Definition df-xrs 17424

Description: The extended real number structure. Unlike df-cnfld 21312, the extended real numbers do not have good algebraic properties, so this is not actually a group or anything higher, even though it has just as many operations as df-cnfld 21312. The main interest in this structure is in its ordering, which is complete and compact. The metric described here is an extension of the absolute value metric, but it is not itself a metric because +∞ is infinitely far from all other points. The topology is based on the order and not the extended metric (which would make +∞ an isolated point since there is nothing else in the 1 -ball around it). All components of this structure agree with ℂ_fld when restricted to ℝ. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
df-xrs	⊢ ℝ*𝑠 = ({⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∪ {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩})

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-xrs

Step	Hyp	Ref	Expression
1		cxrs 17422	. 2 class ℝ*𝑠
2		cnx 17121	. . . . . 6 class ndx
3		cbs 17137	. . . . . 6 class Base
4	2, 3	cfv 6490	. . . . 5 class (Base‘ndx)
5		cxr 11166	. . . . 5 class ℝ*
6	4, 5	cop 4574	. . . 4 class ⟨(Base‘ndx), ℝ*⟩
7		cplusg 17178	. . . . . 6 class +g
8	2, 7	cfv 6490	. . . . 5 class (+g‘ndx)
9		cxad 13025	. . . . 5 class +𝑒
10	8, 9	cop 4574	. . . 4 class ⟨(+g‘ndx), +𝑒 ⟩
11		cmulr 17179	. . . . . 6 class .r
12	2, 11	cfv 6490	. . . . 5 class (.r‘ndx)
13		cxmu 13026	. . . . 5 class ·e
14	12, 13	cop 4574	. . . 4 class ⟨(.r‘ndx), ·e ⟩
15	6, 10, 14	ctp 4572	. . 3 class {⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩}
16		cts 17184	. . . . . 6 class TopSet
17	2, 16	cfv 6490	. . . . 5 class (TopSet‘ndx)
18		cle 11168	. . . . . 6 class ≤
19		cordt 17421	. . . . . 6 class ordTop
20	18, 19	cfv 6490	. . . . 5 class (ordTop‘ ≤ )
21	17, 20	cop 4574	. . . 4 class ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩
22		cple 17185	. . . . . 6 class le
23	2, 22	cfv 6490	. . . . 5 class (le‘ndx)
24	23, 18	cop 4574	. . . 4 class ⟨(le‘ndx), ≤ ⟩
25		cds 17187	. . . . . 6 class dist
26	2, 25	cfv 6490	. . . . 5 class (dist‘ndx)
27		vx	. . . . . 6 setvar 𝑥
28		vy	. . . . . 6 setvar 𝑦
29	27	cv 1541	. . . . . . . 8 class 𝑥
30	28	cv 1541	. . . . . . . 8 class 𝑦
31	29, 30, 18	wbr 5086	. . . . . . 7 wff 𝑥 ≤ 𝑦
32	29	cxne 13024	. . . . . . . 8 class -𝑒𝑥
33	30, 32, 9	co 7358	. . . . . . 7 class (𝑦 +𝑒 -𝑒𝑥)
34	30	cxne 13024	. . . . . . . 8 class -𝑒𝑦
35	29, 34, 9	co 7358	. . . . . . 7 class (𝑥 +𝑒 -𝑒𝑦)
36	31, 33, 35	cif 4467	. . . . . 6 class if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))
37	27, 28, 5, 5, 36	cmpo 7360	. . . . 5 class (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))
38	26, 37	cop 4574	. . . 4 class ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩
39	21, 24, 38	ctp 4572	. . 3 class {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩}
40	15, 39	cun 3888	. 2 class ({⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∪ {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩})
41	1, 40	wceq 1542	1 wff ℝ*𝑠 = ({⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∪ {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩})

This definition is referenced by:  xrsle  17526  xrsbas  17528  xrsstr  21340  xrsex  21341  xrsadd  21342  xrsmul  21343  xrstset  21344  xrsds  21366