redvmptabs - Mathbox for Steven Nguyen

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Theorem redvmptabs 43249

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 6681	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -1, 1)) = ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -1) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 1))
2		reelprrecn 11238	. . . . . . 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 11203	. . . . . . . . . 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 11248	. . . . . 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 11255	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
17	3	dvmptid 26213	. . . . . . 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 25060	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21		eqid 2760	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
22	5	eleq2i 2852	. . . . . . . . . . . . . 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 3454	. . . . . . . . . . . . . 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 11726	. . . . . . . . . . . . . . . 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 11302	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
37		elioomnf 13519	. . . . . . . . . . . 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 2757	. . . . . . . . 9 ⊢ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) = (-∞(,)0)
41		iooretop 25020	. . . . . . . . 9 ⊢ (-∞(,)0) ∈ (topGen‘ran (,))
42	40, 41	eqeltri 2856	. . . . . . . 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 26219	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)) = (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ 1))
45	3, 12, 13, 44	dvmptneg 26222	. . . . 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 43135	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 0 ∈ ℝ)
53		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ)
54	52, 53	lttrid 11394	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (0 < 𝑥 ↔ ¬ (0 = 𝑥 ∨ 𝑥 < 0)))
55		ioran 999	. . . . . . . . . . . . . 14 ⊢ (¬ (0 = 𝑥 ∨ 𝑥 < 0) ↔ (¬ 0 = 𝑥 ∧ ¬ 𝑥 < 0))
56		nesym 3011	. . . . . . . . . . . . . . 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 13521	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
64	61, 62, 63	3bitr4i 306	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↔ 𝑥 ∈ (0(,)+∞))
65	64	eqriv 2757	. . . . . . . 8 ⊢ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) = (0(,)+∞)
66		iooretop 25020	. . . . . . . 8 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
67	65, 66	eqeltri 2856	. . . . . . 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 26219	. . . . 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 11602	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) → -𝑥 ∈ ℂ)
73	72	fmpttd 7110	. . . . 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 7110	. . . . 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 25016	. . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ Top
83		isopn3i 23336	. . . . . . . . 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 23336	. . . . . . . . 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 23157	. . . . . . . . 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 23336	. . . . . . . 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 2786	. . . . . 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 43248	. . . 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 2790	. 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 6681	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -𝑥, 𝑥)) = ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥))
98		elioore 13450	. . . . . . . . . 10 ⊢ (𝑥 ∈ (-∞(,)0) → 𝑥 ∈ ℝ)
99		0red 11257	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (-∞(,)0) → 0 ∈ ℝ)
100	38	simprbi 503	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (-∞(,)0) → 𝑥 < 0)
101	98, 99, 100	ltled 11404	. . . . . . . . . 10 ⊢ (𝑥 ∈ (-∞(,)0) → 𝑥 ≤ 0)
102	98, 101	absnidd 15523	. . . . . . . . 9 ⊢ (𝑥 ∈ (-∞(,)0) → (abs‘𝑥) = -𝑥)
103	102	eqcomd 2766	. . . . . . . 8 ⊢ (𝑥 ∈ (-∞(,)0) → -𝑥 = (abs‘𝑥))
104	103, 40	eleq2s 2878	. . . . . . 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 43210	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘𝑥) = 𝑥)
107	106	eqcomd 2766	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 = (abs‘𝑥))
108		ioorp 13500	. . . . . . . . 9 ⊢ (0(,)+∞) = ℝ⁺
109	107, 108	eleq2s 2878	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)+∞) → 𝑥 = (abs‘𝑥))
110	109, 65	eleq2s 2878	. . . . . . 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 2785	. . 3 ⊢ ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)) = (𝑥 ∈ 𝐷 ↦ (abs‘𝑥))
115	114	oveq2i 7426	. 2 ⊢ (ℝ D ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥))) = (ℝ D (𝑥 ∈ 𝐷 ↦ (abs‘𝑥)))
116		eqid 2760	. . . 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 2791	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 2738 ≠ wne 2955 ∖ 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 5656 ‘cfv 6534 (class class class)co 7415 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 +∞cpnf 11286 -∞cmnf 11287 ℝ^*cxr 11288 < clt 11289 -cneg 11488 ℝ⁺crp 13064 (,)cioo 13420 abscabs 15343 TopOpenctopn 17528 topGenctg 17544 ℂ_fldccnfld 21614 Topctop 23147 intcnt 23271 D cdv 26119

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 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7738 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 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 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7372 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8161 df-frecs 8282 df-wrecs 8313 df-recs 8362 df-rdg 8401 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8909 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fsupp 9336 df-fi 9385 df-sup 9416 df-inf 9417 df-oi 9486 df-card 9966 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11489 df-neg 11490 df-div 11918 df-nn 12280 df-2 12349 df-3 12350 df-4 12351 df-5 12352 df-6 12353 df-7 12354 df-8 12355 df-9 12356 df-n0 12551 df-z 12638 df-dec 12759 df-uz 12910 df-q 13020 df-rp 13065 df-xneg 13185 df-xadd 13186 df-xmul 13187 df-ioo 13424 df-icc 13427 df-fz 13584 df-fzo 13732 df-seq 14088 df-exp 14148 df-hash 14417 df-cj 15208 df-re 15209 df-im 15210 df-sqrt 15344 df-abs 15345 df-struct 17261 df-sets 17278 df-slot 17296 df-ndx 17308 df-base 17324 df-ress 17345 df-plusg 17377 df-mulr 17378 df-starv 17379 df-sca 17380 df-vsca 17381 df-ip 17382 df-tset 17383 df-ple 17384 df-ds 17386 df-unif 17387 df-hom 17388 df-cco 17389 df-rest 17529 df-topn 17530 df-0g 17548 df-gsum 17549 df-topgen 17550 df-pt 17551 df-prds 17554 df-xrs 17610 df-qtop 17615 df-imas 17616 df-xps 17618 df-mre 17692 df-mrc 17693 df-acs 17695 df-mgm 18752 df-sgrp 18844 df-mnd 18860 df-submnd 18915 df-mulg 19214 df-cntz 19467 df-cmn 19932 df-psmet 21606 df-xmet 21607 df-met 21608 df-bl 21609 df-mopn 21610 df-fbas 21611 df-fg 21612 df-cnfld 21615 df-top 23148 df-topon 23165 df-topsp 23187 df-bases 23200 df-cld 23273 df-ntr 23274 df-cls 23275 df-nei 23352 df-lp 23390 df-perf 23391 df-cn 23481 df-cnp 23482 df-haus 23569 df-tx 23817 df-hmeo 24010 df-fil 24101 df-fm 24193 df-flim 24194 df-flf 24195 df-xms 24575 df-ms 24576 df-tms 24577 df-cncf 25135 df-limc 26122 df-dv 26123

This theorem is used by: readvrec 43251

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