dvrelogpow2b - Mathbox for metakunt

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Theorem dvrelogpow2b 42322

Description: Derivative of the power of the binary logarithm. (Contributed by metakunt, 12-Aug-2024.)

**Hypotheses**
Ref	Expression
dvrelogpow2b.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
dvrelogpow2b.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
dvrelogpow2b.3	⊢ (𝜑 → 0 < 𝐴)
dvrelogpow2b.4	⊢ (𝜑 → 𝐴 ≤ 𝐵)
dvrelogpow2b.5	⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑𝑁))
dvrelogpow2b.6	⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)))
dvrelogpow2b.7	⊢ 𝐶 = (𝑁 / ((log‘2)↑𝑁))
dvrelogpow2b.8	⊢ (𝜑 → 𝑁 ∈ ℕ)

**Assertion**
Ref	Expression
dvrelogpow2b	⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝑁 𝜑,𝑥 Allowed substitution hints: 𝐶(𝑥) 𝐹(𝑥) 𝐺(𝑥)

Proof of Theorem dvrelogpow2b Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvrelogpow2b.5	. . . 4 ⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑𝑁))
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑𝑁)))
3	2	oveq2d 7374	. 2 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑𝑁))))
4		reelprrecn 11118	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
5	4	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
6		cnelprrecn 11119	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
7	6	a1i 11	. . . 4 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
8		elioore 13291	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
10	9	recnd 11160	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ)
11		dvrelogpow2b.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
12	11	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
13		dvrelogpow2b.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
14	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
15		dvrelogpow2b.3	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝐴)
16	15	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 𝐴)
17		dvrelogpow2b.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
18	17	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝐵)
19		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
20	12, 14, 16, 18, 19	0nonelalab 42321	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ 𝑥)
21	20	necomd 2987	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
22	10, 21	logcld 26535	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℂ)
23		2cnd 12223	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
24		0ne2 12347	. . . . . . . . 9 ⊢ 0 ≠ 2
25	24	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ 2)
26	25	necomd 2987	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
27	23, 26	logcld 26535	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
28		0red 11135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
29		1lt2 12311	. . . . . . . . . 10 ⊢ 1 < 2
30	29	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2)
31		2rp 12910	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
32		loggt0b 26597	. . . . . . . . . 10 ⊢ (2 ∈ ℝ⁺ → (0 < (log‘2) ↔ 1 < 2))
33	31, 32	ax-mp 5	. . . . . . . . 9 ⊢ (0 < (log‘2) ↔ 1 < 2)
34	30, 33	sylibr 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (log‘2))
35	28, 34	ltned 11269	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ (log‘2))
36	35	necomd 2987	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ≠ 0)
37	22, 27, 36	divcld 11917	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥) / (log‘2)) ∈ ℂ)
38		1red 11133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
39	38, 30	ltned 11269	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2)
40	39	necomd 2987	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1)
41	26, 40	nelprd 4614	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 2 ∈ {0, 1})
42	23, 41	eldifd 3912	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℂ ∖ {0, 1}))
43		necom 2985	. . . . . . . . . . . 12 ⊢ (0 ≠ 𝑥 ↔ 𝑥 ≠ 0)
44	43	imbi2i 336	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ 𝑥) ↔ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0))
45	20, 44	mpbi 230	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
46	45	neneqd 2937	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 = 0)
47		velsn 4596	. . . . . . . . 9 ⊢ (𝑥 ∈ {0} ↔ 𝑥 = 0)
48	46, 47	sylnibr 329	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 ∈ {0})
49	10, 48	eldifd 3912	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (ℂ ∖ {0}))
50		logbval 26732	. . . . . . 7 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
51	42, 49, 50	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
52	51	eleq1d 2821	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥) ∈ ℂ ↔ ((log‘𝑥) / (log‘2)) ∈ ℂ))
53	37, 52	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) ∈ ℂ)
54	31	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ⁺)
55	54	relogcld 26588	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℝ)
56	9, 55	remulcld 11162	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈ ℝ)
57	54	rpne0d 12954	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
58	23, 57	logcld 26535	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
59	10, 58, 21, 36	mulne0d 11789	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ≠ 0)
60	38, 56, 59	redivcld 11969	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / (𝑥 · (log‘2))) ∈ ℝ)
61		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
62		dvrelogpow2b.8	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
63	62	nnnn0d 12462	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
64	63	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈ ℕ₀)
65	61, 64	expcld 14069	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑𝑁) ∈ ℂ)
66	62	nncnd 12161	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
67	66	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈ ℂ)
68		nnm1nn0 12442	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
69	62, 68	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
70	69	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 − 1) ∈ ℕ₀)
71	61, 70	expcld 14069	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑(𝑁 − 1)) ∈ ℂ)
72	67, 71	mulcld 11152	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 · (𝑦↑(𝑁 − 1))) ∈ ℂ)
73	11	rexrd 11182	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
74	13	rexrd 11182	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
75		0red 11135	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
76	75, 11, 15	ltled 11281	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝐴)
77		eqid 2736	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))
78		eqid 2736	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))
79	73, 74, 76, 17, 77, 78	dvrelog2b 42320	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))))
80		dvexp 25913	. . . . 5 ⊢ (𝑁 ∈ ℕ → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑𝑁))) = (𝑦 ∈ ℂ ↦ (𝑁 · (𝑦↑(𝑁 − 1)))))
81	62, 80	syl 17	. . . 4 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑𝑁))) = (𝑦 ∈ ℂ ↦ (𝑁 · (𝑦↑(𝑁 − 1)))))
82		oveq1 7365	. . . 4 ⊢ (𝑦 = (2 log_b 𝑥) → (𝑦↑𝑁) = ((2 log_b 𝑥)↑𝑁))
83		oveq1 7365	. . . . 5 ⊢ (𝑦 = (2 log_b 𝑥) → (𝑦↑(𝑁 − 1)) = ((2 log_b 𝑥)↑(𝑁 − 1)))
84	83	oveq2d 7374	. . . 4 ⊢ (𝑦 = (2 log_b 𝑥) → (𝑁 · (𝑦↑(𝑁 − 1))) = (𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))))
85	5, 7, 53, 60, 65, 72, 79, 81, 82, 84	dvmptco 25932	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑𝑁))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 · (log‘2))))))
86		dvrelogpow2b.6	. . . . . 6 ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)))
87	86	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥))))
88		dvrelogpow2b.7	. . . . . . . . . . . . . 14 ⊢ 𝐶 = (𝑁 / ((log‘2)↑𝑁))
89	88	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐶 = (𝑁 / ((log‘2)↑𝑁)))
90	89	oveq1d 7373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 / ((log‘2)↑𝑁)) · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)))
91	66	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑁 ∈ ℂ)
92	63	nn0zd 12513	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℤ)
93	92	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑁 ∈ ℤ)
94	27, 36, 93	expclzd 14074	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) ∈ ℂ)
95	69	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 − 1) ∈ ℕ₀)
96	22, 95	expcld 14069	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
97	27, 36, 93	expne0d 14075	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) ≠ 0)
98	91, 94, 96, 10, 97, 21	divmuldivd 11958	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 / ((log‘2)↑𝑁)) · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / (((log‘2)↑𝑁) · 𝑥)))
99	94, 10	mulcomd 11153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (𝑥 · ((log‘2)↑𝑁)))
100		1cnd 11127	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 1 ∈ ℂ)
101	100, 66	pncan3d 11495	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1 + (𝑁 − 1)) = 𝑁)
102	101	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑁 = (1 + (𝑁 − 1)))
103	102	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑁 = (1 + (𝑁 − 1)))
104	103	oveq2d 7374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = ((log‘2)↑(1 + (𝑁 − 1))))
105		1nn0 12417	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℕ₀
106	105	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℕ₀)
107	27, 95, 106	expaddd 14071	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(1 + (𝑁 − 1))) = (((log‘2)↑1) · ((log‘2)↑(𝑁 − 1))))
108	104, 107	eqtrd 2771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = (((log‘2)↑1) · ((log‘2)↑(𝑁 − 1))))
109	27	exp1d 14064	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑1) = (log‘2))
110	109	oveq1d 7373	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑1) · ((log‘2)↑(𝑁 − 1))) = ((log‘2) · ((log‘2)↑(𝑁 − 1))))
111	108, 110	eqtrd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = ((log‘2) · ((log‘2)↑(𝑁 − 1))))
112	111	oveq2d 7374	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · ((log‘2)↑𝑁)) = (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))))
113	99, 112	eqtrd 2771	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))))
114	27, 95	expcld 14069	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(𝑁 − 1)) ∈ ℂ)
115	10, 27, 114	mulassd 11155	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))) = (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))))
116	115	eqcomd 2742	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))) = ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))))
117	113, 116	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))))
118	10, 27	mulcld 11152	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈ ℂ)
119	118, 114	mulcomd 11153	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))) = (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2))))
120	117, 119	eqtrd 2771	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2))))
121	120	oveq2d 7374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / (((log‘2)↑𝑁) · 𝑥)) = ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2)))))
122	98, 121	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 / ((log‘2)↑𝑁)) · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2)))))
123	90, 122	eqtrd 2771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2)))))
124	91, 96	mulcld 11152	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((log‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
125		1zzd 12522	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℤ)
126	93, 125	zsubcld 12601	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 − 1) ∈ ℤ)
127	27, 36, 126	expne0d 14075	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(𝑁 − 1)) ≠ 0)
128	124, 114, 118, 127, 59	divdiv1d 11948	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / ((log‘2)↑(𝑁 − 1))) / (𝑥 · (log‘2))) = ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2)))))
129	128	eqcomd 2742	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2)))) = (((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / ((log‘2)↑(𝑁 − 1))) / (𝑥 · (log‘2))))
130	123, 129	eqtrd 2771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = (((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / ((log‘2)↑(𝑁 − 1))) / (𝑥 · (log‘2))))
131	91, 96, 114, 127	divassd 11952	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / ((log‘2)↑(𝑁 − 1))) = (𝑁 · (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1)))))
132	131	oveq1d 7373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / ((log‘2)↑(𝑁 − 1))) / (𝑥 · (log‘2))) = ((𝑁 · (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1)))) / (𝑥 · (log‘2))))
133	130, 132	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1)))) / (𝑥 · (log‘2))))
134	22, 27, 36, 95	expdivd 14083	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥) / (log‘2))↑(𝑁 − 1)) = (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1))))
135	134	eqcomd 2742	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1))) = (((log‘𝑥) / (log‘2))↑(𝑁 − 1)))
136	135	oveq2d 7374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1)))) = (𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))))
137	136	oveq1d 7373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 · (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1)))) / (𝑥 · (log‘2))) = ((𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))) / (𝑥 · (log‘2))))
138	133, 137	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))) / (𝑥 · (log‘2))))
139	51	oveq1d 7373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑(𝑁 − 1)) = (((log‘𝑥) / (log‘2))↑(𝑁 − 1)))
140	139	oveq2d 7374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) = (𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))))
141	140	oveq1d 7373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) / (𝑥 · (log‘2))) = ((𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))) / (𝑥 · (log‘2))))
142	141	eqcomd 2742	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))) / (𝑥 · (log‘2))) = ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) / (𝑥 · (log‘2))))
143	138, 142	eqtrd 2771	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) / (𝑥 · (log‘2))))
144	53, 95	expcld 14069	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑(𝑁 − 1)) ∈ ℂ)
145	91, 144	mulcld 11152	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) ∈ ℂ)
146	145, 118, 59	divrecd 11920	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) / (𝑥 · (log‘2))) = ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 · (log‘2)))))
147	143, 146	eqtrd 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 · (log‘2)))))
148	147	mpteq2dva 5191	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 · (log‘2))))))
149	87, 148	eqtrd 2771	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 · (log‘2))))))
150	149	eqcomd 2742	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 · (log‘2))))) = 𝐺)
151	85, 150	eqtrd 2771	. 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑𝑁))) = 𝐺)
152	3, 151	eqtrd 2771	1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∖ cdif 3898 {csn 4580 {cpr 4582 class class class wbr 5098 ↦ cmpt 5179 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 · cmul 11031 < clt 11166 ≤ cle 11167 − cmin 11364 / cdiv 11794 ℕcn 12145 2c2 12200 ℕ₀cn0 12401 ℤcz 12488 ℝ⁺crp 12905 (,)cioo 13261 ↑cexp 13984 D cdv 25820 logclog 26519 log_b clogb 26730

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 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 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-ef 15990 df-sin 15992 df-cos 15993 df-pi 15995 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824 df-log 26521 df-logb 26731

This theorem is referenced by: aks4d1p1p6 42327 aks4d1p1p5 42329

W3C validator