aks4d1p1p6 - Mathbox for metakunt

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Theorem aks4d1p1p6 41397

Description: Inequality lift to differentiable functions for a term in AKS inequality lemma. (Contributed by metakunt, 19-Aug-2024.)

**Hypotheses**
Ref	Expression
aks4d1p1p6.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
aks4d1p1p6.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
aks4d1p1p6.3	⊢ (𝜑 → 3 ≤ 𝐴)
aks4d1p1p6.4	⊢ (𝜑 → 𝐴 ≤ 𝐵)

**Assertion**
Ref	Expression
aks4d1p1p6	⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 · (2 log_b (((2 log_b 𝑥)↑5) + 1))) + ((2 log_b 𝑥)↑2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 · ((1 / ((((2 log_b 𝑥)↑5) + 1) · (log‘2))) · (((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0))) + ((2 / ((log‘2)↑2)) · (((log‘𝑥)↑(2 − 1)) / 𝑥)))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝜑,𝑥

Proof of Theorem aks4d1p1p6 Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reelprrecn 11197	. . 3 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. 2 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
3		2cnd 12286	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
4		2re 12282	. . . . . 6 ⊢ 2 ∈ ℝ
5	4	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ)
6		2pos 12311	. . . . . 6 ⊢ 0 < 2
7	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 2)
8		elioore 13350	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
10		0red 11213	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
11		aks4d1p1p6.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
12	11	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
13		3re 12288	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
14	13	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 3 ∈ ℝ)
15		3pos 12313	. . . . . . . . . . 11 ⊢ 0 < 3
16	15	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 3)
17		aks4d1p1p6.3	. . . . . . . . . . 11 ⊢ (𝜑 → 3 ≤ 𝐴)
18	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 3 ≤ 𝐴)
19	10, 14, 12, 16, 18	ltletrd 11370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 𝐴)
20		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
21	11	rexrd 11260	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22	21	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
23		aks4d1p1p6.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℝ)
24	23	rexrd 11260	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
25	24	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
26	9	rexrd 11260	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ^*)
27		elioo5 13377	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
28	22, 25, 26, 27	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
29	20, 28	mpbid 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
30	29	simpld 494	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑥)
31	10, 12, 9, 19, 30	lttrd 11371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 𝑥)
32		1red 11211	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
33		1lt2 12379	. . . . . . . . . . 11 ⊢ 1 < 2
34	33	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2)
35	32, 34	ltned 11346	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2)
36	35	necomd 2988	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1)
37	5, 7, 9, 31, 36	relogbcld 41297	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) ∈ ℝ)
38		5nn0 12488	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
39	38	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℕ₀)
40	37, 39	reexpcld 14124	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑5) ∈ ℝ)
41	40, 32	readdcld 11239	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ∈ ℝ)
42	10, 32	readdcld 11239	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (0 + 1) ∈ ℝ)
43	10	ltp1d 12140	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (0 + 1))
44	39	nn0zd 12580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℤ)
45		2cnd 12286	. . . . . . . . . . 11 ⊢ (⊤ → 2 ∈ ℂ)
46		0red 11213	. . . . . . . . . . . . 13 ⊢ (⊤ → 0 ∈ ℝ)
47	6	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 0 < 2)
48	46, 47	ltned 11346	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ≠ 2)
49	48	necomd 2988	. . . . . . . . . . 11 ⊢ (⊤ → 2 ≠ 0)
50		1red 11211	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℝ)
51	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 < 2)
52	50, 51	ltned 11346	. . . . . . . . . . . 12 ⊢ (⊤ → 1 ≠ 2)
53	52	necomd 2988	. . . . . . . . . . 11 ⊢ (⊤ → 2 ≠ 1)
54		logb1 26616	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 1) = 0)
55	45, 49, 53, 54	syl3anc 1368	. . . . . . . . . 10 ⊢ (⊤ → (2 log_b 1) = 0)
56	55	mptru 1540	. . . . . . . . 9 ⊢ (2 log_b 1) = 0
57		2lt3 12380	. . . . . . . . . . . . . 14 ⊢ 2 < 3
58	57	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 < 3)
59	32, 5, 14, 34, 58	lttrd 11371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 3)
60	32, 14, 12, 59, 18	ltletrd 11370	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 𝐴)
61	32, 12, 9, 60, 30	lttrd 11371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 𝑥)
62		2z 12590	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
63	62	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℤ)
64	63	uzidd 12834	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℤ_≥‘2))
65		1rp 12974	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
66	65	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ⁺)
67	9, 31	elrpd 13009	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ⁺)
68		logblt 26631	. . . . . . . . . . 11 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 < 𝑥 ↔ (2 log_b 1) < (2 log_b 𝑥)))
69	64, 66, 67, 68	syl3anc 1368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 < 𝑥 ↔ (2 log_b 1) < (2 log_b 𝑥)))
70	61, 69	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 1) < (2 log_b 𝑥))
71	56, 70	eqbrtrrid 5174	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (2 log_b 𝑥))
72		expgt0 14057	. . . . . . . 8 ⊢ (((2 log_b 𝑥) ∈ ℝ ∧ 5 ∈ ℤ ∧ 0 < (2 log_b 𝑥)) → 0 < ((2 log_b 𝑥)↑5))
73	37, 44, 71, 72	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < ((2 log_b 𝑥)↑5))
74	10, 40, 32, 73	ltadd1dd 11821	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (0 + 1) < (((2 log_b 𝑥)↑5) + 1))
75	10, 42, 41, 43, 74	lttrd 11371	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (((2 log_b 𝑥)↑5) + 1))
76	5, 7, 41, 75, 36	relogbcld 41297	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b (((2 log_b 𝑥)↑5) + 1)) ∈ ℝ)
77		recn 11195	. . . 4 ⊢ ((2 log_b (((2 log_b 𝑥)↑5) + 1)) ∈ ℝ → (2 log_b (((2 log_b 𝑥)↑5) + 1)) ∈ ℂ)
78	76, 77	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b (((2 log_b 𝑥)↑5) + 1)) ∈ ℂ)
79	3, 78	mulcld 11230	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 · (2 log_b (((2 log_b 𝑥)↑5) + 1))) ∈ ℂ)
80		2rp 12975	. . . . . . . 8 ⊢ 2 ∈ ℝ⁺
81	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ⁺)
82	81	relogcld 26472	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℝ)
83	41, 82	remulcld 11240	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((((2 log_b 𝑥)↑5) + 1) · (log‘2)) ∈ ℝ)
84	40	recnd 11238	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑5) ∈ ℂ)
85		1cnd 11205	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
86	84, 85	addcld 11229	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ∈ ℂ)
87	7	gt0ne0d 11774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
88	3, 87	logcld 26420	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
89	75	gt0ne0d 11774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ≠ 0)
90		0red 11213	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
91		loggt0b 26481	. . . . . . . . . . . 12 ⊢ (2 ∈ ℝ⁺ → (0 < (log‘2) ↔ 1 < 2))
92	80, 91	ax-mp 5	. . . . . . . . . . 11 ⊢ (0 < (log‘2) ↔ 1 < 2)
93	33, 92	mpbir 230	. . . . . . . . . 10 ⊢ 0 < (log‘2)
94	93	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < (log‘2))
95	90, 94	ltned 11346	. . . . . . . 8 ⊢ (𝜑 → 0 ≠ (log‘2))
96	95	necomd 2988	. . . . . . 7 ⊢ (𝜑 → (log‘2) ≠ 0)
97	96	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ≠ 0)
98	86, 88, 89, 97	mulne0d 11862	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((((2 log_b 𝑥)↑5) + 1) · (log‘2)) ≠ 0)
99	32, 83, 98	redivcld 12038	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / ((((2 log_b 𝑥)↑5) + 1) · (log‘2))) ∈ ℝ)
100		5re 12295	. . . . . . . 8 ⊢ 5 ∈ ℝ
101	100	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℝ)
102		4nn0 12487	. . . . . . . . 9 ⊢ 4 ∈ ℕ₀
103	102	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 4 ∈ ℕ₀)
104	37, 103	reexpcld 14124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑4) ∈ ℝ)
105	101, 104	remulcld 11240	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (5 · ((2 log_b 𝑥)↑4)) ∈ ℝ)
106	9, 82	remulcld 11240	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈ ℝ)
107	9	recnd 11238	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ)
108	10, 31	gtned 11345	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
109	107, 88, 108, 97	mulne0d 11862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ≠ 0)
110	32, 106, 109	redivcld 12038	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / (𝑥 · (log‘2))) ∈ ℝ)
111	105, 110	remulcld 11240	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) ∈ ℝ)
112	111, 10	readdcld 11239	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0) ∈ ℝ)
113	99, 112	remulcld 11240	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((1 / ((((2 log_b 𝑥)↑5) + 1) · (log‘2))) · (((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0)) ∈ ℝ)
114	5, 113	remulcld 11240	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 · ((1 / ((((2 log_b 𝑥)↑5) + 1) · (log‘2))) · (((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0))) ∈ ℝ)
115	41, 75	elrpd 13009	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ∈ ℝ⁺)
116	4	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ∈ ℝ)
117	6	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 0 < 2)
118		rpre 12978	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
119	118	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
120		rpgt0 12982	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 0 < 𝑦)
121	120	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 0 < 𝑦)
122		1red 11211	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 ∈ ℝ)
123	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 < 2)
124	122, 123	ltned 11346	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 ≠ 2)
125	124	necomd 2988	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ≠ 1)
126	116, 117, 119, 121, 125	relogbcld 41297	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (2 log_b 𝑦) ∈ ℝ)
127	126	recnd 11238	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (2 log_b 𝑦) ∈ ℂ)
128	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ∈ ℝ⁺)
129	128	relogcld 26472	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (log‘2) ∈ ℝ)
130	119, 129	remulcld 11240	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 · (log‘2)) ∈ ℝ)
131	119	recnd 11238	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℂ)
132		2cnd 12286	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ∈ ℂ)
133	128	rpne0d 13017	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ≠ 0)
134	132, 133	logcld 26420	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (log‘2) ∈ ℂ)
135		rpne0 12986	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ≠ 0)
136	135	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ≠ 0)
137	96	necomd 2988	. . . . . . . 8 ⊢ (𝜑 → 0 ≠ (log‘2))
138	137	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 0 ≠ (log‘2))
139	138	necomd 2988	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (log‘2) ≠ 0)
140	131, 134, 136, 139	mulne0d 11862	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 · (log‘2)) ≠ 0)
141	122, 130, 140	redivcld 12038	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (1 / (𝑦 · (log‘2))) ∈ ℝ)
142		cnelprrecn 11198	. . . . . . 7 ⊢ ℂ ∈ {ℝ, ℂ}
143	142	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
144	37	recnd 11238	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) ∈ ℂ)
145		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
146	38	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 5 ∈ ℕ₀)
147	145, 146	expcld 14107	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑5) ∈ ℂ)
148		5cn 12296	. . . . . . . 8 ⊢ 5 ∈ ℂ
149	148	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 5 ∈ ℂ)
150	102	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 4 ∈ ℕ₀)
151	145, 150	expcld 14107	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑4) ∈ ℂ)
152	149, 151	mulcld 11230	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (5 · (𝑧↑4)) ∈ ℂ)
153	13	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 3 ∈ ℝ)
154	15	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 3)
155	90, 153, 11, 154, 17	ltletrd 11370	. . . . . . . 8 ⊢ (𝜑 → 0 < 𝐴)
156	90, 11, 155	ltled 11358	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝐴)
157		aks4d1p1p6.4	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
158		eqid 2724	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))
159		eqid 2724	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))
160	21, 24, 156, 157, 158, 159	dvrelog2b 41390	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))))
161		5nn 12294	. . . . . . . 8 ⊢ 5 ∈ ℕ
162		dvexp 25806	. . . . . . . 8 ⊢ (5 ∈ ℕ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧↑5))) = (𝑧 ∈ ℂ ↦ (5 · (𝑧↑(5 − 1)))))
163	161, 162	ax-mp 5	. . . . . . 7 ⊢ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧↑5))) = (𝑧 ∈ ℂ ↦ (5 · (𝑧↑(5 − 1))))
164		5m1e4 12338	. . . . . . . . . . 11 ⊢ (5 − 1) = 4
165	164	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (5 − 1) = 4)
166	165	oveq2d 7417	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑(5 − 1)) = (𝑧↑4))
167	166	oveq2d 7417	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (5 · (𝑧↑(5 − 1))) = (5 · (𝑧↑4)))
168	167	mpteq2dva 5238	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ (5 · (𝑧↑(5 − 1)))) = (𝑧 ∈ ℂ ↦ (5 · (𝑧↑4))))
169	163, 168	eqtrid 2776	. . . . . 6 ⊢ (𝜑 → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧↑5))) = (𝑧 ∈ ℂ ↦ (5 · (𝑧↑4))))
170		oveq1 7408	. . . . . 6 ⊢ (𝑧 = (2 log_b 𝑥) → (𝑧↑5) = ((2 log_b 𝑥)↑5))
171		oveq1 7408	. . . . . . 7 ⊢ (𝑧 = (2 log_b 𝑥) → (𝑧↑4) = ((2 log_b 𝑥)↑4))
172	171	oveq2d 7417	. . . . . 6 ⊢ (𝑧 = (2 log_b 𝑥) → (5 · (𝑧↑4)) = (5 · ((2 log_b 𝑥)↑4)))
173	2, 143, 144, 110, 147, 152, 160, 169, 170, 172	dvmptco 25825	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑5))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2))))))
174		1cnd 11205	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
175	174	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
176		0red 11213	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
177	2, 174	dvmptc 25811	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
178		ioossre 13381	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ
179	178	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
180		eqid 2724	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
181	180	tgioo2 24640	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
182		iooretop 24603	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
183	182	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
184	2, 175, 176, 177, 179, 181, 180, 183	dvmptres 25816	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 1)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
185	2, 84, 111, 173, 85, 10, 184	dvmptadd 25813	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (((2 log_b 𝑥)↑5) + 1))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0)))
186		dfrp2 13369	. . . . . . . 8 ⊢ ℝ⁺ = (0(,)+∞)
187	186	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ⁺ = (0(,)+∞))
188	187	mpteq1d 5233	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ℝ⁺ ↦ (2 log_b 𝑦)) = (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦)))
189	188	oveq2d 7417	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (2 log_b 𝑦))) = (ℝ D (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))))
190	90	rexrd 11260	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ^*)
191		pnfxr 11264	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
192	191	a1i 11	. . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ^*)
193	90	leidd 11776	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 0)
194		0lepnf 13108	. . . . . . . 8 ⊢ 0 ≤ +∞
195	194	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ≤ +∞)
196		eqid 2724	. . . . . . 7 ⊢ (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦)) = (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))
197		eqid 2724	. . . . . . 7 ⊢ (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2)))) = (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2))))
198	190, 192, 193, 195, 196, 197	dvrelog2b 41390	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))) = (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2)))))
199	187	eqcomd 2730	. . . . . . 7 ⊢ (𝜑 → (0(,)+∞) = ℝ⁺)
200	199	mpteq1d 5233	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2)))) = (𝑦 ∈ ℝ⁺ ↦ (1 / (𝑦 · (log‘2)))))
201	198, 200	eqtrd 2764	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / (𝑦 · (log‘2)))))
202	189, 201	eqtrd 2764	. . . 4 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (2 log_b 𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / (𝑦 · (log‘2)))))
203		oveq2 7409	. . . 4 ⊢ (𝑦 = (((2 log_b 𝑥)↑5) + 1) → (2 log_b 𝑦) = (2 log_b (((2 log_b 𝑥)↑5) + 1)))
204		oveq1 7408	. . . . 5 ⊢ (𝑦 = (((2 log_b 𝑥)↑5) + 1) → (𝑦 · (log‘2)) = ((((2 log_b 𝑥)↑5) + 1) · (log‘2)))
205	204	oveq2d 7417	. . . 4 ⊢ (𝑦 = (((2 log_b 𝑥)↑5) + 1) → (1 / (𝑦 · (log‘2))) = (1 / ((((2 log_b 𝑥)↑5) + 1) · (log‘2))))
206	2, 2, 115, 112, 127, 141, 185, 202, 203, 205	dvmptco 25825	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b (((2 log_b 𝑥)↑5) + 1)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / ((((2 log_b 𝑥)↑5) + 1) · (log‘2))) · (((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0))))
207	4	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ)
208	207	recnd 11238	. . 3 ⊢ (𝜑 → 2 ∈ ℂ)
209	2, 78, 113, 206, 208	dvmptcmul 25817	. 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 · (2 log_b (((2 log_b 𝑥)↑5) + 1))))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 · ((1 / ((((2 log_b 𝑥)↑5) + 1) · (log‘2))) · (((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0)))))
210	144	sqcld 14105	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑2) ∈ ℂ)
211	82	resqcld 14086	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑2) ∈ ℝ)
212	81	rpne0d 13017	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
213	3, 212	logcld 26420	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
214	213, 97, 63	expne0d 14113	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑2) ≠ 0)
215	5, 211, 214	redivcld 12038	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 / ((log‘2)↑2)) ∈ ℝ)
216	67	relogcld 26472	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℝ)
217		2m1e1 12334	. . . . . . 7 ⊢ (2 − 1) = 1
218		1nn0 12484	. . . . . . 7 ⊢ 1 ∈ ℕ₀
219	217, 218	eqeltri 2821	. . . . . 6 ⊢ (2 − 1) ∈ ℕ₀
220	219	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 − 1) ∈ ℕ₀)
221	216, 220	reexpcld 14124	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥)↑(2 − 1)) ∈ ℝ)
222	67	rpne0d 13017	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
223	221, 9, 222	redivcld 12038	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥)↑(2 − 1)) / 𝑥) ∈ ℝ)
224	215, 223	remulcld 11240	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 / ((log‘2)↑2)) · (((log‘𝑥)↑(2 − 1)) / 𝑥)) ∈ ℝ)
225		eqid 2724	. . 3 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑2)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑2))
226		eqid 2724	. . 3 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 / ((log‘2)↑2)) · (((log‘𝑥)↑(2 − 1)) / 𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 / ((log‘2)↑2)) · (((log‘𝑥)↑(2 − 1)) / 𝑥)))
227		eqid 2724	. . 3 ⊢ (2 / ((log‘2)↑2)) = (2 / ((log‘2)↑2))
228		2nn 12281	. . . 4 ⊢ 2 ∈ ℕ
229	228	a1i 11	. . 3 ⊢ (𝜑 → 2 ∈ ℕ)
230	11, 23, 155, 157, 225, 226, 227, 229	dvrelogpow2b 41392	. 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑2))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 / ((log‘2)↑2)) · (((log‘𝑥)↑(2 − 1)) / 𝑥))))
231	2, 79, 114, 209, 210, 224, 230	dvmptadd 25813	1 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 · (2 log_b (((2 log_b 𝑥)↑5) + 1))) + ((2 log_b 𝑥)↑2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 · ((1 / ((((2 log_b 𝑥)↑5) + 1) · (log‘2))) · (((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0))) + ((2 / ((log‘2)↑2)) · (((log‘𝑥)↑(2 − 1)) / 𝑥)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ≠ wne 2932 ⊆ wss 3940 {cpr 4622 class class class wbr 5138 ↦ cmpt 5221 ran crn 5667 ‘cfv 6533 (class class class)co 7401 ℂcc 11103 ℝcr 11104 0cc0 11105 1c1 11106 + caddc 11108 · cmul 11110 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 3c3 12264 4c4 12265 5c5 12266 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ℝ⁺crp 12970 (,)cioo 13320 ↑cexp 14023 TopOpenctopn 17365 topGenctg 17381 ℂ_fldccnfld 21227 D cdv 25713 logclog 26404 log_b clogb 26611

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-starv 17210 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-unif 17218 df-hom 17219 df-cco 17220 df-rest 17366 df-topn 17367 df-0g 17385 df-gsum 17386 df-topgen 17387 df-pt 17388 df-prds 17391 df-xrs 17446 df-qtop 17451 df-imas 17452 df-xps 17454 df-mre 17528 df-mrc 17529 df-acs 17531 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-submnd 18703 df-mulg 18985 df-cntz 19222 df-cmn 19691 df-psmet 21219 df-xmet 21220 df-met 21221 df-bl 21222 df-mopn 21223 df-fbas 21224 df-fg 21225 df-cnfld 21228 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cld 22844 df-ntr 22845 df-cls 22846 df-nei 22923 df-lp 22961 df-perf 22962 df-cn 23052 df-cnp 23053 df-haus 23140 df-cmp 23212 df-tx 23387 df-hmeo 23580 df-fil 23671 df-fm 23763 df-flim 23764 df-flf 23765 df-xms 24147 df-ms 24148 df-tms 24149 df-cncf 24719 df-limc 25716 df-dv 25717 df-log 26406 df-logb 26612

This theorem is referenced by: aks4d1p1p5 41399

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