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Theorem abscxpbnd 26730

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 11769	. . . . 5 ⊢ 1 ≤ 1
2	1	a1i 11	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → 1 ≤ 1)
3		oveq12 7369	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴↑_𝑐𝐵) = (0↑_𝑐0))
4	3	adantll 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (𝐴↑_𝑐𝐵) = (0↑_𝑐0))
5		0cn 11127	. . . . . . . 8 ⊢ 0 ∈ ℂ
6		cxp0 26647	. . . . . . . 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 6838	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘1))
10		abs1 15250	. . . . 5 ⊢ (abs‘1) = 1
11	9, 10	eqtrdi 2788	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘(𝐴↑_𝑐𝐵)) = 1)
12		fveq2 6834	. . . . . . . . 9 ⊢ (𝐵 = 0 → (ℜ‘𝐵) = (ℜ‘0))
13		re0 15105	. . . . . . . . 9 ⊢ (ℜ‘0) = 0
14	12, 13	eqtrdi 2788	. . . . . . . 8 ⊢ (𝐵 = 0 → (ℜ‘𝐵) = 0)
15	14	oveq2d 7376	. . . . . . 7 ⊢ (𝐵 = 0 → (𝑀↑_𝑐(ℜ‘𝐵)) = (𝑀↑_𝑐0))
16		abscxpbnd.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
17	16	recnd 11164	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
18	17	cxp0d 26682	. . . . . . . 8 ⊢ (𝜑 → (𝑀↑_𝑐0) = 1)
19	18	adantr 480	. . . . . . 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 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → 𝐵 = 0)
22	21	abs00bd 15244	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘𝐵) = 0)
23	22	oveq1d 7375	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((abs‘𝐵) · π) = (0 · π))
24		picn 26435	. . . . . . . . . 10 ⊢ π ∈ ℂ
25	24	mul02i 11326	. . . . . . . . 9 ⊢ (0 · π) = 0
26	23, 25	eqtrdi 2788	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((abs‘𝐵) · π) = 0)
27	26	fveq2d 6838	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (exp‘((abs‘𝐵) · π)) = (exp‘0))
28		ef0 16047	. . . . . . 7 ⊢ (exp‘0) = 1
29	27, 28	eqtrdi 2788	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (exp‘((abs‘𝐵) · π)) = 1)
30	20, 29	oveq12d 7378	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) = (1 · 1))
31		1t1e1 12329	. . . . 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 5118	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 = 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
34		simplr 769	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 𝐴 = 0)
35	34	oveq1d 7375	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝐴↑_𝑐𝐵) = (0↑_𝑐𝐵))
36		abscxpbnd.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
37	36	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 0) → 𝐵 ∈ ℂ)
38		0cxp 26643	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (0↑_𝑐𝐵) = 0)
39	37, 38	sylan 581	. . . . . 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 15244	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) = 0)
42		0red 11138	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
43		abscxpbnd.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ)
44	43	abscld 15392	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
45	43	absge0d 15400	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (abs‘𝐴))
46		abscxpbnd.5	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐴) ≤ 𝑀)
47	42, 44, 16, 45, 46	letrd 11294	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑀)
48	36	recld 15147	. . . . . . 7 ⊢ (𝜑 → (ℜ‘𝐵) ∈ ℝ)
49	16, 47, 48	recxpcld 26700	. . . . . 6 ⊢ (𝜑 → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
50	49	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
51	36	abscld 15392	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐵) ∈ ℝ)
52	51	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘𝐵) ∈ ℝ)
53		pire 26434	. . . . . . 7 ⊢ π ∈ ℝ
54		remulcl 11114	. . . . . . 7 ⊢ (((abs‘𝐵) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘𝐵) · π) ∈ ℝ)
55	52, 53, 54	sylancl 587	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → ((abs‘𝐵) · π) ∈ ℝ)
56	55	reefcld 16044	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
57	16, 47, 48	cxpge0d 26701	. . . . . 6 ⊢ (𝜑 → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
58	57	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
59	55	rpefcld 16063	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ⁺)
60	59	rpge0d 12981	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 0 ≤ (exp‘((abs‘𝐵) · π)))
61	50, 56, 58, 60	mulge0d 11718	. . . 4 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → 0 ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
62	41, 61	eqbrtrd 5108	. . 3 ⊢ (((𝜑 ∧ 𝐴 = 0) ∧ 𝐵 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
63	33, 62	pm2.61dane 3020	. 2 ⊢ ((𝜑 ∧ 𝐴 = 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
64	43	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
65		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
66	36	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐵 ∈ ℂ)
67	64, 65, 66	cxpefd 26689	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
68	67	fveq2d 6838	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) = (abs‘(exp‘(𝐵 · (log‘𝐴)))))
69		logcl 26545	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
70	43, 69	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
71	66, 70	mulcld 11156	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐵 · (log‘𝐴)) ∈ ℂ)
72		absef 16155	. . . . 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 15147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘𝐵) ∈ ℝ)
75	70	recld 15147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) ∈ ℝ)
76	74, 75	remulcld 11166	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℝ)
77	76	recnd 11164	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) ∈ ℂ)
78	66	imcld 15148	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘𝐵) ∈ ℝ)
79	70	imcld 15148	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℝ)
80	79	renegcld 11568	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -(ℑ‘(log‘𝐴)) ∈ ℝ)
81	78, 80	remulcld 11166	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℝ)
82	81	recnd 11164	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ∈ ℂ)
83		efadd 16050	. . . . . 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 585	. . . . 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 11166	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℝ)
86	85	recnd 11164	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · (ℑ‘(log‘𝐴))) ∈ ℂ)
87	77, 86	negsubd 11502	. . . . . . 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 11164	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘𝐵) ∈ ℂ)
89	79	recnd 11164	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℂ)
90	88, 89	mulneg2d 11595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) = -((ℑ‘𝐵) · (ℑ‘(log‘𝐴))))
91	90	oveq2d 7376	. . . . . . 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 15171	. . . . . . 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 6838	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘(((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) + ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) = (exp‘(ℜ‘(𝐵 · (log‘𝐴)))))
95		relog 26574	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
96	43, 95	sylan 581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
97	96	oveq2d 7376	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℜ‘𝐵) · (ℜ‘(log‘𝐴))) = ((ℜ‘𝐵) · (log‘(abs‘𝐴))))
98	97	fveq2d 6838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℜ‘𝐵) · (ℜ‘(log‘𝐴)))) = (exp‘((ℜ‘𝐵) · (log‘(abs‘𝐴)))))
99	44	recnd 11164	. . . . . . . . 9 ⊢ (𝜑 → (abs‘𝐴) ∈ ℂ)
100	99	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℂ)
101	43	abs00ad 15243	. . . . . . . . . 10 ⊢ (𝜑 → ((abs‘𝐴) = 0 ↔ 𝐴 = 0))
102	101	necon3bid 2977	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
103	102	biimpar 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≠ 0)
104	74	recnd 11164	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℜ‘𝐵) ∈ ℂ)
105	100, 103, 104	cxpefd 26689	. . . . . . 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 7375	. . . . 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 15392	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ)
111	64	absge0d 15400	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘𝐴))
112	110, 111, 74	recxpcld 26700	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
113	81	reefcld 16044	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ)
114	112, 113	remulcld 11166	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
115	49	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝑀↑_𝑐(ℜ‘𝐵)) ∈ ℝ)
116	115, 113	remulcld 11166	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))))) ∈ ℝ)
117	51, 53, 54	sylancl 587	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐵) · π) ∈ ℝ)
118	117	reefcld 16044	. . . . . 6 ⊢ (𝜑 → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
119	118	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((abs‘𝐵) · π)) ∈ ℝ)
120	115, 119	remulcld 11166	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))) ∈ ℝ)
121	81	rpefcld 16063	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ∈ ℝ⁺)
122	121	rpge0d 12981	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
123	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝑀 ∈ ℝ)
124		abscxpbnd.3	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (ℜ‘𝐵))
125	124	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (ℜ‘𝐵))
126	46	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≤ 𝑀)
127	110, 111, 123, 74, 125, 126	cxple2ad 26702	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑_𝑐(ℜ‘𝐵)) ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
128	112, 115, 113, 122, 127	lemul1ad 12086	. . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (𝑀↑_𝑐(ℜ‘𝐵)))
130	88	abscld 15392	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘𝐵)) ∈ ℝ)
131	80	recnd 11164	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -(ℑ‘(log‘𝐴)) ∈ ℂ)
132	131	abscld 15392	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) ∈ ℝ)
133	130, 132	remulcld 11166	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
134	117	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · π) ∈ ℝ)
135	81	leabsd 15368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))))
136	88, 131	absmuld 15410	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) = ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
137	135, 136	breqtrd 5112	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))))
138	66	abscld 15392	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘𝐵) ∈ ℝ)
139	138, 132	remulcld 11166	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ∈ ℝ)
140	131	absge0d 15400	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘-(ℑ‘(log‘𝐴))))
141		absimle 15262	. . . . . . . . . 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 12086	. . . . . . . 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 15400	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ (abs‘𝐵))
146	89	absnegd 15405	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) = (abs‘(ℑ‘(log‘𝐴))))
147		logimcl 26546	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
148	43, 147	sylan 581	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
149	148	simpld 494	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → -π < (ℑ‘(log‘𝐴)))
150	53	renegcli 11446	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ
151		ltle 11225	. . . . . . . . . . . . 13 ⊢ ((-π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
152	150, 79, 151	sylancr 588	. . . . . . . . . . . 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 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ≤ π)
155		absle 15269	. . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘(log‘𝐴))) ≤ π ↔ (-π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π)))
157	153, 154, 156	mpbir2and 714	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘(log‘𝐴))) ≤ π)
158	146, 157	eqbrtrd 5108	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘-(ℑ‘(log‘𝐴))) ≤ π)
159	132, 144, 138, 145, 158	lemul2ad 12087	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘𝐵) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
160	133, 139, 134, 143, 159	letrd 11294	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((abs‘(ℑ‘𝐵)) · (abs‘-(ℑ‘(log‘𝐴)))) ≤ ((abs‘𝐵) · π))
161	81, 133, 134, 137, 160	letrd 11294	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((ℑ‘𝐵) · -(ℑ‘(log‘𝐴))) ≤ ((abs‘𝐵) · π))
162		efle 16076	. . . . . . 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 585	. . . . . 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 232	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (exp‘((ℑ‘𝐵) · -(ℑ‘(log‘𝐴)))) ≤ (exp‘((abs‘𝐵) · π)))
165	113, 119, 115, 129, 164	lemul2ad 12087	. . . 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 11294	. . 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 5108	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))
168	63, 167	pm2.61dane 3020	1 ⊢ (𝜑 → (abs‘(𝐴↑_𝑐𝐵)) ≤ ((𝑀↑_𝑐(ℜ‘𝐵)) · (exp‘((abs‘𝐵) · π))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5086 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 < clt 11170 ≤ cle 11171 − cmin 11368 -cneg 11369 ℜcre 15050 ℑcim 15051 abscabs 15187 expce 16017 πcpi 16022 logclog 26531 ↑_𝑐ccxp 26532

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-sin 16025 df-cos 16026 df-pi 16028 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-fbas 21341 df-fg 21342 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cld 22994 df-ntr 22995 df-cls 22996 df-nei 23073 df-lp 23111 df-perf 23112 df-cn 23202 df-cnp 23203 df-haus 23290 df-tx 23537 df-hmeo 23730 df-fil 23821 df-fm 23913 df-flim 23914 df-flf 23915 df-xms 24295 df-ms 24296 df-tms 24297 df-cncf 24855 df-limc 25843 df-dv 25844 df-log 26533 df-cxp 26534

This theorem is referenced by: o1cxp 26952

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