readvrec - Mathbox for Steven Nguyen

	Mathbox for Steven Nguyen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > readvrec		Structured version Visualization version GIF version

Theorem readvrec 42343

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 11136	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
3		cnelprrecn 11137	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
4	3	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
5		dfrp2 13331	. . . . . . 7 ⊢ ℝ⁺ = (0(,)+∞)
6		mnfxr 11207	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
7	6	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → -∞ ∈ ℝ^*)
8		0xr 11197	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 0 ∈ ℝ^*)
10		pnfxr 11204	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
11	10	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → +∞ ∈ ℝ^*)
12	7, 9, 11	iocioodisjd 42301	. . . . . . . . . 10 ⊢ (⊤ → ((-∞(,]0) ∩ (0(,)+∞)) = ∅)
13	12	mptru 1547	. . . . . . . . 9 ⊢ ((-∞(,]0) ∩ (0(,)+∞)) = ∅
14	13	ineqcomi 4170	. . . . . . . 8 ⊢ ((0(,)+∞) ∩ (-∞(,]0)) = ∅
15		disjdif2 4439	. . . . . . . 8 ⊢ (((0(,)+∞) ∩ (-∞(,]0)) = ∅ → ((0(,)+∞) ∖ (-∞(,]0)) = (0(,)+∞))
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ ((0(,)+∞) ∖ (-∞(,]0)) = (0(,)+∞)
17	5, 16	eqtr4i 2755	. . . . . 6 ⊢ ℝ⁺ = ((0(,)+∞) ∖ (-∞(,]0))
18		ioosscn 13345	. . . . . . 7 ⊢ (0(,)+∞) ⊆ ℂ
19		ssdif 4103	. . . . . . 7 ⊢ ((0(,)+∞) ⊆ ℂ → ((0(,)+∞) ∖ (-∞(,]0)) ⊆ (ℂ ∖ (-∞(,]0)))
20	18, 19	ax-mp 5	. . . . . 6 ⊢ ((0(,)+∞) ∖ (-∞(,]0)) ⊆ (ℂ ∖ (-∞(,]0))
21	17, 20	eqsstri 3990	. . . . 5 ⊢ ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0))
22		redvabs.d	. . . . . . . . . . 11 ⊢ 𝐷 = (ℝ ∖ {0})
23	22	eleq2i 2820	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (ℝ ∖ {0}))
24		eldifsn 4746	. . . . . . . . . 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 11178	. . . . . . 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 15393	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (abs‘𝑥) ∈ ℝ⁺)
32	21, 31	sselid 3941	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (abs‘𝑥) ∈ (ℂ ∖ (-∞(,]0)))
33		negex 11395	. . . . . 6 ⊢ -1 ∈ V
34		1ex 11146	. . . . . 6 ⊢ 1 ∈ V
35	33, 34	ifex 4535	. . . . 5 ⊢ if(𝑥 < 0, -1, 1) ∈ V
36	35	a1i 11	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → if(𝑥 < 0, -1, 1) ∈ V)
37		eldifi 4090	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → 𝑦 ∈ ℂ)
38	37	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → 𝑦 ∈ ℂ)
39		eldifn 4091	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → ¬ 𝑦 ∈ (-∞(,]0))
40		mnflt0 13061	. . . . . . . . . 10 ⊢ -∞ < 0
41		ubioc1 13336	. . . . . . . . . 10 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ -∞ < 0) → 0 ∈ (-∞(,]0))
42	6, 8, 40, 41	mp3an 1463	. . . . . . . . 9 ⊢ 0 ∈ (-∞(,]0)
43		eleq1 2816	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑦 ∈ (-∞(,]0) ↔ 0 ∈ (-∞(,]0)))
44	42, 43	mpbiri 258	. . . . . . . 8 ⊢ (𝑦 = 0 → 𝑦 ∈ (-∞(,]0))
45	44	necon3bi 2951	. . . . . . 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 26512	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (log‘𝑦) ∈ ℂ)
49		ovexd 7404	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (1 / 𝑦) ∈ V)
50	22	redvmptabs 42341	. . . . 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 26506	. . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
53		f1of 6782	. . . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
56	55	logdmss 26584	. . . . . . . . . 10 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
57	56	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0}))
58	54, 57	feqresmpt 6912	. . . . . . . 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 7380	. . . . . 6 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦)))
61	55	dvlog 26593	. . . . . 6 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑦))
62	60, 61	eqtr3i 2754	. . . . 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 6840	. . . 4 ⊢ (𝑦 = (abs‘𝑥) → (log‘𝑦) = (log‘(abs‘𝑥)))
65		oveq2 7377	. . . 4 ⊢ (𝑦 = (abs‘𝑥) → (1 / 𝑦) = (1 / (abs‘𝑥)))
66	2, 4, 32, 36, 48, 49, 51, 63, 64, 65	dvmptco 25909	. . 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 7468	. . . 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 11178	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ∈ ℂ)
71	70	abscld 15381	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) ∈ ℝ)
72	71	recnd 11178	. . . . . . . . 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 15392	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) ≠ 0)
75	72, 74	reccld 11927	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℂ)
76		neg1cn 12147	. . . . . . . . 9 ⊢ -1 ∈ ℂ
77	76	a1i 11	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -1 ∈ ℂ)
78	75, 77	mulcomd 11171	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → ((1 / (abs‘𝑥)) · -1) = (-1 · (1 / (abs‘𝑥))))
79	75	mulm1d 11606	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (-1 · (1 / (abs‘𝑥))) = -(1 / (abs‘𝑥)))
80		1cnd 11145	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 1 ∈ ℂ)
81	80, 72, 74	divneg2d 11948	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(1 / (abs‘𝑥)) = (1 / -(abs‘𝑥)))
82		0red 11153	. . . . . . . . . . . . 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 11298	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ≤ 0)
85	69, 84	absnidd 15356	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) = -𝑥)
86	85	eqcomd 2735	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -𝑥 = (abs‘𝑥))
87	70, 86	negcon1ad 11504	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(abs‘𝑥) = 𝑥)
88	87	oveq2d 7385	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (1 / -(abs‘𝑥)) = (1 / 𝑥))
89	81, 88	eqtrd 2764	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(1 / (abs‘𝑥)) = (1 / 𝑥))
90	78, 79, 89	3eqtrd 2768	. . . . . 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 11134	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
93	92	abscld 15381	. . . . . . . . . . 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 15392	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (abs‘𝑥) ≠ 0)
98	94, 97	rereccld 11985	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℝ)
99	98	recnd 11178	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℂ)
100	99	mulridd 11167	. . . . . . 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 11153	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) → 0 ∈ ℝ)
103		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) → 𝑥 ∈ ℝ)
104	102, 103	lenltd 11296	. . . . . . . . . 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 15365	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (abs‘𝑥) = 𝑥)
107	106	oveq2d 7385	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) = (1 / 𝑥))
108	100, 107	eqtrd 2764	. . . . . 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 4521	. . . 4 ⊢ (𝑥 ∈ 𝐷 → if(𝑥 < 0, ((1 / (abs‘𝑥)) · -1), ((1 / (abs‘𝑥)) · 1)) = (1 / 𝑥))
111	68, 110	eqtrid 2776	. . 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 2752	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 2925 Vcvv 3444 ∖ cdif 3908 ∩ cin 3910 ⊆ wss 3911 ∅c0 4292 ifcif 4484 {csn 4585 {cpr 4587 class class class wbr 5102 ↦ cmpt 5183 ran crn 5632 ↾ cres 5633 ⟶wf 6495 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 · cmul 11049 +∞cpnf 11181 -∞cmnf 11182 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 -cneg 11382 / cdiv 11811 ℝ⁺crp 12927 (,)cioo 13282 (,]cioc 13283 abscabs 15176 D cdv 25797 logclog 26496

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-tan 16013 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

This theorem is referenced by: readvcot 42345

W3C validator