readvrec - Mathbox for Steven Nguyen

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Theorem readvrec 42976

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 13400	. . . . . . 7 ⊢ ℝ⁺ = (0(,)+∞)
6		mnfxr 11241	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
7	6	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → -∞ ∈ ℝ^*)
8		0xr 11231	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 0 ∈ ℝ^*)
10		pnfxr 11238	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
11	10	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → +∞ ∈ ℝ^*)
12	7, 9, 11	iocioodisjd 42934	. . . . . . . . . 10 ⊢ (⊤ → ((-∞(,]0) ∩ (0(,)+∞)) = ∅)
13	12	mptru 1569	. . . . . . . . 9 ⊢ ((-∞(,]0) ∩ (0(,)+∞)) = ∅
14	13	ineqcomi 4165	. . . . . . . 8 ⊢ ((0(,)+∞) ∩ (-∞(,]0)) = ∅
15		disjdif2 4436	. . . . . . . 8 ⊢ (((0(,)+∞) ∩ (-∞(,]0)) = ∅ → ((0(,)+∞) ∖ (-∞(,]0)) = (0(,)+∞))
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ ((0(,)+∞) ∖ (-∞(,]0)) = (0(,)+∞)
17	5, 16	eqtr4i 2790	. . . . . 6 ⊢ ℝ⁺ = ((0(,)+∞) ∖ (-∞(,]0))
18		ioosscn 13414	. . . . . . 7 ⊢ (0(,)+∞) ⊆ ℂ
19		ssdif 4099	. . . . . . 7 ⊢ ((0(,)+∞) ⊆ ℂ → ((0(,)+∞) ∖ (-∞(,]0)) ⊆ (ℂ ∖ (-∞(,]0)))
20	18, 19	ax-mp 5	. . . . . 6 ⊢ ((0(,)+∞) ∖ (-∞(,]0)) ⊆ (ℂ ∖ (-∞(,]0))
21	17, 20	eqsstri 3984	. . . . 5 ⊢ ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0))
22		redvabs.d	. . . . . . . . . . 11 ⊢ 𝐷 = (ℝ ∖ {0})
23	22	eleq2i 2856	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (ℝ ∖ {0}))
24		eldifsn 4748	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℝ ∖ {0}) ↔ (𝑥 ∈ ℝ ∧ 𝑥 ≠ 0))
25	23, 24	bitri 277	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℝ ∧ 𝑥 ≠ 0))
26	25	simplbi 500	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℝ)
27	26	recnd 11212	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
28	27	adantl 485	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ℂ)
29	25	simprbi 501	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
30	29	adantl 485	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → 𝑥 ≠ 0)
31	28, 30	absrpcld 15480	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (abs‘𝑥) ∈ ℝ⁺)
32	21, 31	sselid 3936	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (abs‘𝑥) ∈ (ℂ ∖ (-∞(,]0)))
33		negex 11430	. . . . . 6 ⊢ -1 ∈ V
34		1ex 11178	. . . . . 6 ⊢ 1 ∈ V
35	33, 34	ifex 4533	. . . . 5 ⊢ if(𝑥 < 0, -1, 1) ∈ V
36	35	a1i 11	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → if(𝑥 < 0, -1, 1) ∈ V)
37		eldifi 4086	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → 𝑦 ∈ ℂ)
38	37	adantl 485	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → 𝑦 ∈ ℂ)
39		eldifn 4087	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → ¬ 𝑦 ∈ (-∞(,]0))
40		mnflt0 13129	. . . . . . . . . 10 ⊢ -∞ < 0
41		ubioc1 13405	. . . . . . . . . 10 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ -∞ < 0) → 0 ∈ (-∞(,]0))
42	6, 8, 40, 41	mp3an 1484	. . . . . . . . 9 ⊢ 0 ∈ (-∞(,]0)
43		eleq1 2852	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑦 ∈ (-∞(,]0) ↔ 0 ∈ (-∞(,]0)))
44	42, 43	mpbiri 260	. . . . . . . 8 ⊢ (𝑦 = 0 → 𝑦 ∈ (-∞(,]0))
45	44	necon3bi 2985	. . . . . . 7 ⊢ (¬ 𝑦 ∈ (-∞(,]0) → 𝑦 ≠ 0)
46	39, 45	syl 17	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → 𝑦 ≠ 0)
47	46	adantl 485	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → 𝑦 ≠ 0)
48	38, 47	logcld 26637	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (log‘𝑦) ∈ ℂ)
49		ovexd 7433	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (1 / 𝑦) ∈ V)
50	22	redvmptabs 42974	. . . . 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 26631	. . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
53		f1of 6808	. . . . . . . . . 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 2764	. . . . . . . . . . 11 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
56	55	logdmss 26709	. . . . . . . . . 10 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
57	56	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0}))
58	54, 57	feqresmpt 6938	. . . . . . . 8 ⊢ (⊤ → (log ↾ (ℂ ∖ (-∞(,]0))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦)))
59	58	mptru 1569	. . . . . . 7 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦))
60	59	oveq2i 7409	. . . . . 6 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦)))
61	55	dvlog 26718	. . . . . 6 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑦))
62	60, 61	eqtr3i 2789	. . . . 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 6869	. . . 4 ⊢ (𝑦 = (abs‘𝑥) → (log‘𝑦) = (log‘(abs‘𝑥)))
65		oveq2 7406	. . . 4 ⊢ (𝑦 = (abs‘𝑥) → (1 / 𝑦) = (1 / (abs‘𝑥)))
66	2, 4, 32, 36, 48, 49, 51, 63, 64, 65	dvmptco 26036	. . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1))))
67	66	mptru 1569	. 2 ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1)))
68		ovif2 7497	. . . 4 ⊢ ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1)) = if(𝑥 < 0, ((1 / (abs‘𝑥)) · -1), ((1 / (abs‘𝑥)) · 1))
69		simpll 776	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ∈ ℝ)
70	69	recnd 11212	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ∈ ℂ)
71	70	abscld 15468	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) ∈ ℝ)
72	71	recnd 11212	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) ∈ ℂ)
73		simplr 778	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ≠ 0)
74	70, 73	absne0d 15479	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) ≠ 0)
75	72, 74	reccld 11962	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℂ)
76		neg1cn 12182	. . . . . . . . 9 ⊢ -1 ∈ ℂ
77	76	a1i 11	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -1 ∈ ℂ)
78	75, 77	mulcomd 11205	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → ((1 / (abs‘𝑥)) · -1) = (-1 · (1 / (abs‘𝑥))))
79	75	mulm1d 11641	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (-1 · (1 / (abs‘𝑥))) = -(1 / (abs‘𝑥)))
80		1cnd 11177	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 1 ∈ ℂ)
81	80, 72, 74	divneg2d 11983	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(1 / (abs‘𝑥)) = (1 / -(abs‘𝑥)))
82		0red 11186	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 0 ∈ ℝ)
83		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 < 0)
84	69, 82, 83	ltled 11333	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ≤ 0)
85	69, 84	absnidd 15443	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) = -𝑥)
86	85	eqcomd 2770	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -𝑥 = (abs‘𝑥))
87	70, 86	negcon1ad 11539	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(abs‘𝑥) = 𝑥)
88	87	oveq2d 7414	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (1 / -(abs‘𝑥)) = (1 / 𝑥))
89	81, 88	eqtrd 2799	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(1 / (abs‘𝑥)) = (1 / 𝑥))
90	78, 79, 89	3eqtrd 2803	. . . . . 6 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → ((1 / (abs‘𝑥)) · -1) = (1 / 𝑥))
91	25, 90	sylanb 590	. . . . 5 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 < 0) → ((1 / (abs‘𝑥)) · -1) = (1 / 𝑥))
92		recn 11165	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
93	92	abscld 15468	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (abs‘𝑥) ∈ ℝ)
94	93	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (abs‘𝑥) ∈ ℝ)
95	92	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → 𝑥 ∈ ℂ)
96		simplr 778	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → 𝑥 ≠ 0)
97	95, 96	absne0d 15479	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (abs‘𝑥) ≠ 0)
98	94, 97	rereccld 12020	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℝ)
99	98	recnd 11212	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℂ)
100	99	mulridd 11201	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → ((1 / (abs‘𝑥)) · 1) = (1 / (abs‘𝑥)))
101		simpll 776	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → 𝑥 ∈ ℝ)
102		0red 11186	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) → 0 ∈ ℝ)
103		simpl 486	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) → 𝑥 ∈ ℝ)
104	102, 103	lenltd 11331	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) → (0 ≤ 𝑥 ↔ ¬ 𝑥 < 0))
105	104	biimpar 481	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → 0 ≤ 𝑥)
106	101, 105	absidd 15452	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (abs‘𝑥) = 𝑥)
107	106	oveq2d 7414	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) = (1 / 𝑥))
108	100, 107	eqtrd 2799	. . . . . 6 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → ((1 / (abs‘𝑥)) · 1) = (1 / 𝑥))
109	25, 108	sylanb 590	. . . . 5 ⊢ ((𝑥 ∈ 𝐷 ∧ ¬ 𝑥 < 0) → ((1 / (abs‘𝑥)) · 1) = (1 / 𝑥))
110	91, 109	ifeqda 4519	. . . 4 ⊢ (𝑥 ∈ 𝐷 → if(𝑥 < 0, ((1 / (abs‘𝑥)) · -1), ((1 / (abs‘𝑥)) · 1)) = (1 / 𝑥))
111	68, 110	eqtrid 2811	. . 3 ⊢ (𝑥 ∈ 𝐷 → ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1)) = (1 / 𝑥))
112	111	mpteq2ia 5197	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1))) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))
113	67, 112	eqtri 2787	1 ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1562 ⊤wtru 1563 ∈ wcel 2144 ≠ wne 2959 Vcvv 3456 ∖ cdif 3903 ∩ cin 3905 ⊆ wss 3906 ∅c0 4287 ifcif 4482 {csn 4584 {cpr 4586 class class class wbr 5102 ↦ cmpt 5183 ran crn 5650 ↾ cres 5651 ⟶wf 6519 –1-1-onto→wf1o 6522 ‘cfv 6523 (class class class)co 7398 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 · cmul 11080 +∞cpnf 11215 -∞cmnf 11216 ℝ^*cxr 11217 < clt 11218 ≤ cle 11219 -cneg 11417 / cdiv 11846 ℝ⁺crp 12995 (,)cioo 13351 (,]cioc 13352 abscabs 15263 D cdv 25927 logclog 26621

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-tan 16103 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

This theorem is referenced by: readvcot 42978

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