aks4d1p1p6 - Mathbox for metakunt

	Mathbox for metakunt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > aks4d1p1p6		Structured version Visualization version GIF version

Theorem aks4d1p1p6 42695

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 11167	. . 3 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. 2 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
3		2cnd 12298	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
4		2re 12294	. . . . . 6 ⊢ 2 ∈ ℝ
5	4	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ)
6		2pos 12324	. . . . . 6 ⊢ 0 < 2
7	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 2)
8		elioore 13381	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
9	8	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
10		0red 11186	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
11		aks4d1p1p6.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
12	11	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
13		3re 12300	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
14	13	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 3 ∈ ℝ)
15		3pos 12328	. . . . . . . . . . 11 ⊢ 0 < 3
16	15	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 3)
17		aks4d1p1p6.3	. . . . . . . . . . 11 ⊢ (𝜑 → 3 ≤ 𝐴)
18	17	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 3 ≤ 𝐴)
19	10, 14, 12, 16, 18	ltletrd 11345	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 𝐴)
20		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
21	11	rexrd 11234	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22	21	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
23		aks4d1p1p6.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℝ)
24	23	rexrd 11234	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
25	24	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
26	9	rexrd 11234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ^*)
27		elioo5 13409	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
28	22, 25, 26, 27	syl3anc 1392	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
29	20, 28	mpbid 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
30	29	simpld 498	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑥)
31	10, 12, 9, 19, 30	lttrd 11346	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 𝑥)
32		1red 11184	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
33		1lt2 12392	. . . . . . . . . . 11 ⊢ 1 < 2
34	33	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2)
35	32, 34	ltned 11321	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2)
36	35	necomd 3014	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1)
37	5, 7, 9, 31, 36	relogbcld 42596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) ∈ ℝ)
38		5nn0 12503	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
39	38	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℕ₀)
40	37, 39	reexpcld 14178	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑5) ∈ ℝ)
41	40, 32	readdcld 11213	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ∈ ℝ)
42	10, 32	readdcld 11213	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (0 + 1) ∈ ℝ)
43	10	ltp1d 12124	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (0 + 1))
44	39	nn0zd 12595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℤ)
45		2cnd 12298	. . . . . . . . . . 11 ⊢ (⊤ → 2 ∈ ℂ)
46		0red 11186	. . . . . . . . . . . . 13 ⊢ (⊤ → 0 ∈ ℝ)
47	6	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 0 < 2)
48	46, 47	ltned 11321	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ≠ 2)
49	48	necomd 3014	. . . . . . . . . . 11 ⊢ (⊤ → 2 ≠ 0)
50		1red 11184	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℝ)
51	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 < 2)
52	50, 51	ltned 11321	. . . . . . . . . . . 12 ⊢ (⊤ → 1 ≠ 2)
53	52	necomd 3014	. . . . . . . . . . 11 ⊢ (⊤ → 2 ≠ 1)
54		logb1 26836	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 1) = 0)
55	45, 49, 53, 54	syl3anc 1392	. . . . . . . . . 10 ⊢ (⊤ → (2 log_b 1) = 0)
56	55	mptru 1569	. . . . . . . . 9 ⊢ (2 log_b 1) = 0
57		2lt3 12393	. . . . . . . . . . . . . 14 ⊢ 2 < 3
58	57	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 < 3)
59	32, 5, 14, 34, 58	lttrd 11346	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 3)
60	32, 14, 12, 59, 18	ltletrd 11345	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 𝐴)
61	32, 12, 9, 60, 30	lttrd 11346	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 𝑥)
62		2z 12605	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
63	62	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℤ)
64	63	uzidd 12857	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℤ_≥‘2))
65		1rp 12999	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
66	65	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ⁺)
67	9, 31	elrpd 13036	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ⁺)
68		logblt 26851	. . . . . . . . . . 11 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 1 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 < 𝑥 ↔ (2 log_b 1) < (2 log_b 𝑥)))
69	64, 66, 67, 68	syl3anc 1392	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 < 𝑥 ↔ (2 log_b 1) < (2 log_b 𝑥)))
70	61, 69	mpbid 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 1) < (2 log_b 𝑥))
71	56, 70	eqbrtrrid 5138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (2 log_b 𝑥))
72		expgt0 14110	. . . . . . . 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 1392	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < ((2 log_b 𝑥)↑5))
74	10, 40, 32, 73	ltadd1dd 11800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (0 + 1) < (((2 log_b 𝑥)↑5) + 1))
75	10, 42, 41, 43, 74	lttrd 11346	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (((2 log_b 𝑥)↑5) + 1))
76	5, 7, 41, 75, 36	relogbcld 42596	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b (((2 log_b 𝑥)↑5) + 1)) ∈ ℝ)
77		recn 11165	. . . 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 11204	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 · (2 log_b (((2 log_b 𝑥)↑5) + 1))) ∈ ℂ)
80		2rp 13000	. . . . . . . 8 ⊢ 2 ∈ ℝ⁺
81	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ⁺)
82	81	relogcld 26690	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℝ)
83	41, 82	remulcld 11214	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((((2 log_b 𝑥)↑5) + 1) · (log‘2)) ∈ ℝ)
84	40	recnd 11212	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑5) ∈ ℂ)
85		1cnd 11177	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
86	84, 85	addcld 11203	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ∈ ℂ)
87	7	gt0ne0d 11753	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
88	3, 87	logcld 26637	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
89	75	gt0ne0d 11753	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ≠ 0)
90		0red 11186	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
91		loggt0b 26699	. . . . . . . . . . . 12 ⊢ (2 ∈ ℝ⁺ → (0 < (log‘2) ↔ 1 < 2))
92	80, 91	ax-mp 5	. . . . . . . . . . 11 ⊢ (0 < (log‘2) ↔ 1 < 2)
93	33, 92	mpbir 233	. . . . . . . . . 10 ⊢ 0 < (log‘2)
94	93	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < (log‘2))
95	90, 94	ltned 11321	. . . . . . . 8 ⊢ (𝜑 → 0 ≠ (log‘2))
96	95	necomd 3014	. . . . . . 7 ⊢ (𝜑 → (log‘2) ≠ 0)
97	96	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ≠ 0)
98	86, 88, 89, 97	mulne0d 11841	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((((2 log_b 𝑥)↑5) + 1) · (log‘2)) ≠ 0)
99	32, 83, 98	redivcld 12021	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / ((((2 log_b 𝑥)↑5) + 1) · (log‘2))) ∈ ℝ)
100		5re 12307	. . . . . . . 8 ⊢ 5 ∈ ℝ
101	100	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℝ)
102		4nn0 12502	. . . . . . . . 9 ⊢ 4 ∈ ℕ₀
103	102	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 4 ∈ ℕ₀)
104	37, 103	reexpcld 14178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑4) ∈ ℝ)
105	101, 104	remulcld 11214	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (5 · ((2 log_b 𝑥)↑4)) ∈ ℝ)
106	9, 82	remulcld 11214	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈ ℝ)
107	9	recnd 11212	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ)
108	10, 31	gtned 11320	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
109	107, 88, 108, 97	mulne0d 11841	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ≠ 0)
110	32, 106, 109	redivcld 12021	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / (𝑥 · (log‘2))) ∈ ℝ)
111	105, 110	remulcld 11214	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) ∈ ℝ)
112	111, 10	readdcld 11213	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0) ∈ ℝ)
113	99, 112	remulcld 11214	. . 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 11214	. 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 13036	. . . 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 13004	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
119	118	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
120		rpgt0 13008	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 0 < 𝑦)
121	120	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 0 < 𝑦)
122		1red 11184	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 ∈ ℝ)
123	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 < 2)
124	122, 123	ltned 11321	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 ≠ 2)
125	124	necomd 3014	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ≠ 1)
126	116, 117, 119, 121, 125	relogbcld 42596	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (2 log_b 𝑦) ∈ ℝ)
127	126	recnd 11212	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (2 log_b 𝑦) ∈ ℂ)
128	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ∈ ℝ⁺)
129	128	relogcld 26690	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (log‘2) ∈ ℝ)
130	119, 129	remulcld 11214	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 · (log‘2)) ∈ ℝ)
131	119	recnd 11212	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℂ)
132		2cnd 12298	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ∈ ℂ)
133	128	rpne0d 13044	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ≠ 0)
134	132, 133	logcld 26637	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (log‘2) ∈ ℂ)
135		rpne0 13012	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ≠ 0)
136	135	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ≠ 0)
137	96	necomd 3014	. . . . . . . 8 ⊢ (𝜑 → 0 ≠ (log‘2))
138	137	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 0 ≠ (log‘2))
139	138	necomd 3014	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (log‘2) ≠ 0)
140	131, 134, 136, 139	mulne0d 11841	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 · (log‘2)) ≠ 0)
141	122, 130, 140	redivcld 12021	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (1 / (𝑦 · (log‘2))) ∈ ℝ)
142		cnelprrecn 11168	. . . . . . 7 ⊢ ℂ ∈ {ℝ, ℂ}
143	142	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
144	37	recnd 11212	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) ∈ ℂ)
145		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
146	38	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 5 ∈ ℕ₀)
147	145, 146	expcld 14161	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑5) ∈ ℂ)
148		5cn 12308	. . . . . . . 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 14161	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑4) ∈ ℂ)
152	149, 151	mulcld 11204	. . . . . 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 11345	. . . . . . . 8 ⊢ (𝜑 → 0 < 𝐴)
156	90, 11, 155	ltled 11333	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝐴)
157		aks4d1p1p6.4	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
158		eqid 2764	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))
159		eqid 2764	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))
160	21, 24, 156, 157, 158, 159	dvrelog2b 42688	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))))
161		5nn 12306	. . . . . . . 8 ⊢ 5 ∈ ℕ
162		dvexp 26017	. . . . . . . 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 12349	. . . . . . . . . . 11 ⊢ (5 − 1) = 4
165	164	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (5 − 1) = 4)
166	165	oveq2d 7414	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑(5 − 1)) = (𝑧↑4))
167	166	oveq2d 7414	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (5 · (𝑧↑(5 − 1))) = (5 · (𝑧↑4)))
168	167	mpteq2dva 5195	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ (5 · (𝑧↑(5 − 1)))) = (𝑧 ∈ ℂ ↦ (5 · (𝑧↑4))))
169	163, 168	eqtrid 2811	. . . . . 6 ⊢ (𝜑 → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧↑5))) = (𝑧 ∈ ℂ ↦ (5 · (𝑧↑4))))
170		oveq1 7405	. . . . . 6 ⊢ (𝑧 = (2 log_b 𝑥) → (𝑧↑5) = ((2 log_b 𝑥)↑5))
171		oveq1 7405	. . . . . . 7 ⊢ (𝑧 = (2 log_b 𝑥) → (𝑧↑4) = ((2 log_b 𝑥)↑4))
172	171	oveq2d 7414	. . . . . 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 26036	. . . . 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 11177	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
175	174	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
176		0red 11186	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
177	2, 174	dvmptc 26022	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
178		ioossre 13413	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ
179	178	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
180		tgioo4 24867	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
181		eqid 2764	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
182		iooretop 24827	. . . . . . 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 26027	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 1)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
185	2, 84, 111, 173, 85, 10, 184	dvmptadd 26024	. . . 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 13400	. . . . . . . 8 ⊢ ℝ⁺ = (0(,)+∞)
187	186	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ⁺ = (0(,)+∞))
188	187	mpteq1d 5192	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ℝ⁺ ↦ (2 log_b 𝑦)) = (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦)))
189	188	oveq2d 7414	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (2 log_b 𝑦))) = (ℝ D (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))))
190	90	rexrd 11234	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ^*)
191		pnfxr 11238	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
192	191	a1i 11	. . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ^*)
193	90	leidd 11755	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 0)
194		0lepnf 13137	. . . . . . . 8 ⊢ 0 ≤ +∞
195	194	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ≤ +∞)
196		eqid 2764	. . . . . . 7 ⊢ (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦)) = (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))
197		eqid 2764	. . . . . . 7 ⊢ (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2)))) = (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2))))
198	190, 192, 193, 195, 196, 197	dvrelog2b 42688	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))) = (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2)))))
199	187	eqcomd 2770	. . . . . . 7 ⊢ (𝜑 → (0(,)+∞) = ℝ⁺)
200	199	mpteq1d 5192	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (0(,)+∞) ↦ (1 / (𝑦 · (log‘2)))) = (𝑦 ∈ ℝ⁺ ↦ (1 / (𝑦 · (log‘2)))))
201	198, 200	eqtrd 2799	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / (𝑦 · (log‘2)))))
202	189, 201	eqtrd 2799	. . . 4 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (2 log_b 𝑦))) = (𝑦 ∈ ℝ⁺ ↦ (1 / (𝑦 · (log‘2)))))
203		oveq2 7406	. . . 4 ⊢ (𝑦 = (((2 log_b 𝑥)↑5) + 1) → (2 log_b 𝑦) = (2 log_b (((2 log_b 𝑥)↑5) + 1)))
204		oveq1 7405	. . . . 5 ⊢ (𝑦 = (((2 log_b 𝑥)↑5) + 1) → (𝑦 · (log‘2)) = ((((2 log_b 𝑥)↑5) + 1) · (log‘2)))
205	204	oveq2d 7414	. . . 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 26036	. . 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 11212	. . 3 ⊢ (𝜑 → 2 ∈ ℂ)
209	2, 78, 113, 206, 208	dvmptcmul 26028	. 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 14159	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑2) ∈ ℂ)
211	82	resqcld 14140	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑2) ∈ ℝ)
212	81	rpne0d 13044	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
213	3, 212	logcld 26637	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
214	213, 97, 63	expne0d 14167	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑2) ≠ 0)
215	5, 211, 214	redivcld 12021	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 / ((log‘2)↑2)) ∈ ℝ)
216	67	relogcld 26690	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℝ)
217		2m1e1 12344	. . . . . . 7 ⊢ (2 − 1) = 1
218		1nn0 12499	. . . . . . 7 ⊢ 1 ∈ ℕ₀
219	217, 218	eqeltri 2860	. . . . . 6 ⊢ (2 − 1) ∈ ℕ₀
220	219	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 − 1) ∈ ℕ₀)
221	216, 220	reexpcld 14178	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥)↑(2 − 1)) ∈ ℝ)
222	67	rpne0d 13044	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
223	221, 9, 222	redivcld 12021	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥)↑(2 − 1)) / 𝑥) ∈ ℝ)
224	215, 223	remulcld 11214	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 / ((log‘2)↑2)) · (((log‘𝑥)↑(2 − 1)) / 𝑥)) ∈ ℝ)
225		eqid 2764	. . 3 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑2)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 log_b 𝑥)↑2))
226		eqid 2764	. . 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 2764	. . 3 ⊢ (2 / ((log‘2)↑2)) = (2 / ((log‘2)↑2))
228		2nn 12293	. . . 4 ⊢ 2 ∈ ℕ
229	228	a1i 11	. . 3 ⊢ (𝜑 → 2 ∈ ℕ)
230	11, 23, 155, 157, 225, 226, 227, 229	dvrelogpow2b 42690	. 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 26024	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 208 ∧ wa 399 = wceq 1562 ⊤wtru 1563 ∈ wcel 2144 ≠ wne 2959 ⊆ wss 3906 {cpr 4586 class class class wbr 5102 ↦ cmpt 5183 ran crn 5650 ‘cfv 6523 (class class class)co 7398 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 +∞cpnf 11215 ℝ^*cxr 11217 < clt 11218 ≤ cle 11219 − cmin 11416 / cdiv 11846 ℕcn 12212 2c2 12274 3c3 12275 4c4 12276 5c5 12277 ℕ₀cn0 12483 ℤcz 12570 ℤ_≥cuz 12841 ℝ⁺crp 12995 (,)cioo 13351 ↑cexp 14076 TopOpenctopn 17452 topGenctg 17468 ℂ_fldccnfld 21426 D cdv 25927 logclog 26621 log_b clogb 26831

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-inf2 9598 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-er 8680 df-map 8812 df-pm 8813 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-q 12952 df-rp 12996 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13515 df-fzo 13662 df-fl 13804 df-mod 13882 df-seq 14017 df-exp 14077 df-fac 14289 df-bc 14318 df-hash 14346 df-shft 15082 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-limsup 15500 df-clim 15517 df-rlim 15518 df-sum 15716 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-starv 17303 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-hom 17312 df-cco 17313 df-rest 17453 df-topn 17454 df-0g 17472 df-gsum 17473 df-topgen 17474 df-pt 17475 df-prds 17478 df-xrs 17534 df-qtop 17539 df-imas 17540 df-xps 17542 df-mre 17616 df-mrc 17617 df-acs 17619 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-submnd 18820 df-mulg 19112 df-cntz 19359 df-cmn 19824 df-psmet 21418 df-xmet 21419 df-met 21420 df-bl 21421 df-mopn 21422 df-fbas 21423 df-fg 21424 df-cnfld 21427 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cld 23081 df-ntr 23082 df-cls 23083 df-nei 23160 df-lp 23198 df-perf 23199 df-cn 23289 df-cnp 23290 df-haus 23377 df-cmp 23449 df-tx 23624 df-hmeo 23817 df-fil 23908 df-fm 24000 df-flim 24001 df-flf 24002 df-xms 24382 df-ms 24383 df-tms 24384 df-cncf 24942 df-limc 25930 df-dv 25931 df-log 26623 df-logb 26832

This theorem is referenced by: aks4d1p1p5 42697

W3C validator