redvmptabs - Mathbox for Steven Nguyen

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Theorem redvmptabs 42368

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 6715	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -1, 1)) = ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -1) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 1))
2		reelprrecn 11244	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
3	2	a1i 11	. . . . . 6 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
4		inss1 4244	. . . . . . . . 9 ⊢ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ⊆ 𝐷
5		redvabs.d	. . . . . . . . . . 11 ⊢ 𝐷 = (ℝ ∖ {0})
6		difss 4145	. . . . . . . . . . 11 ⊢ (ℝ ∖ {0}) ⊆ ℝ
7	5, 6	eqsstri 4029	. . . . . . . . . 10 ⊢ 𝐷 ⊆ ℝ
8		ax-resscn 11209	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
9	7, 8	sstri 4004	. . . . . . . . 9 ⊢ 𝐷 ⊆ ℂ
10	4, 9	sstri 4004	. . . . . . . 8 ⊢ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ⊆ ℂ
11	10	sseli 3990	. . . . . . 7 ⊢ (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) → 𝑥 ∈ ℂ)
12	11	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) → 𝑥 ∈ ℂ)
13		1cnd 11253	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) → 1 ∈ ℂ)
14	8	a1i 11	. . . . . . . 8 ⊢ (⊤ → ℝ ⊆ ℂ)
15	14	sselda 3994	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
16		1red 11259	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
17	3	dvmptid 26009	. . . . . . 7 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
18		ssinss1 4253	. . . . . . . 8 ⊢ (𝐷 ⊆ ℝ → (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ)
19	7, 18	mp1i 13	. . . . . . 7 ⊢ (⊤ → (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ)
20		eqid 2734	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
21	20	tgioo2 24838	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
22	5	eleq2i 2830	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (ℝ ∖ {0}))
23		eldifsn 4790	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (ℝ ∖ {0}) ↔ (𝑥 ∈ ℝ ∧ 𝑥 ≠ 0))
24	22, 23	bitri 275	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℝ ∧ 𝑥 ≠ 0))
25		vex 3481	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
26		breq1 5150	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑦 < 0 ↔ 𝑥 < 0))
27	25, 26	elab 3680	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {𝑦 ∣ 𝑦 < 0} ↔ 𝑥 < 0)
28	24, 27	anbi12i 628	. . . . . . . . . . . 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 413	. . . . . . . . . . . . . . 15 ⊢ (𝑥 < 0 → (𝑥 ∈ ℝ → 𝑥 ≠ 0))
31	30	pm4.71d 561	. . . . . . . . . . . . . 14 ⊢ (𝑥 < 0 → (𝑥 ∈ ℝ ↔ (𝑥 ∈ ℝ ∧ 𝑥 ≠ 0)))
32	31	bicomd 223	. . . . . . . . . . . . 13 ⊢ (𝑥 < 0 → ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ↔ 𝑥 ∈ ℝ))
33	32	pm5.32ri 575	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < 0))
34	28, 33	bitri 275	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < 0))
35		elin 3978	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↔ (𝑥 ∈ 𝐷 ∧ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}))
36		0xr 11305	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
37		elioomnf 13480	. . . . . . . . . . . 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 303	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↔ 𝑥 ∈ (-∞(,)0))
40	39	eqriv 2731	. . . . . . . . 9 ⊢ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) = (-∞(,)0)
41		iooretop 24801	. . . . . . . . 9 ⊢ (-∞(,)0) ∈ (topGen‘ran (,))
42	40, 41	eqeltri 2834	. . . . . . . 8 ⊢ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∈ (topGen‘ran (,))
43	42	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∈ (topGen‘ran (,)))
44	3, 15, 16, 17, 19, 21, 20, 43	dvmptres 26015	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)) = (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ 1))
45	3, 12, 13, 44	dvmptneg 26018	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥)) = (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -1))
46	45	mptru 1543	. . . 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 4156	. . . . . 6 ⊢ (⊤ → (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ)
49	27	notbii 320	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0} ↔ ¬ 𝑥 < 0)
50	24, 49	anbi12i 628	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐷 ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) ↔ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0))
51		anass 468	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) ↔ (𝑥 ∈ ℝ ∧ (𝑥 ≠ 0 ∧ ¬ 𝑥 < 0)))
52		elre0re 42273	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 0 ∈ ℝ)
53		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ)
54	52, 53	lttrid 11396	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (0 < 𝑥 ↔ ¬ (0 = 𝑥 ∨ 𝑥 < 0)))
55		ioran 985	. . . . . . . . . . . . . 14 ⊢ (¬ (0 = 𝑥 ∨ 𝑥 < 0) ↔ (¬ 0 = 𝑥 ∧ ¬ 𝑥 < 0))
56		nesym 2994	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ 0 ↔ ¬ 0 = 𝑥)
57	56	bicomi 224	. . . . . . . . . . . . . 14 ⊢ (¬ 0 = 𝑥 ↔ 𝑥 ≠ 0)
58	55, 57	bianbi 627	. . . . . . . . . . . . 13 ⊢ (¬ (0 = 𝑥 ∨ 𝑥 < 0) ↔ (𝑥 ≠ 0 ∧ ¬ 𝑥 < 0))
59	54, 58	bitr2di 288	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → ((𝑥 ≠ 0 ∧ ¬ 𝑥 < 0) ↔ 0 < 𝑥))
60	59	pm5.32i 574	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ (𝑥 ≠ 0 ∧ ¬ 𝑥 < 0)) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
61	50, 51, 60	3bitri 297	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐷 ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
62		eldif 3972	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↔ (𝑥 ∈ 𝐷 ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}))
63		repos 13482	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
64	61, 62, 63	3bitr4i 303	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↔ 𝑥 ∈ (0(,)+∞))
65	64	eqriv 2731	. . . . . . . 8 ⊢ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) = (0(,)+∞)
66		iooretop 24801	. . . . . . . 8 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
67	65, 66	eqeltri 2834	. . . . . . 7 ⊢ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ∈ (topGen‘ran (,))
68	67	a1i 11	. . . . . 6 ⊢ (⊤ → (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ∈ (topGen‘ran (,)))
69	3, 15, 16, 17, 48, 21, 20, 68	dvmptres 26015	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)) = (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 1))
70	69	mptru 1543	. . . 4 ⊢ (ℝ D (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)) = (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 1)
71	46, 70	uneq12i 4175	. . 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 11604	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) → -𝑥 ∈ ℂ)
73	72	fmpttd 7134	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥):(𝐷 ∩ {𝑦 ∣ 𝑦 < 0})⟶ℂ)
74		ssdifss 4149	. . . . . . . . . 10 ⊢ (𝐷 ⊆ ℝ → (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ)
75	7, 74	ax-mp 5	. . . . . . . . 9 ⊢ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ
76	75, 8	sstri 4004	. . . . . . . 8 ⊢ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℂ
77	76	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℂ)
78	77	sselda 3994	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) → 𝑥 ∈ ℂ)
79	78	fmpttd 7134	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥):(𝐷 ∖ {𝑦 ∣ 𝑦 < 0})⟶ℂ)
80		inindif 4380	. . . . . 6 ⊢ ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∩ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) = ∅
81	80	a1i 11	. . . . 5 ⊢ (⊤ → ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∩ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) = ∅)
82		retop 24797	. . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ Top
83		isopn3i 23105	. . . . . . . . 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 692	. . . . . . . 8 ⊢ ((int‘(topGen‘ran (,)))‘(𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) = (𝐷 ∩ {𝑦 ∣ 𝑦 < 0})
85		isopn3i 23105	. . . . . . . . 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 692	. . . . . . . 8 ⊢ ((int‘(topGen‘ran (,)))‘(𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) = (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})
87	84, 86	uneq12i 4175	. . . . . . 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 22924	. . . . . . . . 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 1460	. . . . . . . 8 ⊢ ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) ∈ (topGen‘ran (,))
90		isopn3i 23105	. . . . . . . 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 692	. . . . . . 7 ⊢ ((int‘(topGen‘ran (,)))‘((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}))) = ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}))
92	87, 91	eqtr4i 2765	. . . . . 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	21, 20, 14, 73, 79, 19, 48, 81, 93	dvun 42367	. . . 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 1543	. . 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 2769	. 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 6715	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -𝑥, 𝑥)) = ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥))
98		elioore 13413	. . . . . . . . . 10 ⊢ (𝑥 ∈ (-∞(,)0) → 𝑥 ∈ ℝ)
99		0red 11261	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (-∞(,)0) → 0 ∈ ℝ)
100	38	simprbi 496	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (-∞(,)0) → 𝑥 < 0)
101	98, 99, 100	ltled 11406	. . . . . . . . . 10 ⊢ (𝑥 ∈ (-∞(,)0) → 𝑥 ≤ 0)
102	98, 101	absnidd 15448	. . . . . . . . 9 ⊢ (𝑥 ∈ (-∞(,)0) → (abs‘𝑥) = -𝑥)
103	102	eqcomd 2740	. . . . . . . 8 ⊢ (𝑥 ∈ (-∞(,)0) → -𝑥 = (abs‘𝑥))
104	103, 40	eleq2s 2856	. . . . . . 7 ⊢ (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) → -𝑥 = (abs‘𝑥))
105	35, 104	sylbir 235	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) → -𝑥 = (abs‘𝑥))
106		rpabsid 42334	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘𝑥) = 𝑥)
107	106	eqcomd 2740	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 = (abs‘𝑥))
108		ioorp 13461	. . . . . . . . 9 ⊢ (0(,)+∞) = ℝ⁺
109	107, 108	eleq2s 2856	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)+∞) → 𝑥 = (abs‘𝑥))
110	109, 65	eleq2s 2856	. . . . . . 7 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) → 𝑥 = (abs‘𝑥))
111	62, 110	sylbir 235	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) → 𝑥 = (abs‘𝑥))
112	105, 111	ifeqda 4566	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -𝑥, 𝑥) = (abs‘𝑥))
113	112	mpteq2ia 5250	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -𝑥, 𝑥)) = (𝑥 ∈ 𝐷 ↦ (abs‘𝑥))
114	97, 113	eqtr3i 2764	. . 3 ⊢ ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)) = (𝑥 ∈ 𝐷 ↦ (abs‘𝑥))
115	114	oveq2i 7441	. 2 ⊢ (ℝ D ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥))) = (ℝ D (𝑥 ∈ 𝐷 ↦ (abs‘𝑥)))
116		eqid 2734	. . . 4 ⊢ 1 = 1
117	27, 116	ifbieq2i 4555	. . 3 ⊢ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -1, 1) = if(𝑥 < 0, -1, 1)
118	117	mpteq2i 5252	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -1, 1)) = (𝑥 ∈ 𝐷 ↦ if(𝑥 < 0, -1, 1))
119	96, 115, 118	3eqtr3i 2770	1 ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (abs‘𝑥))) = (𝑥 ∈ 𝐷 ↦ if(𝑥 < 0, -1, 1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ⊤wtru 1537 ∈ wcel 2105 {cab 2711 ≠ wne 2937 ∖ cdif 3959 ∪ cun 3960 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 ifcif 4530 {csn 4630 {cpr 4632 class class class wbr 5147 ↦ cmpt 5230 ran crn 5689 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 +∞cpnf 11289 -∞cmnf 11290 ℝ^*cxr 11291 < clt 11292 -cneg 11490 ℝ⁺crp 13031 (,)cioo 13383 abscabs 15269 TopOpenctopn 17467 topGenctg 17483 ℂ_fldccnfld 21381 Topctop 22914 intcnt 23040 D cdv 25912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 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 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916

This theorem is referenced by: readvrec 42370

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