)
77	76	recnd 10669	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℂ)
78	66	imcld 14554	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘𝐵) ∈ ℝ)
79	70	imcld 14554	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℝ)
80	79	renegcld 11067	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -(ℑ‘(log‘𝐴)) ∈ ℝ)
81	78, 80	remulcld 10671	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℝ)
82	81	recnd 10669	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℂ)
83		efadd 15447	. . . . . 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 586	. . . . 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 10671	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℝ)
86	85	recnd 10669	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℂ)
87	77, 86	negsubd 11003	. . . . . . 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 10669	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘𝐵) ∈ ℂ)
89	79	recnd 10669	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℂ)
90	88, 89	mulneg2d 11094	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) = -((ℑ‘𝐵) · (ℑ‘(log‘𝐴))))
91	90	oveq2d 7172	. . . . . . 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 14577	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘(𝐵 · (log‘𝐴))) = (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) − ((ℑ‘𝐵) · (ℑ‘(log‘𝐴)))))
93	87, 91, 92	3eqtr4d 2866	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = (ℜ‘(𝐵 · (log‘𝐴))))
94	93	fveq2d 6674	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘(((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
95		relog 25180	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
96	43, 95	sylan 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
97	96	oveq2d 7172	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) = ((ℜ‘𝐵) · (log‘(abs‘𝐴))))
98	97	fveq2d 6674	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
99	44	recnd 10669	. . . . . . . . 9 ⊢ (𝜑 → (abs‘𝐴) ∈ ℂ)
100	99	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℂ)
101	43	abs00ad 14650	. . . . . . . . . 10 ⊢ (𝜑 → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
102	101	necon3bid 3060	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
103	102	biimpar 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≠ 0)
104	74	recnd 10669	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘𝐵) ∈ ℂ)
105	100, 103, 104	cxpefd 25295	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
106	98, 105	eqtr4d 2859	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)))
107	106	oveq1d 7171	. . . . 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 2864	. . . 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 2860	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) = (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
110	64	abscld 14796	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ)
111	64	absge0d 14804	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘𝐴))
112	110, 111, 74	recxpcld 25306	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
113	81	reefcld 15441	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ)
114	112, 113	remulcld 10671	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
115	49	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
116	115, 113	remulcld 10671	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
117	51, 53, 54	sylancl 588	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐵) · π) ∈ ℝ)
118	117	reefcld 15441	. . . . . 6 ⊢ (𝜑 → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
119	118	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
120	115, 119	remulcld 10671	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) ∈ ℝ)
121	81	rpefcld 15458	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ⁺)
122	121	rpge0d 12436	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
123	16	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝑀 ∈ ℝ)
124		abscxpbnd.3	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (ℜ‘𝐵))
125	124	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (ℜ‘𝐵))
126	46	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≤ 𝑀)
127	110, 111, 123, 74, 125, 126	cxple2ad 25308	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
128	112, 115, 113, 122, 127	lemul1ad 11579	. . . 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 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
130	88	abscld 14796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘𝐵)) ∈ ℝ)
131	80	recnd 10669	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -(ℑ‘(log‘𝐴)) ∈ ℂ)
132	131	abscld 14796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) ∈ ℝ)
133	130, 132	remulcld 10671	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
134	117	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · π) ∈ ℝ)
135	81	leabsd 14774	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
136	88, 131	absmuld 14814	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
137	135, 136	breqtrd 5092	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
138	66	abscld 14796	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐵) ∈ ℝ)
139	138, 132	remulcld 10671	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
140	131	absge0d 14804	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘-(ℑ‘(log‘𝐴))))
141		absimle 14669	. . . . . . . . . 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 11579	. . . . . . . 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 14804	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘𝐵))
146	89	absnegd 14809	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) = (abs‘(ℑ‘(log‘𝐴))))
147		logimcl 25153	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
148	43, 147	sylan 582	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
149	148	simpld 497	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -π < (ℑ‘(log‘𝐴)))
150	53	renegcli 10947	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ
151		ltle 10729	. . . . . . . . . . . . 13 ⊢ ((-π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
152	150, 79, 151	sylancr 589	. . . . . . . . . . . 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 498	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ≤ π)
155		absle 14675	. . . . . . . . . . . 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 588	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘(log‘𝐴))) ≤ π ↔ (-π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π)))
157	153, 154, 156	mpbir2and 711	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘(log‘𝐴))) ≤ π)
158	146, 157	eqbrtrd 5088	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) ≤ π)
159	132, 144, 138, 145, 158	lemul2ad 11580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
160	133, 139, 134, 143, 159	letrd 10797	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
161	81, 133, 134, 137, 160	letrd 10797	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘𝐵) · π))
162		efle 15471	. . . . . . 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 586	. . . . . 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 234	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ≤ (exp‘((abs‘𝐵) · π)))
165	113, 119, 115, 129, 164	lemul2ad 11580	. . . 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 10797	. . 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 5088	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
168	63, 167	pm2.61dane 3104	1 ⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 < clt 10675 ≤ cle 10676 − cmin 10870 -cneg 10871 ℜcre 14456 ℑcim 14457 abscabs 14593 expce 15415 πcpi 15420 logclog 25138 ↑_𝑐ccxp 25139

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140 df-cxp 25141

This theorem is referenced by: o1cxp 25552

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