aks4d1p1p6 - Mathbox for metakunt

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Theorem aks4d1p1p6 42500

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 11119	. . 3 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. 2 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
3		2cnd 12248	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
4		2re 12244	. . . . . 6 ⊢ 2 ∈ ℝ
5	4	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ)
6		2pos 12273	. . . . . 6 ⊢ 0 < 2
7	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 2)
8		elioore 13317	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
10		0red 11136	. . . . . . . . 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 12250	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
14	13	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 3 ∈ ℝ)
15		3pos 12275	. . . . . . . . . . 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 11295	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 𝐴)
20		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
21	11	rexrd 11184	. . . . . . . . . . . . 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 11184	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
25	24	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
26	9	rexrd 11184	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ^*)
27		elioo5 13345	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
28	22, 25, 26, 27	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
29	20, 28	mpbid 232	. . . . . . . . . 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 11296	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 𝑥)
32		1red 11134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
33		1lt2 12336	. . . . . . . . . . 11 ⊢ 1 < 2
34	33	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2)
35	32, 34	ltned 11271	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2)
36	35	necomd 2985	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1)
37	5, 7, 9, 31, 36	relogbcld 42401	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) ∈ ℝ)
38		5nn0 12446	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
39	38	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℕ₀)
40	37, 39	reexpcld 14114	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑5) ∈ ℝ)
41	40, 32	readdcld 11163	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ∈ ℝ)
42	10, 32	readdcld 11163	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (0 + 1) ∈ ℝ)
43	10	ltp1d 12075	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (0 + 1))
44	39	nn0zd 12538	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℤ)
45		2cnd 12248	. . . . . . . . . . 11 ⊢ (⊤ → 2 ∈ ℂ)
46		0red 11136	. . . . . . . . . . . . 13 ⊢ (⊤ → 0 ∈ ℝ)
47	6	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 0 < 2)
48	46, 47	ltned 11271	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ≠ 2)
49	48	necomd 2985	. . . . . . . . . . 11 ⊢ (⊤ → 2 ≠ 0)
50		1red 11134	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℝ)
51	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 < 2)
52	50, 51	ltned 11271	. . . . . . . . . . . 12 ⊢ (⊤ → 1 ≠ 2)
53	52	necomd 2985	. . . . . . . . . . 11 ⊢ (⊤ → 2 ≠ 1)
54		logb1 26721	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 1) = 0)
55	45, 49, 53, 54	syl3anc 1374	. . . . . . . . . 10 ⊢ (⊤ → (2 log_b 1) = 0)
56	55	mptru 1549	. . . . . . . . 9 ⊢ (2 log_b 1) = 0
57		2lt3 12337	. . . . . . . . . . . . . 14 ⊢ 2 < 3
58	57	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 < 3)
59	32, 5, 14, 34, 58	lttrd 11296	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 3)
60	32, 14, 12, 59, 18	ltletrd 11295	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 𝐴)
61	32, 12, 9, 60, 30	lttrd 11296	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 𝑥)
62		2z 12548	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
63	62	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℤ)
64	63	uzidd 12793	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℤ_≥‘2))
65		1rp 12935	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
66	65	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ⁺)
67	9, 31	elrpd 12972	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ⁺)
68		logblt 26736	. . . . . . . . . . 11 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 < 𝑥 ↔ (2 log_b 1) < (2 log_b 𝑥)))
69	64, 66, 67, 68	syl3anc 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 < 𝑥 ↔ (2 log_b 1) < (2 log_b 𝑥)))
70	61, 69	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 1) < (2 log_b 𝑥))
71	56, 70	eqbrtrrid 5110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (2 log_b 𝑥))
72		expgt0 14046	. . . . . . . 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 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < ((2 log_b 𝑥)↑5))
74	10, 40, 32, 73	ltadd1dd 11750	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (0 + 1) < (((2 log_b 𝑥)↑5) + 1))
75	10, 42, 41, 43, 74	lttrd 11296	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (((2 log_b 𝑥)↑5) + 1))
76	5, 7, 41, 75, 36	relogbcld 42401	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b (((2 log_b 𝑥)↑5) + 1)) ∈ ℝ)
77		recn 11117	. . . 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 11154	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 · (2 log_b (((2 log_b 𝑥)↑5) + 1))) ∈ ℂ)
80		2rp 12936	. . . . . . . 8 ⊢ 2 ∈ ℝ⁺
81	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ⁺)
82	81	relogcld 26575	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℝ)
83	41, 82	remulcld 11164	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((((2 log_b 𝑥)↑5) + 1) · (log‘2)) ∈ ℝ)
84	40	recnd 11162	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑5) ∈ ℂ)
85		1cnd 11128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
86	84, 85	addcld 11153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ∈ ℂ)
87	7	gt0ne0d 11703	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
88	3, 87	logcld 26522	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
89	75	gt0ne0d 11703	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ≠ 0)
90		0red 11136	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
91		loggt0b 26584	. . . . . . . . . . . 12 ⊢ (2 ∈ ℝ⁺ → (0 < (log‘2) ↔ 1 < 2))
92	80, 91	ax-mp 5	. . . . . . . . . . 11 ⊢ (0 < (log‘2) ↔ 1 < 2)
93	33, 92	mpbir 231	. . . . . . . . . 10 ⊢ 0 < (log‘2)
94	93	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < (log‘2))
95	90, 94	ltned 11271	. . . . . . . 8 ⊢ (𝜑 → 0 ≠ (log‘2))
96	95	necomd 2985	. . . . . . 7 ⊢ (𝜑 → (log‘2) ≠ 0)
97	96	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ≠ 0)
98	86, 88, 89, 97	mulne0d 11791	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((((2 log_b 𝑥)↑5) + 1) · (log‘2)) ≠ 0)
99	32, 83, 98	redivcld 11972	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / ((((2 log_b 𝑥)↑5) + 1) · (log‘2))) ∈ ℝ)
100		5re 12257	. . . . . . . 8 ⊢ 5 ∈ ℝ
101	100	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℝ)
102		4nn0 12445	. . . . . . . . 9 ⊢ 4 ∈ ℕ₀
103	102	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 4 ∈ ℕ₀)
104	37, 103	reexpcld 14114	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑4) ∈ ℝ)
105	101, 104	remulcld 11164	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (5 · ((2 log_b 𝑥)↑4)) ∈ ℝ)
106	9, 82	remulcld 11164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈ ℝ)
107	9	recnd 11162	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ)
108	10, 31	gtned 11270	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
109	107, 88, 108, 97	mulne0d 11791	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ≠ 0)
110	32, 106, 109	redivcld 11972	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / (𝑥 · (log‘2))) ∈ ℝ)
111	105, 110	remulcld 11164	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) ∈ ℝ)
112	111, 10	readdcld 11163	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0) ∈ ℝ)
113	99, 112	remulcld 11164	. . 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 11164	. 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 12972	. . . 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 12940	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
119	118	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
120		rpgt0 12944	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 0 < 𝑦)
121	120	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 0 < 𝑦)
122		1red 11134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 ∈ ℝ)
123	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 < 2)
124	122, 123	ltned 11271	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 ≠ 2)
125	124	necomd 2985	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ≠ 1)
126	116, 117, 119, 121, 125	relogbcld 42401	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (2 log_b 𝑦) ∈ ℝ)
127	126	recnd 11162	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (2 log_b 𝑦) ∈ ℂ)
128	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ∈ ℝ⁺)
129	128	relogcld 26575	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (log‘2) ∈ ℝ)
130	119, 129	remulcld 11164	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 · (log‘2)) ∈ ℝ)
131	119	recnd 11162	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℂ)
132		2cnd 12248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ∈ ℂ)
133	128	rpne0d 12980	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ≠ 0)
134	132, 133	logcld 26522	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (log‘2) ∈ ℂ)
135		rpne0 12948	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ≠ 0)
136	135	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ≠ 0)
137	96	necomd 2985	. . . . . . . 8 ⊢ (𝜑 → 0 ≠ (log‘2))
138	137	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 0 ≠ (log‘2))
139	138	necomd 2985	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (log‘2) ≠ 0)
140	131, 134, 136, 139	mulne0d 11791	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 · (log‘2)) ≠ 0)
141	122, 130, 140	redivcld 11972	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (1 / (𝑦 · (log‘2))) ∈ ℝ)
142		cnelprrecn 11120	. . . . . . 7 ⊢ ℂ ∈ {ℝ, ℂ}
143	142	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
144	37	recnd 11162	. . . . . 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 14097	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑5) ∈ ℂ)
148		5cn 12258	. . . . . . . 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 14097	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑4) ∈ ℂ)
152	149, 151	mulcld 11154	. . . . . 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 11295	. . . . . . . 8 ⊢ (𝜑 → 0 < 𝐴)
156	90, 11, 155	ltled 11283	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝐴)
157		aks4d1p1p6.4	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
158		eqid 2735	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))
159		eqid 2735	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))
160	21, 24, 156, 157, 158, 159	dvrelog2b 42493	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))))
161		5nn 12256	. . . . . . . 8 ⊢ 5 ∈ ℕ
162		dvexp 25908	. . . . . . . 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 12295	. . . . . . . . . . 11 ⊢ (5 − 1) = 4
165	164	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (5 − 1) = 4)
166	165	oveq2d 7372	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑(5 − 1)) = (𝑧↑4))
167	166	oveq2d 7372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (5 · (𝑧↑(5 − 1))) = (5 · (𝑧↑4)))
168	167	mpteq2dva 5167	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ (5 · (𝑧↑(5 − 1)))) = (𝑧 ∈ ℂ ↦ (5 · (𝑧↑4))))
169	163, 168	eqtrid 2782	. . . . . 6 ⊢ (𝜑 → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧↑5))) = (𝑧 ∈ ℂ ↦ (5 · (𝑧↑4))))
170		oveq1 7363	. . . . . 6 ⊢ (𝑧 = (2 log_b 𝑥) → (𝑧↑5) = ((2 log_b 𝑥)↑5))
171		oveq1 7363	. . . . . . 7 ⊢ (𝑧 = (2 log_b 𝑥) → (𝑧↑4) = ((2 log_b 𝑥)↑4))
172	171	oveq2d 7372	. . . . . 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 25927	. . . . 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 11128	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
175	174	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
176		0red 11136	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
177	2, 174	dvmptc 25913	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
178		ioossre 13349	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ
179	178	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
180		tgioo4 24758	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
181		eqid 2735	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
182		iooretop 24718	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
183	182	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
184	2, 175, 176, 177, 179, 180, 181, 183	dvmptres 25918	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 1)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
185	2, 84, 111, 173, 85, 10, 184	dvmptadd 25915	. . . 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 13336	. . . . . . . 8 ⊢ ℝ⁺ = (0(,)+∞)
187	186	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ⁺ = (0(,)+∞))
188	187	mpteq1d 5164	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ℝ⁺ ↦ (2 log_b 𝑦)) = (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦)))
189	188	oveq2d 7372	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (2 log_b 𝑦))) = (ℝ D (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))))
190	90	rexrd 11184	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ^*)
191		pnfxr 11188	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
192	191	a1i 11	. . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ^*)
193	90	leidd 11705	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 0)
194		0lepnf 13073	. . . . . . . 8 ⊢ 0 ≤ +∞
195	194	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ≤ +∞)
196		eqid 2735	. . . . . . 7 ⊢ (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦)) = (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))
197		eqid 2735	. . . . . . 7 ⊢ (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2)))) = (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2))))
198	190, 192, 193, 195, 196, 197	dvrelog2b 42493	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))) = (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2)))))
199	187	eqcomd 2741	. . . . . . 7 ⊢ (𝜑 → (0(,)+∞) = ℝ⁺)
200	199	mpteq1d 5164	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2)))) = (𝑦 ∈ ℝ⁺ ↦ (1 / (𝑦 · (log‘2)))))
201	198, 200	eqtrd 2770	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / (𝑦 · (log‘2)))))
202	189, 201	eqtrd 2770	. . . 4 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (2 log_b 𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / (𝑦 · (log‘2)))))
203		oveq2 7364	. . . 4 ⊢ (𝑦 = (((2 log_b 𝑥)↑5) + 1) → (2 log_b 𝑦) = (2 log_b (((2 log_b 𝑥)↑5) + 1)))
204		oveq1 7363	. . . . 5 ⊢ (𝑦 = (((2 log_b 𝑥)↑5) + 1) → (𝑦 · (log‘2)) = ((((2 log_b 𝑥)↑5) + 1) · (log‘2)))
205	204	oveq2d 7372	. . . 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 25927	. . 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 11162	. . 3 ⊢ (𝜑 → 2 ∈ ℂ)
209	2, 78, 113, 206, 208	dvmptcmul 25919	. 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 14095	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑2) ∈ ℂ)
211	82	resqcld 14076	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑2) ∈ ℝ)
212	81	rpne0d 12980	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
213	3, 212	logcld 26522	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
214	213, 97, 63	expne0d 14103	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑2) ≠ 0)
215	5, 211, 214	redivcld 11972	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 / ((log‘2)↑2)) ∈ ℝ)
216	67	relogcld 26575	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℝ)
217		2m1e1 12291	. . . . . . 7 ⊢ (2 − 1) = 1
218		1nn0 12442	. . . . . . 7 ⊢ 1 ∈ ℕ₀
219	217, 218	eqeltri 2831	. . . . . 6 ⊢ (2 − 1) ∈ ℕ₀
220	219	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 − 1) ∈ ℕ₀)
221	216, 220	reexpcld 14114	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥)↑(2 − 1)) ∈ ℝ)
222	67	rpne0d 12980	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
223	221, 9, 222	redivcld 11972	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥)↑(2 − 1)) / 𝑥) ∈ ℝ)
224	215, 223	remulcld 11164	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 / ((log‘2)↑2)) · (((log‘𝑥)↑(2 − 1)) / 𝑥)) ∈ ℝ)
225		eqid 2735	. . 3 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑2)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑2))
226		eqid 2735	. . 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 2735	. . 3 ⊢ (2 / ((log‘2)↑2)) = (2 / ((log‘2)↑2))
228		2nn 12243	. . . 4 ⊢ 2 ∈ ℕ
229	228	a1i 11	. . 3 ⊢ (𝜑 → 2 ∈ ℕ)
230	11, 23, 155, 157, 225, 226, 227, 229	dvrelogpow2b 42495	. 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 25915	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 206 ∧ wa 395 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2930 ⊆ wss 3885 {cpr 4559 class class class wbr 5074 ↦ cmpt 5155 ran crn 5621 ‘cfv 6487 (class class class)co 7356 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 · cmul 11032 +∞cpnf 11165 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 − cmin 11366 / cdiv 11796 ℕcn 12163 2c2 12225 3c3 12226 4c4 12227 5c5 12228 ℕ₀cn0 12426 ℤcz 12513 ℤ_≥cuz 12777 ℝ⁺crp 12931 (,)cioo 13287 ↑cexp 14012 TopOpenctopn 17373 topGenctg 17389 ℂ_fldccnfld 21341 D cdv 25818 logclog 26506 log_b clogb 26716

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8632 df-map 8764 df-pm 8765 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9264 df-fi 9313 df-sup 9344 df-inf 9345 df-oi 9414 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ioc 13292 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15638 df-ef 16021 df-sin 16023 df-cos 16024 df-pi 16026 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18741 df-mulg 19033 df-cntz 19281 df-cmn 19746 df-psmet 21333 df-xmet 21334 df-met 21335 df-bl 21336 df-mopn 21337 df-fbas 21338 df-fg 21339 df-cnfld 21342 df-top 22847 df-topon 22864 df-topsp 22886 df-bases 22899 df-cld 22972 df-ntr 22973 df-cls 22974 df-nei 23051 df-lp 23089 df-perf 23090 df-cn 23180 df-cnp 23181 df-haus 23268 df-cmp 23340 df-tx 23515 df-hmeo 23708 df-fil 23799 df-fm 23891 df-flim 23892 df-flf 23893 df-xms 24273 df-ms 24274 df-tms 24275 df-cncf 24833 df-limc 25821 df-dv 25822 df-log 26508 df-logb 26717

This theorem is referenced by: aks4d1p1p5 42502

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