redvmptabs - Mathbox for Steven Nguyen

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Theorem redvmptabs 42974

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 6670	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -1, 1)) = ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -1) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 1))
2		reelprrecn 11167	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
3	2	a1i 11	. . . . . 6 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
4		inss1 4190	. . . . . . . . 9 ⊢ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ⊆ 𝐷
5		redvabs.d	. . . . . . . . . . 11 ⊢ 𝐷 = (ℝ ∖ {0})
6		difss 4091	. . . . . . . . . . 11 ⊢ (ℝ ∖ {0}) ⊆ ℝ
7	5, 6	eqsstri 3984	. . . . . . . . . 10 ⊢ 𝐷 ⊆ ℝ
8		ax-resscn 11132	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
9	7, 8	sstri 3947	. . . . . . . . 9 ⊢ 𝐷 ⊆ ℂ
10	4, 9	sstri 3947	. . . . . . . 8 ⊢ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ⊆ ℂ
11	10	sseli 3934	. . . . . . 7 ⊢ (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) → 𝑥 ∈ ℂ)
12	11	adantl 485	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) → 𝑥 ∈ ℂ)
13		1cnd 11177	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) → 1 ∈ ℂ)
14	8	a1i 11	. . . . . . . 8 ⊢ (⊤ → ℝ ⊆ ℂ)
15	14	sselda 3938	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
16		1red 11184	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
17	3	dvmptid 26021	. . . . . . 7 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
18		ssinss1 4199	. . . . . . . 8 ⊢ (𝐷 ⊆ ℝ → (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ)
19	7, 18	mp1i 13	. . . . . . 7 ⊢ (⊤ → (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ)
20		tgioo4 24867	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21		eqid 2764	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
22	5	eleq2i 2856	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (ℝ ∖ {0}))
23		eldifsn 4748	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (ℝ ∖ {0}) ↔ (𝑥 ∈ ℝ ∧ 𝑥 ≠ 0))
24	22, 23	bitri 277	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℝ ∧ 𝑥 ≠ 0))
25		vex 3460	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
26		breq1 5105	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑦 < 0 ↔ 𝑥 < 0))
27	25, 26	elab 3640	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {𝑦 ∣ 𝑦 < 0} ↔ 𝑥 < 0)
28	24, 27	anbi12i 637	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) ↔ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0))
29		lt0ne0 11655	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 < 0) → 𝑥 ≠ 0)
30	29	expcom 417	. . . . . . . . . . . . . . 15 ⊢ (𝑥 < 0 → (𝑥 ∈ ℝ → 𝑥 ≠ 0))
31	30	pm4.71d 569	. . . . . . . . . . . . . 14 ⊢ (𝑥 < 0 → (𝑥 ∈ ℝ ↔ (𝑥 ∈ ℝ ∧ 𝑥 ≠ 0)))
32	31	bicomd 225	. . . . . . . . . . . . 13 ⊢ (𝑥 < 0 → ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ↔ 𝑥 ∈ ℝ))
33	32	pm5.32ri 583	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ 𝑥 < 0) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < 0))
34	28, 33	bitri 277	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < 0))
35		elin 3922	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↔ (𝑥 ∈ 𝐷 ∧ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}))
36		0xr 11231	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ^*
37		elioomnf 13450	. . . . . . . . . . . 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 305	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↔ 𝑥 ∈ (-∞(,)0))
40	39	eqriv 2761	. . . . . . . . 9 ⊢ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) = (-∞(,)0)
41		iooretop 24827	. . . . . . . . 9 ⊢ (-∞(,)0) ∈ (topGen‘ran (,))
42	40, 41	eqeltri 2860	. . . . . . . 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 26027	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)) = (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ 1))
45	3, 12, 13, 44	dvmptneg 26030	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥)) = (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -1))
46	45	mptru 1569	. . . 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 4102	. . . . . 6 ⊢ (⊤ → (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ)
49	27	notbii 322	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0} ↔ ¬ 𝑥 < 0)
50	24, 49	anbi12i 637	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐷 ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) ↔ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0))
51		anass 472	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) ∧ ¬ 𝑥 < 0) ↔ (𝑥 ∈ ℝ ∧ (𝑥 ≠ 0 ∧ ¬ 𝑥 < 0)))
52		elre0re 42875	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 0 ∈ ℝ)
53		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ)
54	52, 53	lttrid 11323	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (0 < 𝑥 ↔ ¬ (0 = 𝑥 ∨ 𝑥 < 0)))
55		ioran 997	. . . . . . . . . . . . . 14 ⊢ (¬ (0 = 𝑥 ∨ 𝑥 < 0) ↔ (¬ 0 = 𝑥 ∧ ¬ 𝑥 < 0))
56		nesym 3015	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ 0 ↔ ¬ 0 = 𝑥)
57	56	bicomi 226	. . . . . . . . . . . . . 14 ⊢ (¬ 0 = 𝑥 ↔ 𝑥 ≠ 0)
58	55, 57	bianbi 636	. . . . . . . . . . . . 13 ⊢ (¬ (0 = 𝑥 ∨ 𝑥 < 0) ↔ (𝑥 ≠ 0 ∧ ¬ 𝑥 < 0))
59	54, 58	bitr2di 290	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → ((𝑥 ≠ 0 ∧ ¬ 𝑥 < 0) ↔ 0 < 𝑥))
60	59	pm5.32i 582	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ (𝑥 ≠ 0 ∧ ¬ 𝑥 < 0)) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
61	50, 51, 60	3bitri 299	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐷 ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
62		eldif 3916	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↔ (𝑥 ∈ 𝐷 ∧ ¬ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}))
63		repos 13452	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥))
64	61, 62, 63	3bitr4i 305	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↔ 𝑥 ∈ (0(,)+∞))
65	64	eqriv 2761	. . . . . . . 8 ⊢ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) = (0(,)+∞)
66		iooretop 24827	. . . . . . . 8 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
67	65, 66	eqeltri 2860	. . . . . . 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 26027	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)) = (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 1))
70	69	mptru 1569	. . . 4 ⊢ (ℝ D (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)) = (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 1)
71	46, 70	uneq12i 4121	. . 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 11531	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) → -𝑥 ∈ ℂ)
73	72	fmpttd 7098	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥):(𝐷 ∩ {𝑦 ∣ 𝑦 < 0})⟶ℂ)
74		ssdifss 4095	. . . . . . . . . 10 ⊢ (𝐷 ⊆ ℝ → (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ)
75	7, 74	ax-mp 5	. . . . . . . . 9 ⊢ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℝ
76	75, 8	sstri 3947	. . . . . . . 8 ⊢ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℂ
77	76	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ⊆ ℂ)
78	77	sselda 3938	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) → 𝑥 ∈ ℂ)
79	78	fmpttd 7098	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥):(𝐷 ∖ {𝑦 ∣ 𝑦 < 0})⟶ℂ)
80		inindif 4330	. . . . . 6 ⊢ ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∩ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) = ∅
81	80	a1i 11	. . . . 5 ⊢ (⊤ → ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∩ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) = ∅)
82		retop 24823	. . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ Top
83		isopn3i 23144	. . . . . . . . 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 702	. . . . . . . 8 ⊢ ((int‘(topGen‘ran (,)))‘(𝐷 ∩ {𝑦 ∣ 𝑦 < 0})) = (𝐷 ∩ {𝑦 ∣ 𝑦 < 0})
85		isopn3i 23144	. . . . . . . . 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 702	. . . . . . . 8 ⊢ ((int‘(topGen‘ran (,)))‘(𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) = (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})
87	84, 86	uneq12i 4121	. . . . . . 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 22965	. . . . . . . . 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 1484	. . . . . . . 8 ⊢ ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0})) ∈ (topGen‘ran (,))
90		isopn3i 23144	. . . . . . . 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 702	. . . . . . 7 ⊢ ((int‘(topGen‘ran (,)))‘((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}))) = ((𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ∪ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}))
92	87, 91	eqtr4i 2790	. . . . . 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 42973	. . . 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 1569	. . 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 2794	. 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 6670	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -𝑥, 𝑥)) = ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥))
98		elioore 13381	. . . . . . . . . 10 ⊢ (𝑥 ∈ (-∞(,)0) → 𝑥 ∈ ℝ)
99		0red 11186	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (-∞(,)0) → 0 ∈ ℝ)
100	38	simprbi 501	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (-∞(,)0) → 𝑥 < 0)
101	98, 99, 100	ltled 11333	. . . . . . . . . 10 ⊢ (𝑥 ∈ (-∞(,)0) → 𝑥 ≤ 0)
102	98, 101	absnidd 15443	. . . . . . . . 9 ⊢ (𝑥 ∈ (-∞(,)0) → (abs‘𝑥) = -𝑥)
103	102	eqcomd 2770	. . . . . . . 8 ⊢ (𝑥 ∈ (-∞(,)0) → -𝑥 = (abs‘𝑥))
104	103, 40	eleq2s 2882	. . . . . . 7 ⊢ (𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) → -𝑥 = (abs‘𝑥))
105	35, 104	sylbir 237	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑥 ∈ {𝑦 ∣ 𝑦 < 0}) → -𝑥 = (abs‘𝑥))
106		rpabsid 42935	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘𝑥) = 𝑥)
107	106	eqcomd 2770	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 = (abs‘𝑥))
108		ioorp 13431	. . . . . . . . 9 ⊢ (0(,)+∞) = ℝ⁺
109	107, 108	eleq2s 2882	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)+∞) → 𝑥 = (abs‘𝑥))
110	109, 65	eleq2s 2882	. . . . . . 7 ⊢ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) → 𝑥 = (abs‘𝑥))
111	62, 110	sylbir 237	. . . . . 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 5197	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -𝑥, 𝑥)) = (𝑥 ∈ 𝐷 ↦ (abs‘𝑥))
114	97, 113	eqtr3i 2789	. . 3 ⊢ ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥)) = (𝑥 ∈ 𝐷 ↦ (abs‘𝑥))
115	114	oveq2i 7409	. 2 ⊢ (ℝ D ((𝑥 ∈ (𝐷 ∩ {𝑦 ∣ 𝑦 < 0}) ↦ -𝑥) ∪ (𝑥 ∈ (𝐷 ∖ {𝑦 ∣ 𝑦 < 0}) ↦ 𝑥))) = (ℝ D (𝑥 ∈ 𝐷 ↦ (abs‘𝑥)))
116		eqid 2764	. . . 4 ⊢ 1 = 1
117	27, 116	ifbieq2i 4508	. . 3 ⊢ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -1, 1) = if(𝑥 < 0, -1, 1)
118	117	mpteq2i 5198	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ if(𝑥 ∈ {𝑦 ∣ 𝑦 < 0}, -1, 1)) = (𝑥 ∈ 𝐷 ↦ if(𝑥 < 0, -1, 1))
119	96, 115, 118	3eqtr3i 2795	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 208 ∧ wa 399 ∨ wo 858 = wceq 1562 ⊤wtru 1563 ∈ wcel 2144 {cab 2742 ≠ wne 2959 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4287 ifcif 4482 {csn 4584 {cpr 4586 class class class wbr 5102 ↦ cmpt 5183 ran crn 5650 ‘cfv 6523 (class class class)co 7398 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 +∞cpnf 11215 -∞cmnf 11216 ℝ^*cxr 11217 < clt 11218 -cneg 11417 ℝ⁺crp 12995 (,)cioo 13351 abscabs 15263 TopOpenctopn 17452 topGenctg 17468 ℂ_fldccnfld 21426 Topctop 22955 intcnt 23079 D cdv 25927

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 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 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-er 8680 df-map 8812 df-pm 8813 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-q 12952 df-rp 12996 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13355 df-icc 13358 df-fz 13515 df-fzo 13662 df-seq 14017 df-exp 14077 df-hash 14346 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-starv 17303 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-hom 17312 df-cco 17313 df-rest 17453 df-topn 17454 df-0g 17472 df-gsum 17473 df-topgen 17474 df-pt 17475 df-prds 17478 df-xrs 17534 df-qtop 17539 df-imas 17540 df-xps 17542 df-mre 17616 df-mrc 17617 df-acs 17619 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-submnd 18820 df-mulg 19112 df-cntz 19359 df-cmn 19824 df-psmet 21418 df-xmet 21419 df-met 21420 df-bl 21421 df-mopn 21422 df-fbas 21423 df-fg 21424 df-cnfld 21427 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cld 23081 df-ntr 23082 df-cls 23083 df-nei 23160 df-lp 23198 df-perf 23199 df-cn 23289 df-cnp 23290 df-haus 23377 df-tx 23624 df-hmeo 23817 df-fil 23908 df-fm 24000 df-flim 24001 df-flf 24002 df-xms 24382 df-ms 24383 df-tms 24384 df-cncf 24942 df-limc 25930 df-dv 25931

This theorem is referenced by: readvrec 42976

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