dvrelogpow2b - Mathbox for metakunt

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Theorem dvrelogpow2b 43384

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 7467	. 2 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑𝑁))))
4		reelprrecn 11331	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
5	4	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
6		cnelprrecn 11332	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
7	6	a1i 11	. . . 4 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
8		elioore 13546	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
9	8	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
10	9	recnd 11376	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ)
11		dvrelogpow2b.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
12	11	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
13		dvrelogpow2b.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
14	13	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
15		dvrelogpow2b.3	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝐴)
16	15	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 𝐴)
17		dvrelogpow2b.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
18	17	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝐵)
19		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
20	12, 14, 16, 18, 19	0nonelalab 43383	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ 𝑥)
21	20	necomd 3014	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
22	10, 21	logcld 26955	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℂ)
23		2cnd 12461	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
24		0ne2 12592	. . . . . . . . 9 ⊢ 0 ≠ 2
25	24	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ 2)
26	25	necomd 3014	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
27	23, 26	logcld 26955	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
28		0red 11350	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
29		1lt2 12555	. . . . . . . . . 10 ⊢ 1 < 2
30	29	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2)
31		2rp 13165	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
32		loggt0b 27017	. . . . . . . . . 10 ⊢ (2 ∈ ℝ⁺ → (0 < (log‘2) ↔ 1 < 2))
33	31, 32	ax-mp 5	. . . . . . . . 9 ⊢ (0 < (log‘2) ↔ 1 < 2)
34	30, 33	sylibr 237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (log‘2))
35	28, 34	ltned 11485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ (log‘2))
36	35	necomd 3014	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ≠ 0)
37	22, 27, 36	divcld 12133	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥) / (log‘2)) ∈ ℂ)
38		1red 11348	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
39	38, 30	ltned 11485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2)
40	39	necomd 3014	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1)
41	26, 40	nelprd 4619	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 2 ∈ {0, 1})
42	23, 41	eldifd 4053	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℂ ∖ {0, 1}))
43		necom 3012	. . . . . . . . . . . 12 ⊢ (0 ≠ 𝑥 ↔ 𝑥 ≠ 0)
44	43	imbi2i 339	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ 𝑥) ↔ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0))
45	20, 44	mpbi 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
46	45	neneqd 2964	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 = 0)
47		velsn 4600	. . . . . . . . 9 ⊢ (𝑥 ∈ {0} ↔ 𝑥 = 0)
48	46, 47	sylnibr 332	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 ∈ {0})
49	10, 48	eldifd 4053	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (ℂ ∖ {0}))
50		logbval 27151	. . . . . . 7 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
51	42, 49, 50	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
52	51	eleq1d 2849	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥) ∈ ℂ ↔ ((log‘𝑥) / (log‘2)) ∈ ℂ))
53	37, 52	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) ∈ ℂ)
54	31	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ⁺)
55	54	relogcld 27008	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℝ)
56	9, 55	remulcld 11378	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈ ℝ)
57	54	rpne0d 13209	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
58	23, 57	logcld 26955	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
59	10, 58, 21, 36	mulne0d 12008	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ≠ 0)
60	38, 56, 59	redivcld 12185	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / (𝑥 · (log‘2))) ∈ ℝ)
61		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
62		dvrelogpow2b.8	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
63	62	nnnn0d 12707	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
64	63	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈ ℕ₀)
65	61, 64	expcld 14329	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑𝑁) ∈ ℂ)
66	62	nncnd 12391	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
67	66	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈ ℂ)
68		nnm1nn0 12687	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
69	62, 68	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
70	69	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 − 1) ∈ ℕ₀)
71	61, 70	expcld 14329	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑(𝑁 − 1)) ∈ ℂ)
72	67, 71	mulcld 11368	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 · (𝑦↑(𝑁 − 1))) ∈ ℂ)
73	11	rexrd 11398	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
74	13	rexrd 11398	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
75		0red 11350	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
76	75, 11, 15	ltled 11497	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝐴)
77		eqid 2763	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))
78		eqid 2763	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))
79	73, 74, 76, 17, 77, 78	dvrelog2b 43382	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))))
80		dvexp 26328	. . . . 5 ⊢ (𝑁 ∈ ℕ → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑𝑁))) = (𝑦 ∈ ℂ ↦ (𝑁 · (𝑦↑(𝑁 − 1)))))
81	62, 80	syl 18	. . . 4 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑𝑁))) = (𝑦 ∈ ℂ ↦ (𝑁 · (𝑦↑(𝑁 − 1)))))
82		oveq1 7458	. . . 4 ⊢ (𝑦 = (2 log_b 𝑥) → (𝑦↑𝑁) = ((2 log_b 𝑥)↑𝑁))
83		oveq1 7458	. . . . 5 ⊢ (𝑦 = (2 log_b 𝑥) → (𝑦↑(𝑁 − 1)) = ((2 log_b 𝑥)↑(𝑁 − 1)))
84	83	oveq2d 7467	. . . 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 26347	. . 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 7466	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 / ((log‘2)↑𝑁)) · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)))
91	66	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑁 ∈ ℂ)
92	63	nn0zd 12758	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℤ)
93	92	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑁 ∈ ℤ)
94	27, 36, 93	expclzd 14334	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) ∈ ℂ)
95	69	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 − 1) ∈ ℕ₀)
96	22, 95	expcld 14329	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
97	27, 36, 93	expne0d 14335	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) ≠ 0)
98	91, 94, 96, 10, 97, 21	divmuldivd 12174	. . . . . . . . . . . . 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 11369	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (𝑥 · ((log‘2)↑𝑁)))
100		1cnd 11341	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 1 ∈ ℂ)
101	100, 66	pncan3d 11711	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1 + (𝑁 − 1)) = 𝑁)
102	101	eqcomd 2769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑁 = (1 + (𝑁 − 1)))
103	102	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑁 = (1 + (𝑁 − 1)))
104	103	oveq2d 7467	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = ((log‘2)↑(1 + (𝑁 − 1))))
105		1nn0 12662	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℕ₀
106	105	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℕ₀)
107	27, 95, 106	expaddd 14331	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(1 + (𝑁 − 1))) = (((log‘2)↑1) · ((log‘2)↑(𝑁 − 1))))
108	104, 107	eqtrd 2798	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = (((log‘2)↑1) · ((log‘2)↑(𝑁 − 1))))
109	27	exp1d 14324	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑1) = (log‘2))
110	109	oveq1d 7466	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑1) · ((log‘2)↑(𝑁 − 1))) = ((log‘2) · ((log‘2)↑(𝑁 − 1))))
111	108, 110	eqtrd 2798	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = ((log‘2) · ((log‘2)↑(𝑁 − 1))))
112	111	oveq2d 7467	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · ((log‘2)↑𝑁)) = (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))))
113	99, 112	eqtrd 2798	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))))
114	27, 95	expcld 14329	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(𝑁 − 1)) ∈ ℂ)
115	10, 27, 114	mulassd 11371	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))) = (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))))
116	115	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))) = ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))))
117	113, 116	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))))
118	10, 27	mulcld 11368	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈ ℂ)
119	118, 114	mulcomd 11369	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))) = (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2))))
120	117, 119	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2))))
121	120	oveq2d 7467	. . . . . . . . . . . . 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 2798	. . . . . . . . . . . 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 2798	. . . . . . . . . . 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 11368	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((log‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
125		1zzd 12767	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℤ)
126	93, 125	zsubcld 12848	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 − 1) ∈ ℤ)
127	27, 36, 126	expne0d 14335	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(𝑁 − 1)) ≠ 0)
128	124, 114, 118, 127, 59	divdiv1d 12164	. . . . . . . . . . . 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 2769	. . . . . . . . . . 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 2798	. . . . . . . . . 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 12168	. . . . . . . . . . 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 7466	. . . . . . . . . 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 2798	. . . . . . . . 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 14343	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥) / (log‘2))↑(𝑁 − 1)) = (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1))))
135	134	eqcomd 2769	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1))) = (((log‘𝑥) / (log‘2))↑(𝑁 − 1)))
136	135	oveq2d 7467	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1)))) = (𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))))
137	136	oveq1d 7466	. . . . . . . . 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 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))) / (𝑥 · (log‘2))))
139	51	oveq1d 7466	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑(𝑁 − 1)) = (((log‘𝑥) / (log‘2))↑(𝑁 − 1)))
140	139	oveq2d 7467	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) = (𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))))
141	140	oveq1d 7466	. . . . . . . . 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 2769	. . . . . . . 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 2798	. . . . . . 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 14329	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑(𝑁 − 1)) ∈ ℂ)
145	91, 144	mulcld 11368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) ∈ ℂ)
146	145, 118, 59	divrecd 12136	. . . . . . 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 2798	. . . . . 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 5204	. . . . 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 2798	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 · (log‘2))))))
150	149	eqcomd 2769	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 · (log‘2))))) = 𝐺)
151	85, 150	eqtrd 2798	. 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑𝑁))) = 𝐺)
152	3, 151	eqtrd 2798	1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∖ cdif 4048 {csn 4588 {cpr 4590 class class class wbr 5108 ↦ cmpt 5192 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 + caddc 11242 · cmul 11244 < clt 11382 ≤ cle 11383 − cmin 11580 / cdiv 12013 ℕcn 12375 2c2 12437 ℕ₀cn0 12646 ℤcz 12733 ℝ⁺crp 13160 (,)cioo 13516 ↑cexp 14244 D cdv 26238 logclog 26939 log_b clogb 27149

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-shft 15263 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-limsup 15682 df-clim 15699 df-rlim 15700 df-sum 15898 df-ef 16277 df-sin 16279 df-cos 16280 df-pi 16282 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-cmp 23760 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-limc 26241 df-dv 26242 df-log 26941 df-logb 27150

This theorem is used by: aks4d1p1p6 43389 aks4d1p1p5 43391

W3C validator