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Theorem abscxpbnd 25342

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 11257	. . . . 5 ⊢ 1 ≤ 1
2	1	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → 1 ≤ 1)
3		oveq12 7144	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴↑_𝑐𝐵) = (0↑_𝑐0))
4	3	adantll 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐴↑_𝑐𝐵) = (0↑_𝑐0))
5		0cn 10622	. . . . . . . 8 ⊢ 0 ∈ ℂ
6		cxp0 25261	. . . . . . . 8 ⊢ (0 ∈ ℂ → (0↑_𝑐0) = 1)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (0↑_𝑐0) = 1
8	4, 7	eqtrdi 2849	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐴↑_𝑐𝐵) = 1)
9	8	fveq2d 6649	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘1))
10		abs1 14649	. . . . 5 ⊢ (abs‘1) = 1
11	9, 10	eqtrdi 2849	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘(𝐴↑_𝑐𝐵)) = 1)
12		fveq2 6645	. . . . . . . . 9 ⊢ (𝐵 = 0 → (ℜ‘𝐵) = (ℜ‘0))
13		re0 14503	. . . . . . . . 9 ⊢ (ℜ‘0) = 0
14	12, 13	eqtrdi 2849	. . . . . . . 8 ⊢ (𝐵 = 0 → (ℜ‘𝐵) = 0)
15	14	oveq2d 7151	. . . . . . 7 ⊢ (𝐵 = 0 → (𝑀↑_𝑐(ℜ‘𝐵)) = (𝑀↑_𝑐0))
16		abscxpbnd.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
17	16	recnd 10658	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
18	17	cxp0d 25296	. . . . . . . 8 ⊢ (𝜑 → (𝑀↑_𝑐0) = 1)
19	18	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 0) → (𝑀↑_𝑐0) = 1)
20	15, 19	sylan9eqr 2855	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝑀↑_𝑐(ℜ‘𝐵)) = 1)
21		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → 𝐵 = 0)
22	21	abs00bd 14643	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘𝐵) = 0)
23	22	oveq1d 7150	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((abs‘𝐵) · π) = (0 · π))
24		picn 25052	. . . . . . . . . 10 ⊢ π ∈ ℂ
25	24	mul02i 10818	. . . . . . . . 9 ⊢ (0 · π) = 0
26	23, 25	eqtrdi 2849	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((abs‘𝐵) · π) = 0)
27	26	fveq2d 6649	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (exp‘((abs‘𝐵) · π)) = (exp‘0))
28		ef0 15436	. . . . . . 7 ⊢ (exp‘0) = 1
29	27, 28	eqtrdi 2849	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (exp‘((abs‘𝐵) · π)) = 1)
30	20, 29	oveq12d 7153	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) = (1 · 1))
31		1t1e1 11787	. . . . 5 ⊢ (1 · 1) = 1
32	30, 31	eqtrdi 2849	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) = 1)
33	2, 11, 32	3brtr4d 5062	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
34		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 𝐴 = 0)
35	34	oveq1d 7150	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑_𝑐𝐵) = (0↑_𝑐𝐵))
36		abscxpbnd.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
37	36	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
38		0cxp 25257	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (0↑_𝑐𝐵) = 0)
39	37, 38	sylan 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (0↑_𝑐𝐵) = 0)
40	35, 39	eqtrd 2833	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑_𝑐𝐵) = 0)
41	40	abs00bd 14643	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) = 0)
42		0red 10633	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
43		abscxpbnd.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
44	43	abscld 14788	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
45	43	absge0d 14796	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘𝐴))
46		abscxpbnd.5	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐴) ≤ 𝑀)
47	42, 44, 16, 45, 46	letrd 10786	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑀)
48	36	recld 14545	. . . . . . 7 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℝ)
49	16, 47, 48	recxpcld 25314	. . . . . 6 ⊢ (𝜑 → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
50	49	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
51	36	abscld 14788	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐵) ∈ ℝ)
52	51	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘𝐵) ∈ ℝ)
53		pire 25051	. . . . . . 7 ⊢ π ∈ ℝ
54		remulcl 10611	. . . . . . 7 ⊢ (((abs‘𝐵) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘𝐵) · π) ∈ ℝ)
55	52, 53, 54	sylancl 589	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → ((abs‘𝐵) · π) ∈ ℝ)
56	55	reefcld 15433	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
57	16, 47, 48	cxpge0d 25315	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
58	57	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
59	55	rpefcld 15450	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ⁺)
60	59	rpge0d 12423	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 0 ≤ (exp‘((abs‘𝐵) · π)))
61	50, 56, 58, 60	mulge0d 11206	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 0 ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
62	41, 61	eqbrtrd 5052	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
63	33, 62	pm2.61dane 3074	. 2 ⊢ ((𝜑 ∧ 𝐴 = 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
64	43	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
65		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
66	36	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐵 ∈ ℂ)
67	64, 65, 66	cxpefd 25303	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
68	67	fveq2d 6649	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘(exp‘(𝐵 · (log‘𝐴)))))
69		logcl 25160	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
70	43, 69	sylan 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
71	66, 70	mulcld 10650	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐵 · (log‘𝐴)) ∈ ℂ)
72		absef 15542	. . . . 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 14545	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘𝐵) ∈ ℝ)
75	70	recld 14545	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) ∈ ℝ)
76	74, 75	remulcld 10660	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℝ)
77	76	recnd 10658	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℂ)
78	66	imcld 14546	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘𝐵) ∈ ℝ)
79	70	imcld 14546	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℝ)
80	79	renegcld 11056	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -(ℑ‘(log‘𝐴)) ∈ ℝ)
81	78, 80	remulcld 10660	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℝ)
82	81	recnd 10658	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℂ)
83		efadd 15439	. . . . . 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 587	. . . . 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 10660	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℝ)
86	85	recnd 10658	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℂ)
87	77, 86	negsubd 10992	. . . . . . 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 10658	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘𝐵) ∈ ℂ)
89	79	recnd 10658	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℂ)
90	88, 89	mulneg2d 11083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) = -((ℑ‘𝐵) · (ℑ‘(log‘𝐴))))
91	90	oveq2d 7151	. . . . . . 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 14569	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘(𝐵 · (log‘𝐴))) = (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) − ((ℑ‘𝐵) · (ℑ‘(log‘𝐴)))))
93	87, 91, 92	3eqtr4d 2843	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = (ℜ‘(𝐵 · (log‘𝐴))))
94	93	fveq2d 6649	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘(((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
95		relog 25188	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
96	43, 95	sylan 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
97	96	oveq2d 7151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) = ((ℜ‘𝐵) · (log‘(abs‘𝐴))))
98	97	fveq2d 6649	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
99	44	recnd 10658	. . . . . . . . 9 ⊢ (𝜑 → (abs‘𝐴) ∈ ℂ)
100	99	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℂ)
101	43	abs00ad 14642	. . . . . . . . . 10 ⊢ (𝜑 → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
102	101	necon3bid 3031	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
103	102	biimpar 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≠ 0)
104	74	recnd 10658	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘𝐵) ∈ ℂ)
105	100, 103, 104	cxpefd 25303	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
106	98, 105	eqtr4d 2836	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)))
107	106	oveq1d 7150	. . . . 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 2841	. . . 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 2837	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) = (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
110	64	abscld 14788	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ)
111	64	absge0d 14796	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘𝐴))
112	110, 111, 74	recxpcld 25314	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
113	81	reefcld 15433	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ)
114	112, 113	remulcld 10660	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
115	49	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
116	115, 113	remulcld 10660	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
117	51, 53, 54	sylancl 589	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐵) · π) ∈ ℝ)
118	117	reefcld 15433	. . . . . 6 ⊢ (𝜑 → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
119	118	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
120	115, 119	remulcld 10660	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) ∈ ℝ)
121	81	rpefcld 15450	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ⁺)
122	121	rpge0d 12423	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
123	16	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝑀 ∈ ℝ)
124		abscxpbnd.3	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (ℜ‘𝐵))
125	124	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (ℜ‘𝐵))
126	46	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≤ 𝑀)
127	110, 111, 123, 74, 125, 126	cxple2ad 25316	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
128	112, 115, 113, 122, 127	lemul1ad 11568	. . . 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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
130	88	abscld 14788	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘𝐵)) ∈ ℝ)
131	80	recnd 10658	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -(ℑ‘(log‘𝐴)) ∈ ℂ)
132	131	abscld 14788	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) ∈ ℝ)
133	130, 132	remulcld 10660	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
134	117	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · π) ∈ ℝ)
135	81	leabsd 14766	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
136	88, 131	absmuld 14806	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
137	135, 136	breqtrd 5056	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
138	66	abscld 14788	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐵) ∈ ℝ)
139	138, 132	remulcld 10660	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
140	131	absge0d 14796	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘-(ℑ‘(log‘𝐴))))
141		absimle 14661	. . . . . . . . . 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 11568	. . . . . . . 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 14796	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘𝐵))
146	89	absnegd 14801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) = (abs‘(ℑ‘(log‘𝐴))))
147		logimcl 25161	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
148	43, 147	sylan 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
149	148	simpld 498	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -π < (ℑ‘(log‘𝐴)))
150	53	renegcli 10936	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ
151		ltle 10718	. . . . . . . . . . . . 13 ⊢ ((-π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
152	150, 79, 151	sylancr 590	. . . . . . . . . . . 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 499	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ≤ π)
155		absle 14667	. . . . . . . . . . . 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 589	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘(log‘𝐴))) ≤ π ↔ (-π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π)))
157	153, 154, 156	mpbir2and 712	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘(log‘𝐴))) ≤ π)
158	146, 157	eqbrtrd 5052	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) ≤ π)
159	132, 144, 138, 145, 158	lemul2ad 11569	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
160	133, 139, 134, 143, 159	letrd 10786	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
161	81, 133, 134, 137, 160	letrd 10786	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘𝐵) · π))
162		efle 15463	. . . . . . 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 587	. . . . . 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 235	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ≤ (exp‘((abs‘𝐵) · π)))
165	113, 119, 115, 129, 164	lemul2ad 11569	. . . 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 10786	. . 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 5052	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
168	63, 167	pm2.61dane 3074	1 ⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 − cmin 10859 -cneg 10860 ℜcre 14448 ℑcim 14449 abscabs 14585 expce 15407 πcpi 15412 logclog 25146 ↑_𝑐ccxp 25147

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-log 25148 df-cxp 25149

This theorem is referenced by: o1cxp 25560

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