redvmptabs - Mathbox for Steven Nguyen

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Theorem redvmptabs 43411

Description: The derivative of the absolute value, for real numbers. (Contributed by SN, 30-Sep-2025.)

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

**Assertion**
Ref	Expression
redvmptabs	⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (abs‘𝑥))) = (𝑥 ∈ 𝐷 ↦ if(𝑥 < 0, -1, 1))

Distinct variable group: 𝑥,𝐷

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

Step	Hyp	Ref	Expression
1		partfun 6686	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -1, 1)) = ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -1) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 1))
2		reelprrecn 11292	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
3	2	a1i 11	. . . . . 6 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
4		inss1 4182	. . . . . . . . 9 ⊢ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ⊆ 𝐷
5		redvabs.d	. . . . . . . . . . 11 ⊢ 𝐷 = (ℝ ∖ {0})
6		difss 4083	. . . . . . . . . . 11 ⊢ (ℝ ∖ {0}) ⊆ ℝ
7	5, 6	eqsstri 3977	. . . . . . . . . 10 ⊢ 𝐷 ⊆ ℝ
8		ax-resscn 11257	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
9	7, 8	sstri 3940	. . . . . . . . 9 ⊢ 𝐷 ⊆ ℂ
10	4, 9	sstri 3940	. . . . . . . 8 ⊢ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ⊆ ℂ
11	10	sseli 3927	. . . . . . 7 ⊢ (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) → 𝑥 ∈ ℂ)
12	11	adantl 487	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) → 𝑥 ∈ ℂ)
13		1cnd 11302	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) → 1 ∈ ℂ)
14	8	a1i 11	. . . . . . . 8 ⊢ (⊤ → ℝ ⊆ ℂ)
15	14	sselda 3931	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
16		1red 11309	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
17	3	dvmptid 26277	. . . . . . 7 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
18		ssinss1 4191	. . . . . . . 8 ⊢ (𝐷 ⊆ ℝ → (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ)
19	7, 18	mp1i 14	. . . . . . 7 ⊢ (⊤ → (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ)
20		tgioo4 25124	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21		eqid 2761	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
22	5	eleq2i 2853	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (ℝ ∖ {0}))
23		eldifsn 4748	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (ℝ ∖ {0}) ↔ (𝑥 ∈ ℝ ∧ 𝑥 ≠ 0))
24	22, 23	bitri 278	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℝ ∧ 𝑥 ≠ 0))
25		vex 3455	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
26		breq1 5106	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑦 < 0 ↔ 𝑥 < 0))
27	25, 26	elab 3633	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {𝑦 ∣ 𝑦 < 0} ↔ 𝑥 < 0)
28	24, 27	anbi12i 640	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) ↔ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0))
29		lt0ne0 11782	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 < 0) → 𝑥 ≠ 0)
30	29	expcom 419	. . . . . . . . . . . . . . 15 ⊢ (𝑥 < 0 → (𝑥 ∈ ℝ → 𝑥 ≠ 0))
31	30	pm4.71d 571	. . . . . . . . . . . . . 14 ⊢ (𝑥 < 0 → (𝑥 ∈ ℝ ↔ (𝑥 ∈ ℝ ∧ 𝑥 ≠ 0)))
32	31	bicomd 226	. . . . . . . . . . . . 13 ⊢ (𝑥 < 0 → ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ↔ 𝑥 ∈ ℝ))
33	32	pm5.32ri 586	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < 0))
34	28, 33	bitri 278	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < 0))
35		elin 3915	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↔ (𝑥 ∈ 𝐷 ∧ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}))
36		0xr 11356	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
37		elioomnf 13575	. . . . . . . . . . . 12 ⊢ (0 ∈ ℝ^* → (𝑥 ∈ (-∞(,)0) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < 0)))
38	36, 37	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (-∞(,)0) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < 0))
39	34, 35, 38	3bitr4i 306	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↔ 𝑥 ∈ (-∞(,)0))
40	39	eqriv 2758	. . . . . . . . 9 ⊢ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) = (-∞(,)0)
41		iooretop 25084	. . . . . . . . 9 ⊢ (-∞(,)0) ∈ (topGen‘ran (,))
42	40, 41	eqeltri 2857	. . . . . . . 8 ⊢ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∈ (topGen‘ran (,))
43	42	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∈ (topGen‘ran (,)))
44	3, 15, 16, 17, 19, 20, 21, 43	dvmptres 26283	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)) = (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ 1))
45	3, 12, 13, 44	dvmptneg 26286	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥)) = (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -1))
46	45	mptru 1577	. . . 4 ⊢ (ℝ D (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥)) = (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -1)
47	7	a1i 11	. . . . . . 7 ⊢ (⊤ → 𝐷 ⊆ ℝ)
48	47	ssdifssd 4094	. . . . . 6 ⊢ (⊤ → (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ)
49	27	notbii 323	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0} ↔ ¬ 𝑥 < 0)
50	24, 49	anbi12i 640	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐷 ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) ↔ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0))
51		anass 474	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) ↔ (𝑥 ∈ ℝ ∧ (𝑥 ≠ 0 ∧ ¬ 𝑥 < 0)))
52		elre0re 43305	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 0 ∈ ℝ)
53		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ)
54	52, 53	lttrid 11448	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (0 < 𝑥 ↔ ¬ (0 = 𝑥 ∨ 𝑥 < 0)))
55		ioran 999	. . . . . . . . . . . . . 14 ⊢ (¬ (0 = 𝑥 ∨ 𝑥 < 0) ↔ (¬ 0 = 𝑥 ∧ ¬ 𝑥 < 0))
56		nesym 3012	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ 0 ↔ ¬ 0 = 𝑥)
57	56	bicomi 227	. . . . . . . . . . . . . 14 ⊢ (¬ 0 = 𝑥 ↔ 𝑥 ≠ 0)
58	55, 57	bianbi 639	. . . . . . . . . . . . 13 ⊢ (¬ (0 = 𝑥 ∨ 𝑥 < 0) ↔ (𝑥 ≠ 0 ∧ ¬ 𝑥 < 0))
59	54, 58	bitr2di 291	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → ((𝑥 ≠ 0 ∧ ¬ 𝑥 < 0) ↔ 0 < 𝑥))
60	59	pm5.32i 585	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ (𝑥 ≠ 0 ∧ ¬ 𝑥 < 0)) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
61	50, 51, 60	3bitri 300	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐷 ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
62		eldif 3909	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↔ (𝑥 ∈ 𝐷 ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}))
63		repos 13577	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
64	61, 62, 63	3bitr4i 306	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↔ 𝑥 ∈ (0(,)+∞))
65	64	eqriv 2758	. . . . . . . 8 ⊢ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) = (0(,)+∞)
66		iooretop 25084	. . . . . . . 8 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
67	65, 66	eqeltri 2857	. . . . . . 7 ⊢ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ∈ (topGen‘ran (,))
68	67	a1i 11	. . . . . 6 ⊢ (⊤ → (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ∈ (topGen‘ran (,)))
69	3, 15, 16, 17, 48, 20, 21, 68	dvmptres 26283	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)) = (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 1))
70	69	mptru 1577	. . . 4 ⊢ (ℝ D (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)) = (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 1)
71	46, 70	uneq12i 4113	. . 3 ⊢ ((ℝ D (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥)) ∪ (ℝ D (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥))) = ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -1) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 1))
72	12	negcld 11656	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) → -𝑥 ∈ ℂ)
73	72	fmpttd 7115	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥):(𝐷 ∩ {𝑦 ∣ 𝑦 < 0})⟶ℂ)
74		ssdifss 4087	. . . . . . . . . 10 ⊢ (𝐷 ⊆ ℝ → (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ)
75	7, 74	ax-mp 5	. . . . . . . . 9 ⊢ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ
76	75, 8	sstri 3940	. . . . . . . 8 ⊢ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℂ
77	76	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℂ)
78	77	sselda 3931	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) → 𝑥 ∈ ℂ)
79	78	fmpttd 7115	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥):(𝐷 ∖ {𝑦 ∣ 𝑦 < 0})⟶ℂ)
80		inindif 4324	. . . . . 6 ⊢ ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∩ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) = ∅
81	80	a1i 11	. . . . 5 ⊢ (⊤ → ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∩ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) = ∅)
82		retop 25080	. . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ Top
83		isopn3i 23400	. . . . . . . . 9 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∈ (topGen‘ran (,))) → ((int‘(topGen‘ran (,)))‘(𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) = (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}))
84	82, 42, 83	mp2an 705	. . . . . . . 8 ⊢ ((int‘(topGen‘ran (,)))‘(𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) = (𝐷 ∩ {𝑦 ∣ 𝑦 < 0})
85		isopn3i 23400	. . . . . . . . 9 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ∈ (topGen‘ran (,))) → ((int‘(topGen‘ran (,)))‘(𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) = (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}))
86	82, 67, 85	mp2an 705	. . . . . . . 8 ⊢ ((int‘(topGen‘ran (,)))‘(𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) = (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})
87	84, 86	uneq12i 4113	. . . . . . 7 ⊢ (((int‘(topGen‘ran (,)))‘(𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) ∪ ((int‘(topGen‘ran (,)))‘(𝐷 ∖ {𝑦 ∣ 𝑦 < 0}))) = ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}))
88		unopn 23221	. . . . . . . . 9 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∈ (topGen‘ran (,)) ∧ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ∈ (topGen‘ran (,))) → ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) ∈ (topGen‘ran (,)))
89	82, 42, 67, 88	mp3an 1490	. . . . . . . 8 ⊢ ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) ∈ (topGen‘ran (,))
90		isopn3i 23400	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ Top ∧ ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) ∈ (topGen‘ran (,))) → ((int‘(topGen‘ran (,)))‘((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}))) = ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})))
91	82, 89, 90	mp2an 705	. . . . . . 7 ⊢ ((int‘(topGen‘ran (,)))‘((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}))) = ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}))
92	87, 91	eqtr4i 2787	. . . . . 6 ⊢ (((int‘(topGen‘ran (,)))‘(𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) ∪ ((int‘(topGen‘ran (,)))‘(𝐷 ∖ {𝑦 ∣ 𝑦 < 0}))) = ((int‘(topGen‘ran (,)))‘((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})))
93	92	a1i 11	. . . . 5 ⊢ (⊤ → (((int‘(topGen‘ran (,)))‘(𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) ∪ ((int‘(topGen‘ran (,)))‘(𝐷 ∖ {𝑦 ∣ 𝑦 < 0}))) = ((int‘(topGen‘ran (,)))‘((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}))))
94	20, 21, 14, 73, 79, 19, 48, 81, 93	dvun 43410	. . . 4 ⊢ (⊤ → ((ℝ D (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥)) ∪ (ℝ D (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥))) = (ℝ D ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥))))
95	94	mptru 1577	. . 3 ⊢ ((ℝ D (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥)) ∪ (ℝ D (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥))) = (ℝ D ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)))
96	1, 71, 95	3eqtr2ri 2791	. 2 ⊢ (ℝ D ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥))) = (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -1, 1))
97		partfun 6686	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -𝑥, 𝑥)) = ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥))
98		elioore 13506	. . . . . . . . . 10 ⊢ (𝑥 ∈ (-∞(,)0) → 𝑥 ∈ ℝ)
99		0red 11311	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (-∞(,)0) → 0 ∈ ℝ)
100	38	simprbi 503	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (-∞(,)0) → 𝑥 < 0)
101	98, 99, 100	ltled 11458	. . . . . . . . . 10 ⊢ (𝑥 ∈ (-∞(,)0) → 𝑥 ≤ 0)
102	98, 101	absnidd 15581	. . . . . . . . 9 ⊢ (𝑥 ∈ (-∞(,)0) → (abs‘𝑥) = -𝑥)
103	102	eqcomd 2767	. . . . . . . 8 ⊢ (𝑥 ∈ (-∞(,)0) → -𝑥 = (abs‘𝑥))
104	103, 40	eleq2s 2879	. . . . . . 7 ⊢ (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) → -𝑥 = (abs‘𝑥))
105	35, 104	sylbir 238	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) → -𝑥 = (abs‘𝑥))
106		rpabsid 43378	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘𝑥) = 𝑥)
107	106	eqcomd 2767	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 = (abs‘𝑥))
108		ioorp 13556	. . . . . . . . 9 ⊢ (0(,)+∞) = ℝ⁺
109	107, 108	eleq2s 2879	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)+∞) → 𝑥 = (abs‘𝑥))
110	109, 65	eleq2s 2879	. . . . . . 7 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) → 𝑥 = (abs‘𝑥))
111	62, 110	sylbir 238	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) → 𝑥 = (abs‘𝑥))
112	105, 111	ifeqda 4519	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -𝑥, 𝑥) = (abs‘𝑥))
113	112	mpteq2ia 5200	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -𝑥, 𝑥)) = (𝑥 ∈ 𝐷 ↦ (abs‘𝑥))
114	97, 113	eqtr3i 2786	. . 3 ⊢ ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)) = (𝑥 ∈ 𝐷 ↦ (abs‘𝑥))
115	114	oveq2i 7431	. 2 ⊢ (ℝ D ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥))) = (ℝ D (𝑥 ∈ 𝐷 ↦ (abs‘𝑥)))
116		eqid 2761	. . . 4 ⊢ 1 = 1
117	27, 116	ifbieq2i 4508	. . 3 ⊢ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -1, 1) = if(𝑥 < 0, -1, 1)
118	117	mpteq2i 5201	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -1, 1)) = (𝑥 ∈ 𝐷 ↦ if(𝑥 < 0, -1, 1))
119	96, 115, 118	3eqtr3i 2792	1 ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (abs‘𝑥))) = (𝑥 ∈ 𝐷 ↦ if(𝑥 < 0, -1, 1))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ⊤wtru 1571 ∈ wcel 2145 {cab 2739 ≠ wne 2956 ∖ cdif 3896 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 ifcif 4482 {csn 4584 {cpr 4586 class class class wbr 5103 ↦ cmpt 5186 ran crn 5652 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 +∞cpnf 11340 -∞cmnf 11341 ℝ^*cxr 11342 < clt 11343 -cneg 11542 ℝ⁺crp 13120 (,)cioo 13476 abscabs 15401 TopOpenctopn 17592 topGenctg 17608 ℂ_fldccnfld 21678 Topctop 23211 intcnt 23335 D cdv 26183

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 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 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-icc 13483 df-fz 13640 df-fzo 13789 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-limc 26186 df-dv 26187

This theorem is used by: readvrec 43413

W3C validator