dvrelogpow2b - Mathbox for metakunt

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Theorem dvrelogpow2b 39717

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 7188	. 2 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑𝑁))))
4		reelprrecn 10709	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
5	4	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
6		cnelprrecn 10710	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
7	6	a1i 11	. . . 4 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
8		elioore 12853	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
9	8	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
10	9	recnd 10749	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ)
11		dvrelogpow2b.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
12	11	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
13		dvrelogpow2b.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
14	13	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
15		dvrelogpow2b.3	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝐴)
16	15	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 𝐴)
17		dvrelogpow2b.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
18	17	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝐵)
19		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
20	12, 14, 16, 18, 19	0nonelalab 39716	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ 𝑥)
21	20	necomd 2989	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
22	10, 21	logcld 25316	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℂ)
23		2cnd 11796	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
24		0ne2 11925	. . . . . . . . 9 ⊢ 0 ≠ 2
25	24	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ 2)
26	25	necomd 2989	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
27	23, 26	logcld 25316	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
28		0red 10724	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
29		1lt2 11889	. . . . . . . . . 10 ⊢ 1 < 2
30	29	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2)
31		2rp 12479	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
32		loggt0b 25377	. . . . . . . . . 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 10856	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ (log‘2))
36	35	necomd 2989	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ≠ 0)
37	22, 27, 36	divcld 11496	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥) / (log‘2)) ∈ ℂ)
38		1red 10722	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
39	38, 30	ltned 10856	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2)
40	39	necomd 2989	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1)
41	26, 40	nelprd 4547	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 2 ∈ {0, 1})
42	23, 41	eldifd 3854	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℂ ∖ {0, 1}))
43		necom 2987	. . . . . . . . . . . 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 2939	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 = 0)
47		velsn 4532	. . . . . . . . 9 ⊢ (𝑥 ∈ {0} ↔ 𝑥 = 0)
48	46, 47	sylnibr 332	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 ∈ {0})
49	10, 48	eldifd 3854	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (ℂ ∖ {0}))
50		logbval 25506	. . . . . . 7 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
51	42, 49, 50	syl2anc 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
52	51	eleq1d 2817	. . . . 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 25368	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℝ)
56	9, 55	remulcld 10751	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈ ℝ)
57	54	rpne0d 12521	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
58	23, 57	logcld 25316	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
59	10, 58, 21, 36	mulne0d 11372	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ≠ 0)
60	38, 56, 59	redivcld 11548	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / (𝑥 · (log‘2))) ∈ ℝ)
61		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
62		dvrelogpow2b.8	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
63	62	nnnn0d 12038	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
64	63	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈ ℕ₀)
65	61, 64	expcld 13604	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑𝑁) ∈ ℂ)
66	62	nncnd 11734	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
67	66	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈ ℂ)
68		nnm1nn0 12019	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
69	62, 68	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
70	69	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 − 1) ∈ ℕ₀)
71	61, 70	expcld 13604	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑(𝑁 − 1)) ∈ ℂ)
72	67, 71	mulcld 10741	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 · (𝑦↑(𝑁 − 1))) ∈ ℂ)
73	11	rexrd 10771	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
74	13	rexrd 10771	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
75		0red 10724	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
76	75, 11, 15	ltled 10868	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝐴)
77		eqid 2738	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))
78		eqid 2738	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))
79	73, 74, 76, 17, 77, 78	dvrelog2b 39715	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))))
80		dvexp 24707	. . . . 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 7179	. . . 4 ⊢ (𝑦 = (2 log_b 𝑥) → (𝑦↑𝑁) = ((2 log_b 𝑥)↑𝑁))
83		oveq1 7179	. . . . 5 ⊢ (𝑦 = (2 log_b 𝑥) → (𝑦↑(𝑁 − 1)) = ((2 log_b 𝑥)↑(𝑁 − 1)))
84	83	oveq2d 7188	. . . 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 24726	. . 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 7187	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 / ((log‘2)↑𝑁)) · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)))
91	66	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑁 ∈ ℂ)
92	63	nn0zd 12168	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℤ)
93	92	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑁 ∈ ℤ)
94	27, 36, 93	expclzd 13609	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) ∈ ℂ)
95	69	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 − 1) ∈ ℕ₀)
96	22, 95	expcld 13604	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
97	27, 36, 93	expne0d 13610	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) ≠ 0)
98	91, 94, 96, 10, 97, 21	divmuldivd 11537	. . . . . . . . . . . . 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 10742	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (𝑥 · ((log‘2)↑𝑁)))
100		1cnd 10716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 1 ∈ ℂ)
101	100, 66	pncan3d 11080	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1 + (𝑁 − 1)) = 𝑁)
102	101	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑁 = (1 + (𝑁 − 1)))
103	102	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑁 = (1 + (𝑁 − 1)))
104	103	oveq2d 7188	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = ((log‘2)↑(1 + (𝑁 − 1))))
105		1nn0 11994	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℕ₀
106	105	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℕ₀)
107	27, 95, 106	expaddd 13606	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(1 + (𝑁 − 1))) = (((log‘2)↑1) · ((log‘2)↑(𝑁 − 1))))
108	104, 107	eqtrd 2773	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = (((log‘2)↑1) · ((log‘2)↑(𝑁 − 1))))
109	27	exp1d 13599	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑1) = (log‘2))
110	109	oveq1d 7187	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑1) · ((log‘2)↑(𝑁 − 1))) = ((log‘2) · ((log‘2)↑(𝑁 − 1))))
111	108, 110	eqtrd 2773	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = ((log‘2) · ((log‘2)↑(𝑁 − 1))))
112	111	oveq2d 7188	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · ((log‘2)↑𝑁)) = (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))))
113	99, 112	eqtrd 2773	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))))
114	27, 95	expcld 13604	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(𝑁 − 1)) ∈ ℂ)
115	10, 27, 114	mulassd 10744	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))) = (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))))
116	115	eqcomd 2744	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · ((log‘2) · ((log‘2)↑(𝑁 − 1)))) = ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))))
117	113, 116	eqtrd 2773	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))))
118	10, 27	mulcld 10741	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈ ℂ)
119	118, 114	mulcomd 10742	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑥 · (log‘2)) · ((log‘2)↑(𝑁 − 1))) = (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2))))
120	117, 119	eqtrd 2773	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2))))
121	120	oveq2d 7188	. . . . . . . . . . . . 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 2773	. . . . . . . . . . . 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 2773	. . . . . . . . . . 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 10741	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((log‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
125		1zzd 12096	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℤ)
126	93, 125	zsubcld 12175	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 − 1) ∈ ℤ)
127	27, 36, 126	expne0d 13610	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(𝑁 − 1)) ≠ 0)
128	124, 114, 118, 127, 59	divdiv1d 11527	. . . . . . . . . . . 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 2744	. . . . . . . . . . 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 2773	. . . . . . . . . 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 11531	. . . . . . . . . . 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 7187	. . . . . . . . . 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 2773	. . . . . . . . 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 13618	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥) / (log‘2))↑(𝑁 − 1)) = (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1))))
135	134	eqcomd 2744	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1))) = (((log‘𝑥) / (log‘2))↑(𝑁 − 1)))
136	135	oveq2d 7188	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1)))) = (𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))))
137	136	oveq1d 7187	. . . . . . . . 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 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))) / (𝑥 · (log‘2))))
139	51	oveq1d 7187	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑(𝑁 − 1)) = (((log‘𝑥) / (log‘2))↑(𝑁 − 1)))
140	139	oveq2d 7188	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) = (𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))))
141	140	oveq1d 7187	. . . . . . . . 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 2744	. . . . . . . 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 2773	. . . . . . 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 13604	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑(𝑁 − 1)) ∈ ℂ)
145	91, 144	mulcld 10741	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) ∈ ℂ)
146	145, 118, 59	divrecd 11499	. . . . . . 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 2773	. . . . . 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 5125	. . . . 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 2773	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 · (log‘2))))))
150	149	eqcomd 2744	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝑁 · ((2 log_b 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 · (log‘2))))) = 𝐺)
151	85, 150	eqtrd 2773	. 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑𝑁))) = 𝐺)
152	3, 151	eqtrd 2773	1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∖ cdif 3840 {csn 4516 {cpr 4518 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6339 (class class class)co 7172 ℂcc 10615 ℝcr 10616 0cc0 10617 1c1 10618 + caddc 10620 · cmul 10622 < clt 10755 ≤ cle 10756 − cmin 10950 / cdiv 11377 ℕcn 11718 2c2 11773 ℕ₀cn0 11978 ℤcz 12064 ℝ⁺crp 12474 (,)cioo 12823 ↑cexp 13523 D cdv 24617 logclog 25300 log_b clogb 25504

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-inf2 9179 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-pre-sup 10695 ax-addf 10696 ax-mulf 10697

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-of 7427 df-om 7602 df-1st 7716 df-2nd 7717 df-supp 7859 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-2o 8134 df-er 8322 df-map 8441 df-pm 8442 df-ixp 8510 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-fsupp 8909 df-fi 8950 df-sup 8981 df-inf 8982 df-oi 9049 df-card 9443 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-div 11378 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-7 11786 df-8 11787 df-9 11788 df-n0 11979 df-z 12065 df-dec 12182 df-uz 12327 df-q 12433 df-rp 12475 df-xneg 12592 df-xadd 12593 df-xmul 12594 df-ioo 12827 df-ioc 12828 df-ico 12829 df-icc 12830 df-fz 12984 df-fzo 13127 df-fl 13255 df-mod 13331 df-seq 13463 df-exp 13524 df-fac 13728 df-bc 13757 df-hash 13785 df-shft 14518 df-cj 14550 df-re 14551 df-im 14552 df-sqrt 14686 df-abs 14687 df-limsup 14920 df-clim 14937 df-rlim 14938 df-sum 15138 df-ef 15515 df-sin 15517 df-cos 15518 df-pi 15520 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-ress 16596 df-plusg 16683 df-mulr 16684 df-starv 16685 df-sca 16686 df-vsca 16687 df-ip 16688 df-tset 16689 df-ple 16690 df-ds 16692 df-unif 16693 df-hom 16694 df-cco 16695 df-rest 16801 df-topn 16802 df-0g 16820 df-gsum 16821 df-topgen 16822 df-pt 16823 df-prds 16826 df-xrs 16880 df-qtop 16885 df-imas 16886 df-xps 16888 df-mre 16962 df-mrc 16963 df-acs 16965 df-mgm 17970 df-sgrp 18019 df-mnd 18030 df-submnd 18075 df-mulg 18345 df-cntz 18567 df-cmn 19028 df-psmet 20211 df-xmet 20212 df-met 20213 df-bl 20214 df-mopn 20215 df-fbas 20216 df-fg 20217 df-cnfld 20220 df-top 21647 df-topon 21664 df-topsp 21686 df-bases 21699 df-cld 21772 df-ntr 21773 df-cls 21774 df-nei 21851 df-lp 21889 df-perf 21890 df-cn 21980 df-cnp 21981 df-haus 22068 df-cmp 22140 df-tx 22315 df-hmeo 22508 df-fil 22599 df-fm 22691 df-flim 22692 df-flf 22693 df-xms 23075 df-ms 23076 df-tms 23077 df-cncf 23632 df-limc 24620 df-dv 24621 df-log 25302 df-logb 25505

This theorem is referenced by: aks4d1p1p6 39722 aks4d1p1p5 39724

W3C validator