dvrelogpow2b - Mathbox for metakunt

	Mathbox for metakunt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dvrelogpow2b		Structured version Visualization version GIF version

Theorem dvrelogpow2b 42025

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 7464	. 2 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑𝑁))))
4		reelprrecn 11276	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
5	4	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
6		cnelprrecn 11277	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
7	6	a1i 11	. . . 4 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
8		elioore 13437	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
10	9	recnd 11318	. . . . . . 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 42024	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ 𝑥)
21	20	necomd 3002	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
22	10, 21	logcld 26630	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℂ)
23		2cnd 12371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
24		0ne2 12500	. . . . . . . . 9 ⊢ 0 ≠ 2
25	24	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ 2)
26	25	necomd 3002	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
27	23, 26	logcld 26630	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
28		0red 11293	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
29		1lt2 12464	. . . . . . . . . 10 ⊢ 1 < 2
30	29	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2)
31		2rp 13062	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
32		loggt0b 26692	. . . . . . . . . 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 11426	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ (log‘2))
36	35	necomd 3002	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ≠ 0)
37	22, 27, 36	divcld 12070	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥) / (log‘2)) ∈ ℂ)
38		1red 11291	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
39	38, 30	ltned 11426	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2)
40	39	necomd 3002	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1)
41	26, 40	nelprd 4679	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 2 ∈ {0, 1})
42	23, 41	eldifd 3987	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℂ ∖ {0, 1}))
43		necom 3000	. . . . . . . . . . . 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 2951	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 = 0)
47		velsn 4664	. . . . . . . . 9 ⊢ (𝑥 ∈ {0} ↔ 𝑥 = 0)
48	46, 47	sylnibr 329	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 ∈ {0})
49	10, 48	eldifd 3987	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (ℂ ∖ {0}))
50		logbval 26827	. . . . . . 7 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
51	42, 49, 50	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
52	51	eleq1d 2829	. . . . 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 26683	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℝ)
56	9, 55	remulcld 11320	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈ ℝ)
57	54	rpne0d 13104	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
58	23, 57	logcld 26630	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
59	10, 58, 21, 36	mulne0d 11942	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ≠ 0)
60	38, 56, 59	redivcld 12122	. . . 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 12613	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
64	63	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈ ℕ₀)
65	61, 64	expcld 14196	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑𝑁) ∈ ℂ)
66	62	nncnd 12309	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
67	66	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈ ℂ)
68		nnm1nn0 12594	. . . . . . . 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 14196	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑(𝑁 − 1)) ∈ ℂ)
72	67, 71	mulcld 11310	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 · (𝑦↑(𝑁 − 1))) ∈ ℂ)
73	11	rexrd 11340	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
74	13	rexrd 11340	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
75		0red 11293	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
76	75, 11, 15	ltled 11438	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝐴)
77		eqid 2740	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))
78		eqid 2740	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))
79	73, 74, 76, 17, 77, 78	dvrelog2b 42023	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))))
80		dvexp 26011	. . . . 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 7455	. . . 4 ⊢ (𝑦 = (2 log_b 𝑥) → (𝑦↑𝑁) = ((2 log_b 𝑥)↑𝑁))
83		oveq1 7455	. . . . 5 ⊢ (𝑦 = (2 log_b 𝑥) → (𝑦↑(𝑁 − 1)) = ((2 log_b 𝑥)↑(𝑁 − 1)))
84	83	oveq2d 7464	. . . 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 26030	. . 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 7463	. . . . . . . . . . . 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 12665	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℤ)
93	92	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑁 ∈ ℤ)
94	27, 36, 93	expclzd 14201	. . . . . . . . . . . . . 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 14196	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
97	27, 36, 93	expne0d 14202	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) ≠ 0)
98	91, 94, 96, 10, 97, 21	divmuldivd 12111	. . . . . . . . . . . . 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 11311	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (𝑥 · ((log‘2)↑𝑁)))
100		1cnd 11285	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 1 ∈ ℂ)
101	100, 66	pncan3d 11650	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1 + (𝑁 − 1)) = 𝑁)
102	101	eqcomd 2746	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑁 = (1 + (𝑁 − 1)))
103	102	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑁 = (1 + (𝑁 − 1)))
104	103	oveq2d 7464	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = ((log‘2)↑(1 + (𝑁 − 1))))
105		1nn0 12569	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℕ₀
106	105	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℕ₀)
107	27, 95, 106	expaddd 14198	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(1 + (𝑁 − 1))) = (((log‘2)↑1) · ((log‘2)↑(𝑁 − 1))))
108	104, 107	eqtrd 2780	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = (((log‘2)↑1) · ((log‘2)↑(𝑁 − 1))))
109	27	exp1d 14191	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑1) = (log‘2))
110	109	oveq1d 7463	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑1) · ((log‘2)↑(𝑁 − 1))) = ((log‘2) · ((log‘2)↑(𝑁 − 1))))
111	108, 110	eqtrd 2780	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = ((log‘2) · ((log‘2)↑(𝑁 − 1))))
112	111	oveq2d 7464	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · ((log‘2)↑𝑁)) = (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))))
113	99, 112	eqtrd 2780	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))))
114	27, 95	expcld 14196	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(𝑁 − 1)) ∈ ℂ)
115	10, 27, 114	mulassd 11313	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))) = (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))))
116	115	eqcomd 2746	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))) = ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))))
117	113, 116	eqtrd 2780	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))))
118	10, 27	mulcld 11310	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈ ℂ)
119	118, 114	mulcomd 11311	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))) = (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2))))
120	117, 119	eqtrd 2780	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2))))
121	120	oveq2d 7464	. . . . . . . . . . . . 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 2780	. . . . . . . . . . . 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 2780	. . . . . . . . . . 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 11310	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((log‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
125		1zzd 12674	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℤ)
126	93, 125	zsubcld 12752	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 − 1) ∈ ℤ)
127	27, 36, 126	expne0d 14202	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(𝑁 − 1)) ≠ 0)
128	124, 114, 118, 127, 59	divdiv1d 12101	. . . . . . . . . . . 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 2746	. . . . . . . . . . 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 2780	. . . . . . . . . 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 12105	. . . . . . . . . . 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 7463	. . . . . . . . . 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 2780	. . . . . . . . 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 14210	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥) / (log‘2))↑(𝑁 − 1)) = (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1))))
135	134	eqcomd 2746	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1))) = (((log‘𝑥) / (log‘2))↑(𝑁 − 1)))
136	135	oveq2d 7464	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1)))) = (𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))))
137	136	oveq1d 7463	. . . . . . . . 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 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))) / (𝑥 · (log‘2))))
139	51	oveq1d 7463	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑(𝑁 − 1)) = (((log‘𝑥) / (log‘2))↑(𝑁 − 1)))
140	139	oveq2d 7464	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) = (𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))))
141	140	oveq1d 7463	. . . . . . . . 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 2746	. . . . . . . 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 2780	. . . . . . 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 14196	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑(𝑁 − 1)) ∈ ℂ)
145	91, 144	mulcld 11310	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) ∈ ℂ)
146	145, 118, 59	divrecd 12073	. . . . . . 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 2780	. . . . . 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 5266	. . . . 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 2780	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 · (log‘2))))))
150	149	eqcomd 2746	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 · (log‘2))))) = 𝐺)
151	85, 150	eqtrd 2780	. 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑𝑁))) = 𝐺)
152	3, 151	eqtrd 2780	1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∖ cdif 3973 {csn 4648 {cpr 4650 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℝ⁺crp 13057 (,)cioo 13407 ↑cexp 14112 D cdv 25918 logclog 26614 log_b clogb 26825

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616 df-logb 26826

This theorem is referenced by: aks4d1p1p6 42030 aks4d1p1p5 42032

W3C validator