aks4d1p1p6 - Mathbox for metakunt

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Theorem aks4d1p1p6 42054

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 11136	. . 3 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. 2 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
3		2cnd 12240	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
4		2re 12236	. . . . . 6 ⊢ 2 ∈ ℝ
5	4	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ)
6		2pos 12265	. . . . . 6 ⊢ 0 < 2
7	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 2)
8		elioore 13312	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
10		0red 11153	. . . . . . . . 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 12242	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
14	13	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 3 ∈ ℝ)
15		3pos 12267	. . . . . . . . . . 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 11310	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 𝐴)
20		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵))
21	11	rexrd 11200	. . . . . . . . . . . . 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 11200	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
25	24	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
26	9	rexrd 11200	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ^*)
27		elioo5 13340	. . . . . . . . . . . 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 11311	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 𝑥)
32		1red 11151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
33		1lt2 12328	. . . . . . . . . . 11 ⊢ 1 < 2
34	33	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2)
35	32, 34	ltned 11286	. . . . . . . . 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 41954	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) ∈ ℝ)
38		5nn0 12438	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
39	38	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℕ₀)
40	37, 39	reexpcld 14104	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑5) ∈ ℝ)
41	40, 32	readdcld 11179	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ∈ ℝ)
42	10, 32	readdcld 11179	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (0 + 1) ∈ ℝ)
43	10	ltp1d 12089	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (0 + 1))
44	39	nn0zd 12531	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℤ)
45		2cnd 12240	. . . . . . . . . . 11 ⊢ (⊤ → 2 ∈ ℂ)
46		0red 11153	. . . . . . . . . . . . 13 ⊢ (⊤ → 0 ∈ ℝ)
47	6	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 0 < 2)
48	46, 47	ltned 11286	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ≠ 2)
49	48	necomd 2980	. . . . . . . . . . 11 ⊢ (⊤ → 2 ≠ 0)
50		1red 11151	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 ∈ ℝ)
51	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → 1 < 2)
52	50, 51	ltned 11286	. . . . . . . . . . . 12 ⊢ (⊤ → 1 ≠ 2)
53	52	necomd 2980	. . . . . . . . . . 11 ⊢ (⊤ → 2 ≠ 1)
54		logb1 26712	. . . . . . . . . . 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 12329	. . . . . . . . . . . . . 14 ⊢ 2 < 3
58	57	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 < 3)
59	32, 5, 14, 34, 58	lttrd 11311	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 3)
60	32, 14, 12, 59, 18	ltletrd 11310	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 𝐴)
61	32, 12, 9, 60, 30	lttrd 11311	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 𝑥)
62		2z 12541	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
63	62	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℤ)
64	63	uzidd 12785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℤ_≥‘2))
65		1rp 12931	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
66	65	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ⁺)
67	9, 31	elrpd 12968	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ⁺)
68		logblt 26727	. . . . . . . . . . 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 5138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (2 log_b 𝑥))
72		expgt0 14036	. . . . . . . 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 11765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (0 + 1) < (((2 log_b 𝑥)↑5) + 1))
75	10, 42, 41, 43, 74	lttrd 11311	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < (((2 log_b 𝑥)↑5) + 1))
76	5, 7, 41, 75, 36	relogbcld 41954	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b (((2 log_b 𝑥)↑5) + 1)) ∈ ℝ)
77		recn 11134	. . . 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 11170	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 · (2 log_b (((2 log_b 𝑥)↑5) + 1))) ∈ ℂ)
80		2rp 12932	. . . . . . . 8 ⊢ 2 ∈ ℝ⁺
81	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℝ⁺)
82	81	relogcld 26565	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℝ)
83	41, 82	remulcld 11180	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((((2 log_b 𝑥)↑5) + 1) · (log‘2)) ∈ ℝ)
84	40	recnd 11178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑5) ∈ ℂ)
85		1cnd 11145	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
86	84, 85	addcld 11169	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ∈ ℂ)
87	7	gt0ne0d 11718	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
88	3, 87	logcld 26512	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
89	75	gt0ne0d 11718	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((2 log_b 𝑥)↑5) + 1) ≠ 0)
90		0red 11153	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
91		loggt0b 26574	. . . . . . . . . . . 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 11286	. . . . . . . 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 11806	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((((2 log_b 𝑥)↑5) + 1) · (log‘2)) ≠ 0)
99	32, 83, 98	redivcld 11986	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / ((((2 log_b 𝑥)↑5) + 1) · (log‘2))) ∈ ℝ)
100		5re 12249	. . . . . . . 8 ⊢ 5 ∈ ℝ
101	100	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 5 ∈ ℝ)
102		4nn0 12437	. . . . . . . . 9 ⊢ 4 ∈ ℕ₀
103	102	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 4 ∈ ℕ₀)
104	37, 103	reexpcld 14104	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑4) ∈ ℝ)
105	101, 104	remulcld 11180	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (5 · ((2 log_b 𝑥)↑4)) ∈ ℝ)
106	9, 82	remulcld 11180	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈ ℝ)
107	9	recnd 11178	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ)
108	10, 31	gtned 11285	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
109	107, 88, 108, 97	mulne0d 11806	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ≠ 0)
110	32, 106, 109	redivcld 11986	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / (𝑥 · (log‘2))) ∈ ℝ)
111	105, 110	remulcld 11180	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) ∈ ℝ)
112	111, 10	readdcld 11179	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((5 · ((2 log_b 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0) ∈ ℝ)
113	99, 112	remulcld 11180	. . 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 11180	. 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 12968	. . . 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 12936	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
119	118	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
120		rpgt0 12940	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ → 0 < 𝑦)
121	120	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 0 < 𝑦)
122		1red 11151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 ∈ ℝ)
123	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 < 2)
124	122, 123	ltned 11286	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 1 ≠ 2)
125	124	necomd 2980	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ≠ 1)
126	116, 117, 119, 121, 125	relogbcld 41954	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (2 log_b 𝑦) ∈ ℝ)
127	126	recnd 11178	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (2 log_b 𝑦) ∈ ℂ)
128	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ∈ ℝ⁺)
129	128	relogcld 26565	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (log‘2) ∈ ℝ)
130	119, 129	remulcld 11180	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 · (log‘2)) ∈ ℝ)
131	119	recnd 11178	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℂ)
132		2cnd 12240	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ∈ ℂ)
133	128	rpne0d 12976	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 2 ≠ 0)
134	132, 133	logcld 26512	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (log‘2) ∈ ℂ)
135		rpne0 12944	. . . . . . 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 11806	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 · (log‘2)) ≠ 0)
141	122, 130, 140	redivcld 11986	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (1 / (𝑦 · (log‘2))) ∈ ℝ)
142		cnelprrecn 11137	. . . . . . 7 ⊢ ℂ ∈ {ℝ, ℂ}
143	142	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
144	37	recnd 11178	. . . . . 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 14087	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑5) ∈ ℂ)
148		5cn 12250	. . . . . . . 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 14087	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑4) ∈ ℂ)
152	149, 151	mulcld 11170	. . . . . 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 11310	. . . . . . . 8 ⊢ (𝜑 → 0 < 𝐴)
156	90, 11, 155	ltled 11298	. . . . . . 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 42047	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))))
161		5nn 12248	. . . . . . . 8 ⊢ 5 ∈ ℕ
162		dvexp 25890	. . . . . . . 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 12287	. . . . . . . . . . 11 ⊢ (5 − 1) = 4
165	164	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (5 − 1) = 4)
166	165	oveq2d 7385	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝑧↑(5 − 1)) = (𝑧↑4))
167	166	oveq2d 7385	. . . . . . . 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 2776	. . . . . 6 ⊢ (𝜑 → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧↑5))) = (𝑧 ∈ ℂ ↦ (5 · (𝑧↑4))))
170		oveq1 7376	. . . . . 6 ⊢ (𝑧 = (2 log_b 𝑥) → (𝑧↑5) = ((2 log_b 𝑥)↑5))
171		oveq1 7376	. . . . . . 7 ⊢ (𝑧 = (2 log_b 𝑥) → (𝑧↑4) = ((2 log_b 𝑥)↑4))
172	171	oveq2d 7385	. . . . . 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 25909	. . . . 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 11145	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
175	174	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
176		0red 11153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
177	2, 174	dvmptc 25895	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
178		ioossre 13344	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ
179	178	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
180		tgioo4 24726	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
181		eqid 2729	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
182		iooretop 24686	. . . . . . 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 25900	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 1)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
185	2, 84, 111, 173, 85, 10, 184	dvmptadd 25897	. . . 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 13331	. . . . . . . 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 7385	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ⁺ ↦ (2 log_b 𝑦))) = (ℝ D (𝑦 ∈ (0(,)+∞) ↦ (2 log_b 𝑦))))
190	90	rexrd 11200	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ^*)
191		pnfxr 11204	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
192	191	a1i 11	. . . . . . 7 ⊢ (𝜑 → +∞ ∈ ℝ^*)
193	90	leidd 11720	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 0)
194		0lepnf 13069	. . . . . . . 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 42047	. . . . . 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 5192	. . . . . 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 7377	. . . 4 ⊢ (𝑦 = (((2 log_b 𝑥)↑5) + 1) → (2 log_b 𝑦) = (2 log_b (((2 log_b 𝑥)↑5) + 1)))
204		oveq1 7376	. . . . 5 ⊢ (𝑦 = (((2 log_b 𝑥)↑5) + 1) → (𝑦 · (log‘2)) = ((((2 log_b 𝑥)↑5) + 1) · (log‘2)))
205	204	oveq2d 7385	. . . 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 25909	. . 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 11178	. . 3 ⊢ (𝜑 → 2 ∈ ℂ)
209	2, 78, 113, 206, 208	dvmptcmul 25901	. 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 14085	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 log_b 𝑥)↑2) ∈ ℂ)
211	82	resqcld 14066	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑2) ∈ ℝ)
212	81	rpne0d 12976	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
213	3, 212	logcld 26512	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
214	213, 97, 63	expne0d 14093	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑2) ≠ 0)
215	5, 211, 214	redivcld 11986	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 / ((log‘2)↑2)) ∈ ℝ)
216	67	relogcld 26565	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℝ)
217		2m1e1 12283	. . . . . . 7 ⊢ (2 − 1) = 1
218		1nn0 12434	. . . . . . 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 14104	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥)↑(2 − 1)) ∈ ℝ)
222	67	rpne0d 12976	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
223	221, 9, 222	redivcld 11986	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥)↑(2 − 1)) / 𝑥) ∈ ℝ)
224	215, 223	remulcld 11180	. 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 12235	. . . 4 ⊢ 2 ∈ ℕ
229	228	a1i 11	. . 3 ⊢ (𝜑 → 2 ∈ ℕ)
230	11, 23, 155, 157, 225, 226, 227, 229	dvrelogpow2b 42049	. 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 25897	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 3911 {cpr 4587 class class class wbr 5102 ↦ cmpt 5183 ran crn 5632 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 +∞cpnf 11181 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 ℕcn 12162 2c2 12217 3c3 12218 4c4 12219 5c5 12220 ℕ₀cn0 12418 ℤcz 12505 ℤ_≥cuz 12769 ℝ⁺crp 12927 (,)cioo 13282 ↑cexp 14002 TopOpenctopn 17360 topGenctg 17376 ℂ_fldccnfld 21296 D cdv 25797 logclog 26496 log_b clogb 26707

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-ef 16009 df-sin 16011 df-cos 16012 df-pi 16014 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-mulg 18982 df-cntz 19231 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-cmp 23307 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-limc 25800 df-dv 25801 df-log 26498 df-logb 26708

This theorem is referenced by: aks4d1p1p5 42056

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