aks4d1p1p6 - Mathbox for metakunt

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Theorem aks4d1p1p6 42061

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 11160	. . 3 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. 2 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
3		2cnd 12264	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
4		2re 12260	. . . . . 6 ⊢ 2 ∈ ℝ
5	4	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ)
6		2pos 12289	. . . . . 6 ⊢ 0 < 2
7	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 2)
8		elioore 13336	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
10		0red 11177	. . . . . . . . 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 12266	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
14	13	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 3 ∈ ℝ)
15		3pos 12291	. . . . . . . . . . 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 11334	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 𝐴)
20		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
21	11	rexrd 11224	. . . . . . . . . . . . 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 11224	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
25	24	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
26	9	rexrd 11224	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ^*)
27		elioo5 13364	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
28	22, 25, 26, 27	syl3anc 1373	. . . . . . . . . . 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 11335	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 𝑥)
32		1red 11175	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
33		1lt2 12352	. . . . . . . . . . 11 ⊢ 1 < 2
34	33	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2)
35	32, 34	ltned 11310	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2)
36	35	necomd 2980	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1)
37	5, 7, 9, 31, 36	relogbcld 41961	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) ∈ ℝ)
38		5nn0 12462	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
39	38	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℕ₀)
40	37, 39	reexpcld 14128	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑5) ∈ ℝ)
41	40, 32	readdcld 11203	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ∈ ℝ)
42	10, 32	readdcld 11203	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (0 + 1) ∈ ℝ)
43	10	ltp1d 12113	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (0 + 1))
44	39	nn0zd 12555	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℤ)
45		2cnd 12264	. . . . . . . . . . 11 ⊢ (⊤ → 2 ∈ ℂ)
46		0red 11177	. . . . . . . . . . . . 13 ⊢ (⊤ → 0 ∈ ℝ)
47	6	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 0 < 2)
48	46, 47	ltned 11310	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ≠ 2)
49	48	necomd 2980	. . . . . . . . . . 11 ⊢ (⊤ → 2 ≠ 0)
50		1red 11175	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℝ)
51	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 < 2)
52	50, 51	ltned 11310	. . . . . . . . . . . 12 ⊢ (⊤ → 1 ≠ 2)
53	52	necomd 2980	. . . . . . . . . . 11 ⊢ (⊤ → 2 ≠ 1)
54		logb1 26679	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 1) = 0)
55	45, 49, 53, 54	syl3anc 1373	. . . . . . . . . 10 ⊢ (⊤ → (2 log_b 1) = 0)
56	55	mptru 1547	. . . . . . . . 9 ⊢ (2 log_b 1) = 0
57		2lt3 12353	. . . . . . . . . . . . . 14 ⊢ 2 < 3
58	57	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 < 3)
59	32, 5, 14, 34, 58	lttrd 11335	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 3)
60	32, 14, 12, 59, 18	ltletrd 11334	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 𝐴)
61	32, 12, 9, 60, 30	lttrd 11335	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 𝑥)
62		2z 12565	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
63	62	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℤ)
64	63	uzidd 12809	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℤ_≥‘2))
65		1rp 12955	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
66	65	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ⁺)
67	9, 31	elrpd 12992	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ⁺)
68		logblt 26694	. . . . . . . . . . 11 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 < 𝑥 ↔ (2 log_b 1) < (2 log_b 𝑥)))
69	64, 66, 67, 68	syl3anc 1373	. . . . . . . . . 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 5143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (2 log_b 𝑥))
72		expgt0 14060	. . . . . . . 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 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < ((2 log_b 𝑥)↑5))
74	10, 40, 32, 73	ltadd1dd 11789	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (0 + 1) < (((2 log_b 𝑥)↑5) + 1))
75	10, 42, 41, 43, 74	lttrd 11335	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (((2 log_b 𝑥)↑5) + 1))
76	5, 7, 41, 75, 36	relogbcld 41961	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b (((2 log_b 𝑥)↑5) + 1)) ∈ ℝ)
77		recn 11158	. . . 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 11194	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 · (2 log_b (((2 log_b 𝑥)↑5) + 1))) ∈ ℂ)
80		2rp 12956	. . . . . . . 8 ⊢ 2 ∈ ℝ⁺
81	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ⁺)
82	81	relogcld 26532	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℝ)
83	41, 82	remulcld 11204	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((((2 log_b 𝑥)↑5) + 1) · (log‘2)) ∈ ℝ)
84	40	recnd 11202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑5) ∈ ℂ)
85		1cnd 11169	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
86	84, 85	addcld 11193	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ∈ ℂ)
87	7	gt0ne0d 11742	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
88	3, 87	logcld 26479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
89	75	gt0ne0d 11742	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ≠ 0)
90		0red 11177	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
91		loggt0b 26541	. . . . . . . . . . . 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 11310	. . . . . . . 8 ⊢ (𝜑 → 0 ≠ (log‘2))
96	95	necomd 2980	. . . . . . 7 ⊢ (𝜑 → (log‘2) ≠ 0)
97	96	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ≠ 0)
98	86, 88, 89, 97	mulne0d 11830	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((((2 log_b 𝑥)↑5) + 1) · (log‘2)) ≠ 0)
99	32, 83, 98	redivcld 12010	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / ((((2 log_b 𝑥)↑5) + 1) · (log‘2))) ∈ ℝ)
100		5re 12273	. . . . . . . 8 ⊢ 5 ∈ ℝ
101	100	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℝ)
102		4nn0 12461	. . . . . . . . 9 ⊢ 4 ∈ ℕ₀
103	102	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 4 ∈ ℕ₀)
104	37, 103	reexpcld 14128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑4) ∈ ℝ)
105	101, 104	remulcld 11204	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (5 · ((2 log_b 𝑥)↑4)) ∈ ℝ)
106	9, 82	remulcld 11204	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈ ℝ)
107	9	recnd 11202	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ)
108	10, 31	gtned 11309	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
109	107, 88, 108, 97	mulne0d 11830	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ≠ 0)
110	32, 106, 109	redivcld 12010	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / (𝑥 · (log‘2))) ∈ ℝ)
111	105, 110	remulcld 11204	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) ∈ ℝ)
112	111, 10	readdcld 11203	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0) ∈ ℝ)
113	99, 112	remulcld 11204	. . 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 11204	. 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 12992	. . . 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 12960	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
119	118	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
120		rpgt0 12964	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 0 < 𝑦)
121	120	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 0 < 𝑦)
122		1red 11175	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 ∈ ℝ)
123	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 < 2)
124	122, 123	ltned 11310	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 ≠ 2)
125	124	necomd 2980	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ≠ 1)
126	116, 117, 119, 121, 125	relogbcld 41961	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (2 log_b 𝑦) ∈ ℝ)
127	126	recnd 11202	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (2 log_b 𝑦) ∈ ℂ)
128	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ∈ ℝ⁺)
129	128	relogcld 26532	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (log‘2) ∈ ℝ)
130	119, 129	remulcld 11204	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 · (log‘2)) ∈ ℝ)
131	119	recnd 11202	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℂ)
132		2cnd 12264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ∈ ℂ)
133	128	rpne0d 13000	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ≠ 0)
134	132, 133	logcld 26479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (log‘2) ∈ ℂ)
135		rpne0 12968	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ≠ 0)
136	135	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ≠ 0)
137	96	necomd 2980	. . . . . . . 8 ⊢ (𝜑 → 0 ≠ (log‘2))
138	137	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 0 ≠ (log‘2))
139	138	necomd 2980	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (log‘2) ≠ 0)
140	131, 134, 136, 139	mulne0d 11830	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 · (log‘2)) ≠ 0)
141	122, 130, 140	redivcld 12010	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (1 / (𝑦 · (log‘2))) ∈ ℝ)
142		cnelprrecn 11161	. . . . . . 7 ⊢ ℂ ∈ {ℝ, ℂ}
143	142	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
144	37	recnd 11202	. . . . . 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 14111	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑5) ∈ ℂ)
148		5cn 12274	. . . . . . . 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 14111	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑4) ∈ ℂ)
152	149, 151	mulcld 11194	. . . . . 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 11334	. . . . . . . 8 ⊢ (𝜑 → 0 < 𝐴)
156	90, 11, 155	ltled 11322	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝐴)
157		aks4d1p1p6.4	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
158		eqid 2729	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))
159		eqid 2729	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))
160	21, 24, 156, 157, 158, 159	dvrelog2b 42054	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))))
161		5nn 12272	. . . . . . . 8 ⊢ 5 ∈ ℕ
162		dvexp 25857	. . . . . . . 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 12311	. . . . . . . . . . 11 ⊢ (5 − 1) = 4
165	164	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (5 − 1) = 4)
166	165	oveq2d 7403	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑(5 − 1)) = (𝑧↑4))
167	166	oveq2d 7403	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (5 · (𝑧↑(5 − 1))) = (5 · (𝑧↑4)))
168	167	mpteq2dva 5200	. . . . . . 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 7394	. . . . . 6 ⊢ (𝑧 = (2 log_b 𝑥) → (𝑧↑5) = ((2 log_b 𝑥)↑5))
171		oveq1 7394	. . . . . . 7 ⊢ (𝑧 = (2 log_b 𝑥) → (𝑧↑4) = ((2 log_b 𝑥)↑4))
172	171	oveq2d 7403	. . . . . 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 25876	. . . . 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 11169	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
175	174	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
176		0red 11177	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
177	2, 174	dvmptc 25862	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
178		ioossre 13368	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ
179	178	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
180		tgioo4 24693	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
181		eqid 2729	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
182		iooretop 24653	. . . . . . 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 25867	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 1)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
185	2, 84, 111, 173, 85, 10, 184	dvmptadd 25864	. . . 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 13355	. . . . . . . 8 ⊢ ℝ⁺ = (0(,)+∞)
187	186	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ⁺ = (0(,)+∞))
188	187	mpteq1d 5197	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ℝ⁺ ↦ (2 log_b 𝑦)) = (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦)))
189	188	oveq2d 7403	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (2 log_b 𝑦))) = (ℝ D (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))))
190	90	rexrd 11224	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ^*)
191		pnfxr 11228	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
192	191	a1i 11	. . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ^*)
193	90	leidd 11744	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 0)
194		0lepnf 13093	. . . . . . . 8 ⊢ 0 ≤ +∞
195	194	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ≤ +∞)
196		eqid 2729	. . . . . . 7 ⊢ (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦)) = (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))
197		eqid 2729	. . . . . . 7 ⊢ (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2)))) = (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2))))
198	190, 192, 193, 195, 196, 197	dvrelog2b 42054	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))) = (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2)))))
199	187	eqcomd 2735	. . . . . . 7 ⊢ (𝜑 → (0(,)+∞) = ℝ⁺)
200	199	mpteq1d 5197	. . . . . 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 7395	. . . 4 ⊢ (𝑦 = (((2 log_b 𝑥)↑5) + 1) → (2 log_b 𝑦) = (2 log_b (((2 log_b 𝑥)↑5) + 1)))
204		oveq1 7394	. . . . 5 ⊢ (𝑦 = (((2 log_b 𝑥)↑5) + 1) → (𝑦 · (log‘2)) = ((((2 log_b 𝑥)↑5) + 1) · (log‘2)))
205	204	oveq2d 7403	. . . 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 25876	. . 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 11202	. . 3 ⊢ (𝜑 → 2 ∈ ℂ)
209	2, 78, 113, 206, 208	dvmptcmul 25868	. 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 14109	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑2) ∈ ℂ)
211	82	resqcld 14090	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑2) ∈ ℝ)
212	81	rpne0d 13000	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
213	3, 212	logcld 26479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
214	213, 97, 63	expne0d 14117	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑2) ≠ 0)
215	5, 211, 214	redivcld 12010	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 / ((log‘2)↑2)) ∈ ℝ)
216	67	relogcld 26532	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℝ)
217		2m1e1 12307	. . . . . . 7 ⊢ (2 − 1) = 1
218		1nn0 12458	. . . . . . 7 ⊢ 1 ∈ ℕ₀
219	217, 218	eqeltri 2824	. . . . . 6 ⊢ (2 − 1) ∈ ℕ₀
220	219	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 − 1) ∈ ℕ₀)
221	216, 220	reexpcld 14128	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥)↑(2 − 1)) ∈ ℝ)
222	67	rpne0d 13000	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
223	221, 9, 222	redivcld 12010	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥)↑(2 − 1)) / 𝑥) ∈ ℝ)
224	215, 223	remulcld 11204	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 / ((log‘2)↑2)) · (((log‘𝑥)↑(2 − 1)) / 𝑥)) ∈ ℝ)
225		eqid 2729	. . 3 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑2)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑2))
226		eqid 2729	. . 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 2729	. . 3 ⊢ (2 / ((log‘2)↑2)) = (2 / ((log‘2)↑2))
228		2nn 12259	. . . 4 ⊢ 2 ∈ ℕ
229	228	a1i 11	. . 3 ⊢ (𝜑 → 2 ∈ ℕ)
230	11, 23, 155, 157, 225, 226, 227, 229	dvrelogpow2b 42056	. 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 25864	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 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2925 ⊆ wss 3914 {cpr 4591 class class class wbr 5107 ↦ cmpt 5188 ran crn 5639 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 +∞cpnf 11205 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 ℕcn 12186 2c2 12241 3c3 12242 4c4 12243 5c5 12244 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ℝ⁺crp 12951 (,)cioo 13306 ↑cexp 14026 TopOpenctopn 17384 topGenctg 17400 ℂ_fldccnfld 21264 D cdv 25764 logclog 26463 log_b clogb 26674

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-cmp 23274 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-log 26465 df-logb 26675

This theorem is referenced by: aks4d1p1p5 42063

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