readvrec - Mathbox for Steven Nguyen

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Theorem readvrec 42357

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 11167	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
3		cnelprrecn 11168	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
4	3	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
5		dfrp2 13362	. . . . . . 7 ⊢ ℝ⁺ = (0(,)+∞)
6		mnfxr 11238	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
7	6	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → -∞ ∈ ℝ^*)
8		0xr 11228	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 0 ∈ ℝ^*)
10		pnfxr 11235	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
11	10	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → +∞ ∈ ℝ^*)
12	7, 9, 11	iocioodisjd 42315	. . . . . . . . . 10 ⊢ (⊤ → ((-∞(,]0) ∩ (0(,)+∞)) = ∅)
13	12	mptru 1547	. . . . . . . . 9 ⊢ ((-∞(,]0) ∩ (0(,)+∞)) = ∅
14	13	ineqcomi 4177	. . . . . . . 8 ⊢ ((0(,)+∞) ∩ (-∞(,]0)) = ∅
15		disjdif2 4446	. . . . . . . 8 ⊢ (((0(,)+∞) ∩ (-∞(,]0)) = ∅ → ((0(,)+∞) ∖ (-∞(,]0)) = (0(,)+∞))
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ ((0(,)+∞) ∖ (-∞(,]0)) = (0(,)+∞)
17	5, 16	eqtr4i 2756	. . . . . 6 ⊢ ℝ⁺ = ((0(,)+∞) ∖ (-∞(,]0))
18		ioosscn 13376	. . . . . . 7 ⊢ (0(,)+∞) ⊆ ℂ
19		ssdif 4110	. . . . . . 7 ⊢ ((0(,)+∞) ⊆ ℂ → ((0(,)+∞) ∖ (-∞(,]0)) ⊆ (ℂ ∖ (-∞(,]0)))
20	18, 19	ax-mp 5	. . . . . 6 ⊢ ((0(,)+∞) ∖ (-∞(,]0)) ⊆ (ℂ ∖ (-∞(,]0))
21	17, 20	eqsstri 3996	. . . . 5 ⊢ ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0))
22		redvabs.d	. . . . . . . . . . 11 ⊢ 𝐷 = (ℝ ∖ {0})
23	22	eleq2i 2821	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (ℝ ∖ {0}))
24		eldifsn 4753	. . . . . . . . . 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 11209	. . . . . . 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 15424	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (abs‘𝑥) ∈ ℝ⁺)
32	21, 31	sselid 3947	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (abs‘𝑥) ∈ (ℂ ∖ (-∞(,]0)))
33		negex 11426	. . . . . 6 ⊢ -1 ∈ V
34		1ex 11177	. . . . . 6 ⊢ 1 ∈ V
35	33, 34	ifex 4542	. . . . 5 ⊢ if(𝑥 < 0, -1, 1) ∈ V
36	35	a1i 11	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → if(𝑥 < 0, -1, 1) ∈ V)
37		eldifi 4097	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → 𝑦 ∈ ℂ)
38	37	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → 𝑦 ∈ ℂ)
39		eldifn 4098	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → ¬ 𝑦 ∈ (-∞(,]0))
40		mnflt0 13092	. . . . . . . . . 10 ⊢ -∞ < 0
41		ubioc1 13367	. . . . . . . . . 10 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ -∞ < 0) → 0 ∈ (-∞(,]0))
42	6, 8, 40, 41	mp3an 1463	. . . . . . . . 9 ⊢ 0 ∈ (-∞(,]0)
43		eleq1 2817	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑦 ∈ (-∞(,]0) ↔ 0 ∈ (-∞(,]0)))
44	42, 43	mpbiri 258	. . . . . . . 8 ⊢ (𝑦 = 0 → 𝑦 ∈ (-∞(,]0))
45	44	necon3bi 2952	. . . . . . 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 26486	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (log‘𝑦) ∈ ℂ)
49		ovexd 7425	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (1 / 𝑦) ∈ V)
50	22	redvmptabs 42355	. . . . 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 26480	. . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
53		f1of 6803	. . . . . . . . . 10 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
54	52, 53	mp1i 13	. . . . . . . . 9 ⊢ (⊤ → log:(ℂ ∖ {0})⟶ran log)
55		eqid 2730	. . . . . . . . . . 11 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
56	55	logdmss 26558	. . . . . . . . . 10 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
57	56	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0}))
58	54, 57	feqresmpt 6933	. . . . . . . 8 ⊢ (⊤ → (log ↾ (ℂ ∖ (-∞(,]0))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦)))
59	58	mptru 1547	. . . . . . 7 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦))
60	59	oveq2i 7401	. . . . . 6 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦)))
61	55	dvlog 26567	. . . . . 6 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑦))
62	60, 61	eqtr3i 2755	. . . . 5 ⊢ (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑦))
63	62	a1i 11	. . . 4 ⊢ (⊤ → (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑦)))
64		fveq2 6861	. . . 4 ⊢ (𝑦 = (abs‘𝑥) → (log‘𝑦) = (log‘(abs‘𝑥)))
65		oveq2 7398	. . . 4 ⊢ (𝑦 = (abs‘𝑥) → (1 / 𝑦) = (1 / (abs‘𝑥)))
66	2, 4, 32, 36, 48, 49, 51, 63, 64, 65	dvmptco 25883	. . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1))))
67	66	mptru 1547	. 2 ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1)))
68		ovif2 7491	. . . 4 ⊢ ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1)) = if(𝑥 < 0, ((1 / (abs‘𝑥)) · -1), ((1 / (abs‘𝑥)) · 1))
69		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ∈ ℝ)
70	69	recnd 11209	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ∈ ℂ)
71	70	abscld 15412	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) ∈ ℝ)
72	71	recnd 11209	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) ∈ ℂ)
73		simplr 768	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ≠ 0)
74	70, 73	absne0d 15423	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) ≠ 0)
75	72, 74	reccld 11958	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℂ)
76		neg1cn 12178	. . . . . . . . 9 ⊢ -1 ∈ ℂ
77	76	a1i 11	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -1 ∈ ℂ)
78	75, 77	mulcomd 11202	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → ((1 / (abs‘𝑥)) · -1) = (-1 · (1 / (abs‘𝑥))))
79	75	mulm1d 11637	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (-1 · (1 / (abs‘𝑥))) = -(1 / (abs‘𝑥)))
80		1cnd 11176	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 1 ∈ ℂ)
81	80, 72, 74	divneg2d 11979	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(1 / (abs‘𝑥)) = (1 / -(abs‘𝑥)))
82		0red 11184	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 0 ∈ ℝ)
83		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 < 0)
84	69, 82, 83	ltled 11329	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ≤ 0)
85	69, 84	absnidd 15387	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) = -𝑥)
86	85	eqcomd 2736	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -𝑥 = (abs‘𝑥))
87	70, 86	negcon1ad 11535	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(abs‘𝑥) = 𝑥)
88	87	oveq2d 7406	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (1 / -(abs‘𝑥)) = (1 / 𝑥))
89	81, 88	eqtrd 2765	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(1 / (abs‘𝑥)) = (1 / 𝑥))
90	78, 79, 89	3eqtrd 2769	. . . . . 6 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → ((1 / (abs‘𝑥)) · -1) = (1 / 𝑥))
91	25, 90	sylanb 581	. . . . 5 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 < 0) → ((1 / (abs‘𝑥)) · -1) = (1 / 𝑥))
92		recn 11165	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
93	92	abscld 15412	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (abs‘𝑥) ∈ ℝ)
94	93	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (abs‘𝑥) ∈ ℝ)
95	92	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → 𝑥 ∈ ℂ)
96		simplr 768	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → 𝑥 ≠ 0)
97	95, 96	absne0d 15423	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (abs‘𝑥) ≠ 0)
98	94, 97	rereccld 12016	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℝ)
99	98	recnd 11209	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℂ)
100	99	mulridd 11198	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → ((1 / (abs‘𝑥)) · 1) = (1 / (abs‘𝑥)))
101		simpll 766	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → 𝑥 ∈ ℝ)
102		0red 11184	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) → 0 ∈ ℝ)
103		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) → 𝑥 ∈ ℝ)
104	102, 103	lenltd 11327	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) → (0 ≤ 𝑥 ↔ ¬ 𝑥 < 0))
105	104	biimpar 477	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → 0 ≤ 𝑥)
106	101, 105	absidd 15396	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (abs‘𝑥) = 𝑥)
107	106	oveq2d 7406	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) = (1 / 𝑥))
108	100, 107	eqtrd 2765	. . . . . 6 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → ((1 / (abs‘𝑥)) · 1) = (1 / 𝑥))
109	25, 108	sylanb 581	. . . . 5 ⊢ ((𝑥 ∈ 𝐷 ∧ ¬ 𝑥 < 0) → ((1 / (abs‘𝑥)) · 1) = (1 / 𝑥))
110	91, 109	ifeqda 4528	. . . 4 ⊢ (𝑥 ∈ 𝐷 → if(𝑥 < 0, ((1 / (abs‘𝑥)) · -1), ((1 / (abs‘𝑥)) · 1)) = (1 / 𝑥))
111	68, 110	eqtrid 2777	. . 3 ⊢ (𝑥 ∈ 𝐷 → ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1)) = (1 / 𝑥))
112	111	mpteq2ia 5205	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1))) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))
113	67, 112	eqtri 2753	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 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2926 Vcvv 3450 ∖ cdif 3914 ∩ cin 3916 ⊆ wss 3917 ∅c0 4299 ifcif 4491 {csn 4592 {cpr 4594 class class class wbr 5110 ↦ cmpt 5191 ran crn 5642 ↾ cres 5643 ⟶wf 6510 –1-1-onto→wf1o 6513 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 · cmul 11080 +∞cpnf 11212 -∞cmnf 11213 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 -cneg 11413 / cdiv 11842 ℝ⁺crp 12958 (,)cioo 13313 (,]cioc 13314 abscabs 15207 D cdv 25771 logclog 26470

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 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 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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ioc 13318 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-fac 14246 df-bc 14275 df-hash 14303 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15444 df-clim 15461 df-rlim 15462 df-sum 15660 df-ef 16040 df-sin 16042 df-cos 16043 df-tan 16044 df-pi 16045 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-haus 23209 df-cmp 23281 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-limc 25774 df-dv 25775 df-log 26472

This theorem is referenced by: readvcot 42359

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