readvrec - Mathbox for Steven Nguyen

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Theorem readvrec 42370

Description: For real numbers, the antiderivative of 1/x is ln|x|. (Contributed by SN, 30-Sep-2025.)

**Hypothesis**
Ref	Expression
redvabs.d	⊢ 𝐷 = (ℝ ∖ {0})

**Assertion**
Ref	Expression
readvrec	⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))

Distinct variable group: 𝑥,𝐷

Proof of Theorem readvrec Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reelprrecn 11244	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
3		cnelprrecn 11245	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
4	3	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
5		dfrp2 13432	. . . . . . 7 ⊢ ℝ⁺ = (0(,)+∞)
6		mnfxr 11315	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
7	6	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → -∞ ∈ ℝ^*)
8		0xr 11305	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 0 ∈ ℝ^*)
10		pnfxr 11312	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
11	10	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → +∞ ∈ ℝ^*)
12	7, 9, 11	iocioodisjd 42333	. . . . . . . . . 10 ⊢ (⊤ → ((-∞(,]0) ∩ (0(,)+∞)) = ∅)
13	12	mptru 1543	. . . . . . . . 9 ⊢ ((-∞(,]0) ∩ (0(,)+∞)) = ∅
14	13	ineqcomi 4218	. . . . . . . 8 ⊢ ((0(,)+∞) ∩ (-∞(,]0)) = ∅
15		disjdif2 4485	. . . . . . . 8 ⊢ (((0(,)+∞) ∩ (-∞(,]0)) = ∅ → ((0(,)+∞) ∖ (-∞(,]0)) = (0(,)+∞))
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ ((0(,)+∞) ∖ (-∞(,]0)) = (0(,)+∞)
17	5, 16	eqtr4i 2765	. . . . . 6 ⊢ ℝ⁺ = ((0(,)+∞) ∖ (-∞(,]0))
18		ioosscn 13445	. . . . . . 7 ⊢ (0(,)+∞) ⊆ ℂ
19		ssdif 4153	. . . . . . 7 ⊢ ((0(,)+∞) ⊆ ℂ → ((0(,)+∞) ∖ (-∞(,]0)) ⊆ (ℂ ∖ (-∞(,]0)))
20	18, 19	ax-mp 5	. . . . . 6 ⊢ ((0(,)+∞) ∖ (-∞(,]0)) ⊆ (ℂ ∖ (-∞(,]0))
21	17, 20	eqsstri 4029	. . . . 5 ⊢ ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0))
22		redvabs.d	. . . . . . . . . . 11 ⊢ 𝐷 = (ℝ ∖ {0})
23	22	eleq2i 2830	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (ℝ ∖ {0}))
24		eldifsn 4790	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℝ ∖ {0}) ↔ (𝑥 ∈ ℝ ∧ 𝑥 ≠ 0))
25	23, 24	bitri 275	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℝ ∧ 𝑥 ≠ 0))
26	25	simplbi 497	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℝ)
27	26	recnd 11286	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
28	27	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ℂ)
29	25	simprbi 496	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
30	29	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → 𝑥 ≠ 0)
31	28, 30	absrpcld 15483	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (abs‘𝑥) ∈ ℝ⁺)
32	21, 31	sselid 3992	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (abs‘𝑥) ∈ (ℂ ∖ (-∞(,]0)))
33		negex 11503	. . . . . 6 ⊢ -1 ∈ V
34		1ex 11254	. . . . . 6 ⊢ 1 ∈ V
35	33, 34	ifex 4580	. . . . 5 ⊢ if(𝑥 < 0, -1, 1) ∈ V
36	35	a1i 11	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → if(𝑥 < 0, -1, 1) ∈ V)
37		eldifi 4140	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → 𝑦 ∈ ℂ)
38	37	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → 𝑦 ∈ ℂ)
39		eldifn 4141	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → ¬ 𝑦 ∈ (-∞(,]0))
40		mnflt0 13164	. . . . . . . . . 10 ⊢ -∞ < 0
41		ubioc1 13436	. . . . . . . . . 10 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ -∞ < 0) → 0 ∈ (-∞(,]0))
42	6, 8, 40, 41	mp3an 1460	. . . . . . . . 9 ⊢ 0 ∈ (-∞(,]0)
43		eleq1 2826	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑦 ∈ (-∞(,]0) ↔ 0 ∈ (-∞(,]0)))
44	42, 43	mpbiri 258	. . . . . . . 8 ⊢ (𝑦 = 0 → 𝑦 ∈ (-∞(,]0))
45	44	necon3bi 2964	. . . . . . 7 ⊢ (¬ 𝑦 ∈ (-∞(,]0) → 𝑦 ≠ 0)
46	39, 45	syl 17	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → 𝑦 ≠ 0)
47	46	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → 𝑦 ≠ 0)
48	38, 47	logcld 26626	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (log‘𝑦) ∈ ℂ)
49		ovexd 7465	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (1 / 𝑦) ∈ V)
50	22	redvmptabs 42368	. . . . 5 ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (abs‘𝑥))) = (𝑥 ∈ 𝐷 ↦ if(𝑥 < 0, -1, 1))
51	50	a1i 11	. . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ 𝐷 ↦ (abs‘𝑥))) = (𝑥 ∈ 𝐷 ↦ if(𝑥 < 0, -1, 1)))
52		logf1o 26620	. . . . . . . . . . . . 13 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
53		f1of 6848	. . . . . . . . . . . . 13 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
54	52, 53	ax-mp 5	. . . . . . . . . . . 12 ⊢ log:(ℂ ∖ {0})⟶ran log
55	54	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → log:(ℂ ∖ {0})⟶ran log)
56	55	feqmptd 6976	. . . . . . . . . 10 ⊢ (⊤ → log = (𝑦 ∈ (ℂ ∖ {0}) ↦ (log‘𝑦)))
57	56	mptru 1543	. . . . . . . . 9 ⊢ log = (𝑦 ∈ (ℂ ∖ {0}) ↦ (log‘𝑦))
58	57	reseq1i 5995	. . . . . . . 8 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))) = ((𝑦 ∈ (ℂ ∖ {0}) ↦ (log‘𝑦)) ↾ (ℂ ∖ (-∞(,]0)))
59		c0ex 11252	. . . . . . . . . . 11 ⊢ 0 ∈ V
60	59	snss 4789	. . . . . . . . . 10 ⊢ (0 ∈ (-∞(,]0) ↔ {0} ⊆ (-∞(,]0))
61	42, 60	mpbi 230	. . . . . . . . 9 ⊢ {0} ⊆ (-∞(,]0)
62		sscon 4152	. . . . . . . . 9 ⊢ ({0} ⊆ (-∞(,]0) → (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0}))
63		resmpt 6056	. . . . . . . . 9 ⊢ ((ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0}) → ((𝑦 ∈ (ℂ ∖ {0}) ↦ (log‘𝑦)) ↾ (ℂ ∖ (-∞(,]0))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦)))
64	61, 62, 63	mp2b 10	. . . . . . . 8 ⊢ ((𝑦 ∈ (ℂ ∖ {0}) ↦ (log‘𝑦)) ↾ (ℂ ∖ (-∞(,]0))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦))
65	58, 64	eqtr2i 2763	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦)) = (log ↾ (ℂ ∖ (-∞(,]0)))
66	65	oveq2i 7441	. . . . . 6 ⊢ (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦))) = (ℂ D (log ↾ (ℂ ∖ (-∞(,]0))))
67		eqid 2734	. . . . . . 7 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
68	67	dvlog 26707	. . . . . 6 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑦))
69	66, 68	eqtri 2762	. . . . 5 ⊢ (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑦))
70	69	a1i 11	. . . 4 ⊢ (⊤ → (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑦)))
71		fveq2 6906	. . . 4 ⊢ (𝑦 = (abs‘𝑥) → (log‘𝑦) = (log‘(abs‘𝑥)))
72		oveq2 7438	. . . 4 ⊢ (𝑦 = (abs‘𝑥) → (1 / 𝑦) = (1 / (abs‘𝑥)))
73	2, 4, 32, 36, 48, 49, 51, 70, 71, 72	dvmptco 26024	. . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1))))
74	73	mptru 1543	. 2 ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1)))
75		ovif2 7531	. . . 4 ⊢ ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1)) = if(𝑥 < 0, ((1 / (abs‘𝑥)) · -1), ((1 / (abs‘𝑥)) · 1))
76		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ∈ ℝ)
77	76	recnd 11286	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ∈ ℂ)
78	77	abscld 15471	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) ∈ ℝ)
79	78	recnd 11286	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) ∈ ℂ)
80		simplr 769	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ≠ 0)
81	77, 80	absne0d 15482	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) ≠ 0)
82	79, 81	reccld 12033	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℂ)
83		neg1cn 12377	. . . . . . . . 9 ⊢ -1 ∈ ℂ
84	83	a1i 11	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -1 ∈ ℂ)
85	82, 84	mulcomd 11279	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → ((1 / (abs‘𝑥)) · -1) = (-1 · (1 / (abs‘𝑥))))
86	82	mulm1d 11712	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (-1 · (1 / (abs‘𝑥))) = -(1 / (abs‘𝑥)))
87		1cnd 11253	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 1 ∈ ℂ)
88	87, 79, 81	divneg2d 12054	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(1 / (abs‘𝑥)) = (1 / -(abs‘𝑥)))
89		0red 11261	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 0 ∈ ℝ)
90		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 < 0)
91	76, 89, 90	ltled 11406	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ≤ 0)
92	76, 91	absnidd 15448	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) = -𝑥)
93	92	eqcomd 2740	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -𝑥 = (abs‘𝑥))
94	77, 93	negcon1ad 11612	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(abs‘𝑥) = 𝑥)
95	94	oveq2d 7446	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (1 / -(abs‘𝑥)) = (1 / 𝑥))
96	88, 95	eqtrd 2774	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(1 / (abs‘𝑥)) = (1 / 𝑥))
97	85, 86, 96	3eqtrd 2778	. . . . . 6 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → ((1 / (abs‘𝑥)) · -1) = (1 / 𝑥))
98	25, 97	sylanb 581	. . . . 5 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 < 0) → ((1 / (abs‘𝑥)) · -1) = (1 / 𝑥))
99		recn 11242	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
100	99	abscld 15471	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (abs‘𝑥) ∈ ℝ)
101	100	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (abs‘𝑥) ∈ ℝ)
102	99	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → 𝑥 ∈ ℂ)
103		simplr 769	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → 𝑥 ≠ 0)
104	102, 103	absne0d 15482	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (abs‘𝑥) ≠ 0)
105	101, 104	rereccld 12091	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℝ)
106	105	recnd 11286	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℂ)
107	106	mulridd 11275	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → ((1 / (abs‘𝑥)) · 1) = (1 / (abs‘𝑥)))
108		simpll 767	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → 𝑥 ∈ ℝ)
109		0red 11261	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) → 0 ∈ ℝ)
110		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) → 𝑥 ∈ ℝ)
111	109, 110	lenltd 11404	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) → (0 ≤ 𝑥 ↔ ¬ 𝑥 < 0))
112	111	biimpar 477	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → 0 ≤ 𝑥)
113	108, 112	absidd 15457	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (abs‘𝑥) = 𝑥)
114	113	oveq2d 7446	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) = (1 / 𝑥))
115	107, 114	eqtrd 2774	. . . . . 6 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → ((1 / (abs‘𝑥)) · 1) = (1 / 𝑥))
116	25, 115	sylanb 581	. . . . 5 ⊢ ((𝑥 ∈ 𝐷 ∧ ¬ 𝑥 < 0) → ((1 / (abs‘𝑥)) · 1) = (1 / 𝑥))
117	98, 116	ifeqda 4566	. . . 4 ⊢ (𝑥 ∈ 𝐷 → if(𝑥 < 0, ((1 / (abs‘𝑥)) · -1), ((1 / (abs‘𝑥)) · 1)) = (1 / 𝑥))
118	75, 117	eqtrid 2786	. . 3 ⊢ (𝑥 ∈ 𝐷 → ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1)) = (1 / 𝑥))
119	118	mpteq2ia 5250	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1))) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))
120	74, 119	eqtri 2762	1 ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1536 ⊤wtru 1537 ∈ wcel 2105 ≠ wne 2937 Vcvv 3477 ∖ cdif 3959 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 ifcif 4530 {csn 4630 {cpr 4632 class class class wbr 5147 ↦ cmpt 5230 ran crn 5689 ↾ cres 5690 ⟶wf 6558 –1-1-onto→wf1o 6561 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 · cmul 11157 +∞cpnf 11289 -∞cmnf 11290 ℝ^*cxr 11291 < clt 11292 ≤ cle 11293 -cneg 11490 / cdiv 11917 ℝ⁺crp 13031 (,)cioo 13383 (,]cioc 13384 abscabs 15269 D cdv 25912 logclog 26610

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-tan 16103 df-pi 16104 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-cmp 23410 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916 df-log 26612

This theorem is referenced by: (None)

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