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 42392

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 11247	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
3		cnelprrecn 11248	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
4	3	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
5		dfrp2 13436	. . . . . . 7 ⊢ ℝ⁺ = (0(,)+∞)
6		mnfxr 11318	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
7	6	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → -∞ ∈ ℝ^*)
8		0xr 11308	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 0 ∈ ℝ^*)
10		pnfxr 11315	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
11	10	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → +∞ ∈ ℝ^*)
12	7, 9, 11	iocioodisjd 42355	. . . . . . . . . 10 ⊢ (⊤ → ((-∞(,]0) ∩ (0(,)+∞)) = ∅)
13	12	mptru 1547	. . . . . . . . 9 ⊢ ((-∞(,]0) ∩ (0(,)+∞)) = ∅
14	13	ineqcomi 4211	. . . . . . . 8 ⊢ ((0(,)+∞) ∩ (-∞(,]0)) = ∅
15		disjdif2 4480	. . . . . . . 8 ⊢ (((0(,)+∞) ∩ (-∞(,]0)) = ∅ → ((0(,)+∞) ∖ (-∞(,]0)) = (0(,)+∞))
16	14, 15	ax-mp 5	. . . . . . 7 ⊢ ((0(,)+∞) ∖ (-∞(,]0)) = (0(,)+∞)
17	5, 16	eqtr4i 2768	. . . . . 6 ⊢ ℝ⁺ = ((0(,)+∞) ∖ (-∞(,]0))
18		ioosscn 13449	. . . . . . 7 ⊢ (0(,)+∞) ⊆ ℂ
19		ssdif 4144	. . . . . . 7 ⊢ ((0(,)+∞) ⊆ ℂ → ((0(,)+∞) ∖ (-∞(,]0)) ⊆ (ℂ ∖ (-∞(,]0)))
20	18, 19	ax-mp 5	. . . . . 6 ⊢ ((0(,)+∞) ∖ (-∞(,]0)) ⊆ (ℂ ∖ (-∞(,]0))
21	17, 20	eqsstri 4030	. . . . 5 ⊢ ℝ⁺ ⊆ (ℂ ∖ (-∞(,]0))
22		redvabs.d	. . . . . . . . . . 11 ⊢ 𝐷 = (ℝ ∖ {0})
23	22	eleq2i 2833	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (ℝ ∖ {0}))
24		eldifsn 4786	. . . . . . . . . 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 11289	. . . . . . 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 15487	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (abs‘𝑥) ∈ ℝ⁺)
32	21, 31	sselid 3981	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (abs‘𝑥) ∈ (ℂ ∖ (-∞(,]0)))
33		negex 11506	. . . . . 6 ⊢ -1 ∈ V
34		1ex 11257	. . . . . 6 ⊢ 1 ∈ V
35	33, 34	ifex 4576	. . . . 5 ⊢ if(𝑥 < 0, -1, 1) ∈ V
36	35	a1i 11	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → if(𝑥 < 0, -1, 1) ∈ V)
37		eldifi 4131	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → 𝑦 ∈ ℂ)
38	37	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → 𝑦 ∈ ℂ)
39		eldifn 4132	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)) → ¬ 𝑦 ∈ (-∞(,]0))
40		mnflt0 13167	. . . . . . . . . 10 ⊢ -∞ < 0
41		ubioc1 13440	. . . . . . . . . 10 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ -∞ < 0) → 0 ∈ (-∞(,]0))
42	6, 8, 40, 41	mp3an 1463	. . . . . . . . 9 ⊢ 0 ∈ (-∞(,]0)
43		eleq1 2829	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝑦 ∈ (-∞(,]0) ↔ 0 ∈ (-∞(,]0)))
44	42, 43	mpbiri 258	. . . . . . . 8 ⊢ (𝑦 = 0 → 𝑦 ∈ (-∞(,]0))
45	44	necon3bi 2967	. . . . . . 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 26612	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (log‘𝑦) ∈ ℂ)
49		ovexd 7466	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ (-∞(,]0))) → (1 / 𝑦) ∈ V)
50	22	redvmptabs 42390	. . . . 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 26606	. . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
53		f1of 6848	. . . . . . . . . 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 2737	. . . . . . . . . . 11 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
56	55	logdmss 26684	. . . . . . . . . 10 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})
57	56	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0}))
58	54, 57	feqresmpt 6978	. . . . . . . 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 7442	. . . . . 6 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (ℂ D (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (log‘𝑦)))
61	55	dvlog 26693	. . . . . 6 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (𝑦 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑦))
62	60, 61	eqtr3i 2767	. . . . 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 6906	. . . 4 ⊢ (𝑦 = (abs‘𝑥) → (log‘𝑦) = (log‘(abs‘𝑥)))
65		oveq2 7439	. . . 4 ⊢ (𝑦 = (abs‘𝑥) → (1 / 𝑦) = (1 / (abs‘𝑥)))
66	2, 4, 32, 36, 48, 49, 51, 63, 64, 65	dvmptco 26010	. . 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 7532	. . . 4 ⊢ ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1)) = if(𝑥 < 0, ((1 / (abs‘𝑥)) · -1), ((1 / (abs‘𝑥)) · 1))
69		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ∈ ℝ)
70	69	recnd 11289	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ∈ ℂ)
71	70	abscld 15475	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) ∈ ℝ)
72	71	recnd 11289	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) ∈ ℂ)
73		simplr 769	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ≠ 0)
74	70, 73	absne0d 15486	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) ≠ 0)
75	72, 74	reccld 12036	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℂ)
76		neg1cn 12380	. . . . . . . . 9 ⊢ -1 ∈ ℂ
77	76	a1i 11	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -1 ∈ ℂ)
78	75, 77	mulcomd 11282	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → ((1 / (abs‘𝑥)) · -1) = (-1 · (1 / (abs‘𝑥))))
79	75	mulm1d 11715	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (-1 · (1 / (abs‘𝑥))) = -(1 / (abs‘𝑥)))
80		1cnd 11256	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 1 ∈ ℂ)
81	80, 72, 74	divneg2d 12057	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(1 / (abs‘𝑥)) = (1 / -(abs‘𝑥)))
82		0red 11264	. . . . . . . . . . . . 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 11409	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → 𝑥 ≤ 0)
85	69, 84	absnidd 15452	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (abs‘𝑥) = -𝑥)
86	85	eqcomd 2743	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -𝑥 = (abs‘𝑥))
87	70, 86	negcon1ad 11615	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(abs‘𝑥) = 𝑥)
88	87	oveq2d 7447	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → (1 / -(abs‘𝑥)) = (1 / 𝑥))
89	81, 88	eqtrd 2777	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) → -(1 / (abs‘𝑥)) = (1 / 𝑥))
90	78, 79, 89	3eqtrd 2781	. . . . . 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 11245	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
93	92	abscld 15475	. . . . . . . . . . 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 769	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → 𝑥 ≠ 0)
97	95, 96	absne0d 15486	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (abs‘𝑥) ≠ 0)
98	94, 97	rereccld 12094	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℝ)
99	98	recnd 11289	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) ∈ ℂ)
100	99	mulridd 11278	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → ((1 / (abs‘𝑥)) · 1) = (1 / (abs‘𝑥)))
101		simpll 767	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → 𝑥 ∈ ℝ)
102		0red 11264	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) → 0 ∈ ℝ)
103		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) → 𝑥 ∈ ℝ)
104	102, 103	lenltd 11407	. . . . . . . . . 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 15461	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (abs‘𝑥) = 𝑥)
107	106	oveq2d 7447	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) → (1 / (abs‘𝑥)) = (1 / 𝑥))
108	100, 107	eqtrd 2777	. . . . . 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 4562	. . . 4 ⊢ (𝑥 ∈ 𝐷 → if(𝑥 < 0, ((1 / (abs‘𝑥)) · -1), ((1 / (abs‘𝑥)) · 1)) = (1 / 𝑥))
111	68, 110	eqtrid 2789	. . 3 ⊢ (𝑥 ∈ 𝐷 → ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1)) = (1 / 𝑥))
112	111	mpteq2ia 5245	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ ((1 / (abs‘𝑥)) · if(𝑥 < 0, -1, 1))) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))
113	67, 112	eqtri 2765	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 2108 ≠ wne 2940 Vcvv 3480 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ifcif 4525 {csn 4626 {cpr 4628 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ↾ cres 5687 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 +∞cpnf 11292 -∞cmnf 11293 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 -cneg 11493 / cdiv 11920 ℝ⁺crp 13034 (,)cioo 13387 (,]cioc 13388 abscabs 15273 D cdv 25898 logclog 26596

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-tan 16107 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 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 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598

This theorem is referenced by: readvcot 42394

W3C validator