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Theorem abscxpbnd 26268

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 11844	. . . . 5 ⊢ 1 ≤ 1
2	1	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → 1 ≤ 1)
3		oveq12 7420	. . . . . . . 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 11208	. . . . . . . 8 ⊢ 0 ∈ ℂ
6		cxp0 26185	. . . . . . . 8 ⊢ (0 ∈ ℂ → (0↑_𝑐0) = 1)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (0↑_𝑐0) = 1
8	4, 7	eqtrdi 2788	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐴↑_𝑐𝐵) = 1)
9	8	fveq2d 6895	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘1))
10		abs1 15246	. . . . 5 ⊢ (abs‘1) = 1
11	9, 10	eqtrdi 2788	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘(𝐴↑_𝑐𝐵)) = 1)
12		fveq2 6891	. . . . . . . . 9 ⊢ (𝐵 = 0 → (ℜ‘𝐵) = (ℜ‘0))
13		re0 15101	. . . . . . . . 9 ⊢ (ℜ‘0) = 0
14	12, 13	eqtrdi 2788	. . . . . . . 8 ⊢ (𝐵 = 0 → (ℜ‘𝐵) = 0)
15	14	oveq2d 7427	. . . . . . 7 ⊢ (𝐵 = 0 → (𝑀↑_𝑐(ℜ‘𝐵)) = (𝑀↑_𝑐0))
16		abscxpbnd.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
17	16	recnd 11244	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
18	17	cxp0d 26220	. . . . . . . 8 ⊢ (𝜑 → (𝑀↑_𝑐0) = 1)
19	18	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 0) → (𝑀↑_𝑐0) = 1)
20	15, 19	sylan9eqr 2794	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝑀↑_𝑐(ℜ‘𝐵)) = 1)
21		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → 𝐵 = 0)
22	21	abs00bd 15240	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘𝐵) = 0)
23	22	oveq1d 7426	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((abs‘𝐵) · π) = (0 · π))
24		picn 25976	. . . . . . . . . 10 ⊢ π ∈ ℂ
25	24	mul02i 11405	. . . . . . . . 9 ⊢ (0 · π) = 0
26	23, 25	eqtrdi 2788	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((abs‘𝐵) · π) = 0)
27	26	fveq2d 6895	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (exp‘((abs‘𝐵) · π)) = (exp‘0))
28		ef0 16036	. . . . . . 7 ⊢ (exp‘0) = 1
29	27, 28	eqtrdi 2788	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (exp‘((abs‘𝐵) · π)) = 1)
30	20, 29	oveq12d 7429	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) = (1 · 1))
31		1t1e1 12376	. . . . 5 ⊢ (1 · 1) = 1
32	30, 31	eqtrdi 2788	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) = 1)
33	2, 11, 32	3brtr4d 5180	. . 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 7426	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑_𝑐𝐵) = (0↑_𝑐𝐵))
36		abscxpbnd.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
37	36	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
38		0cxp 26181	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (0↑_𝑐𝐵) = 0)
39	37, 38	sylan 580	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (0↑_𝑐𝐵) = 0)
40	35, 39	eqtrd 2772	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑_𝑐𝐵) = 0)
41	40	abs00bd 15240	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) = 0)
42		0red 11219	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
43		abscxpbnd.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
44	43	abscld 15385	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
45	43	absge0d 15393	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘𝐴))
46		abscxpbnd.5	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐴) ≤ 𝑀)
47	42, 44, 16, 45, 46	letrd 11373	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑀)
48	36	recld 15143	. . . . . . 7 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℝ)
49	16, 47, 48	recxpcld 26238	. . . . . 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 15385	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐵) ∈ ℝ)
52	51	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘𝐵) ∈ ℝ)
53		pire 25975	. . . . . . 7 ⊢ π ∈ ℝ
54		remulcl 11197	. . . . . . 7 ⊢ (((abs‘𝐵) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘𝐵) · π) ∈ ℝ)
55	52, 53, 54	sylancl 586	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → ((abs‘𝐵) · π) ∈ ℝ)
56	55	reefcld 16033	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
57	16, 47, 48	cxpge0d 26239	. . . . . 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 16050	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ⁺)
60	59	rpge0d 13022	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 0 ≤ (exp‘((abs‘𝐵) · π)))
61	50, 56, 58, 60	mulge0d 11793	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 0 ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
62	41, 61	eqbrtrd 5170	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
63	33, 62	pm2.61dane 3029	. 2 ⊢ ((𝜑 ∧ 𝐴 = 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
64	43	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
65		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
66	36	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐵 ∈ ℂ)
67	64, 65, 66	cxpefd 26227	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
68	67	fveq2d 6895	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘(exp‘(𝐵 · (log‘𝐴)))))
69		logcl 26084	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
70	43, 69	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
71	66, 70	mulcld 11236	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐵 · (log‘𝐴)) ∈ ℂ)
72		absef 16142	. . . . 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 15143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘𝐵) ∈ ℝ)
75	70	recld 15143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) ∈ ℝ)
76	74, 75	remulcld 11246	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℝ)
77	76	recnd 11244	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℂ)
78	66	imcld 15144	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘𝐵) ∈ ℝ)
79	70	imcld 15144	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℝ)
80	79	renegcld 11643	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -(ℑ‘(log‘𝐴)) ∈ ℝ)
81	78, 80	remulcld 11246	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℝ)
82	81	recnd 11244	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℂ)
83		efadd 16039	. . . . . 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 584	. . . . 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 11246	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℝ)
86	85	recnd 11244	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℂ)
87	77, 86	negsubd 11579	. . . . . . 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 11244	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘𝐵) ∈ ℂ)
89	79	recnd 11244	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℂ)
90	88, 89	mulneg2d 11670	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) = -((ℑ‘𝐵) · (ℑ‘(log‘𝐴))))
91	90	oveq2d 7427	. . . . . . 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 15167	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘(𝐵 · (log‘𝐴))) = (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) − ((ℑ‘𝐵) · (ℑ‘(log‘𝐴)))))
93	87, 91, 92	3eqtr4d 2782	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = (ℜ‘(𝐵 · (log‘𝐴))))
94	93	fveq2d 6895	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘(((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
95		relog 26112	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
96	43, 95	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
97	96	oveq2d 7427	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) = ((ℜ‘𝐵) · (log‘(abs‘𝐴))))
98	97	fveq2d 6895	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
99	44	recnd 11244	. . . . . . . . 9 ⊢ (𝜑 → (abs‘𝐴) ∈ ℂ)
100	99	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℂ)
101	43	abs00ad 15239	. . . . . . . . . 10 ⊢ (𝜑 → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
102	101	necon3bid 2985	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
103	102	biimpar 478	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≠ 0)
104	74	recnd 11244	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘𝐵) ∈ ℂ)
105	100, 103, 104	cxpefd 26227	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
106	98, 105	eqtr4d 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)))
107	106	oveq1d 7426	. . . . 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 2780	. . . 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 2776	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) = (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))))
110	64	abscld 15385	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ)
111	64	absge0d 15393	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘𝐴))
112	110, 111, 74	recxpcld 26238	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
113	81	reefcld 16033	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ)
114	112, 113	remulcld 11246	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
115	49	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
116	115, 113	remulcld 11246	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
117	51, 53, 54	sylancl 586	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐵) · π) ∈ ℝ)
118	117	reefcld 16033	. . . . . 6 ⊢ (𝜑 → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
119	118	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
120	115, 119	remulcld 11246	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) ∈ ℝ)
121	81	rpefcld 16050	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ⁺)
122	121	rpge0d 13022	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
123	16	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝑀 ∈ ℝ)
124		abscxpbnd.3	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (ℜ‘𝐵))
125	124	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (ℜ‘𝐵))
126	46	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≤ 𝑀)
127	110, 111, 123, 74, 125, 126	cxple2ad 26240	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
128	112, 115, 113, 122, 127	lemul1ad 12155	. . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
130	88	abscld 15385	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘𝐵)) ∈ ℝ)
131	80	recnd 11244	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -(ℑ‘(log‘𝐴)) ∈ ℂ)
132	131	abscld 15385	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) ∈ ℝ)
133	130, 132	remulcld 11246	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
134	117	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · π) ∈ ℝ)
135	81	leabsd 15363	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
136	88, 131	absmuld 15403	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
137	135, 136	breqtrd 5174	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
138	66	abscld 15385	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐵) ∈ ℝ)
139	138, 132	remulcld 11246	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
140	131	absge0d 15393	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘-(ℑ‘(log‘𝐴))))
141		absimle 15258	. . . . . . . . . 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 12155	. . . . . . . 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 15393	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘𝐵))
146	89	absnegd 15398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) = (abs‘(ℑ‘(log‘𝐴))))
147		logimcl 26085	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
148	43, 147	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
149	148	simpld 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -π < (ℑ‘(log‘𝐴)))
150	53	renegcli 11523	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ
151		ltle 11304	. . . . . . . . . . . . 13 ⊢ ((-π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
152	150, 79, 151	sylancr 587	. . . . . . . . . . . 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 496	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ≤ π)
155		absle 15264	. . . . . . . . . . . 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 586	. . . . . . . . . . 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 5170	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) ≤ π)
159	132, 144, 138, 145, 158	lemul2ad 12156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
160	133, 139, 134, 143, 159	letrd 11373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
161	81, 133, 134, 137, 160	letrd 11373	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘𝐵) · π))
162		efle 16063	. . . . . . 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 584	. . . . . 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 231	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ≤ (exp‘((abs‘𝐵) · π)))
165	113, 119, 115, 129, 164	lemul2ad 12156	. . . 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 11373	. . 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 5170	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
168	63, 167	pm2.61dane 3029	1 ⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 < clt 11250 ≤ cle 11251 − cmin 11446 -cneg 11447 ℜcre 15046 ℑcim 15047 abscabs 15183 expce 16007 πcpi 16012 logclog 26070 ↑_𝑐ccxp 26071

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ioc 13331 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-seq 13969 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15016 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-sum 15635 df-ef 16013 df-sin 16015 df-cos 16016 df-pi 16018 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-pt 17392 df-prds 17395 df-xrs 17450 df-qtop 17455 df-imas 17456 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-mulg 18953 df-cntz 19183 df-cmn 19652 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-fbas 20947 df-fg 20948 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-nei 22609 df-lp 22647 df-perf 22648 df-cn 22738 df-cnp 22739 df-haus 22826 df-tx 23073 df-hmeo 23266 df-fil 23357 df-fm 23449 df-flim 23450 df-flf 23451 df-xms 23833 df-ms 23834 df-tms 23835 df-cncf 24401 df-limc 25390 df-dv 25391 df-log 26072 df-cxp 26073

This theorem is referenced by: o1cxp 26486

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