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Definition df-xrs 16370
Description: The extended real number structure. Unlike df-cnfld 19958, 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 19958. 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 fld when restricted to . (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
df-xrs *𝑠 = ({⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∪ {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩})
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-xrs
StepHypRef Expression
1 cxrs 16368 . 2 class *𝑠
2 cnx 16068 . . . . . 6 class ndx
3 cbs 16071 . . . . . 6 class Base
42, 3cfv 6104 . . . . 5 class (Base‘ndx)
5 cxr 10361 . . . . 5 class *
64, 5cop 4383 . . . 4 class ⟨(Base‘ndx), ℝ*
7 cplusg 16156 . . . . . 6 class +g
82, 7cfv 6104 . . . . 5 class (+g‘ndx)
9 cxad 12163 . . . . 5 class +𝑒
108, 9cop 4383 . . . 4 class ⟨(+g‘ndx), +𝑒
11 cmulr 16157 . . . . . 6 class .r
122, 11cfv 6104 . . . . 5 class (.r‘ndx)
13 cxmu 12164 . . . . 5 class ·e
1412, 13cop 4383 . . . 4 class ⟨(.r‘ndx), ·e
156, 10, 14ctp 4381 . . 3 class {⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩}
16 cts 16162 . . . . . 6 class TopSet
172, 16cfv 6104 . . . . 5 class (TopSet‘ndx)
18 cle 10363 . . . . . 6 class
19 cordt 16367 . . . . . 6 class ordTop
2018, 19cfv 6104 . . . . 5 class (ordTop‘ ≤ )
2117, 20cop 4383 . . . 4 class ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩
22 cple 16163 . . . . . 6 class le
232, 22cfv 6104 . . . . 5 class (le‘ndx)
2423, 18cop 4383 . . . 4 class ⟨(le‘ndx), ≤ ⟩
25 cds 16165 . . . . . 6 class dist
262, 25cfv 6104 . . . . 5 class (dist‘ndx)
27 vx . . . . . 6 setvar 𝑥
28 vy . . . . . 6 setvar 𝑦
2927cv 1636 . . . . . . . 8 class 𝑥
3028cv 1636 . . . . . . . 8 class 𝑦
3129, 30, 18wbr 4851 . . . . . . 7 wff 𝑥𝑦
3229cxne 12162 . . . . . . . 8 class -𝑒𝑥
3330, 32, 9co 6877 . . . . . . 7 class (𝑦 +𝑒 -𝑒𝑥)
3430cxne 12162 . . . . . . . 8 class -𝑒𝑦
3529, 34, 9co 6877 . . . . . . 7 class (𝑥 +𝑒 -𝑒𝑦)
3631, 33, 35cif 4286 . . . . . 6 class if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))
3727, 28, 5, 5, 36cmpt2 6879 . . . . 5 class (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))
3826, 37cop 4383 . . . 4 class ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩
3921, 24, 38ctp 4381 . . 3 class {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩}
4015, 39cun 3774 . 2 class ({⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∪ {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩})
411, 40wceq 1637 1 wff *𝑠 = ({⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩} ∪ {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩})
Colors of variables: wff setvar class
This definition is referenced by:  xrsstr  19971  xrsex  19972  xrsbas  19973  xrsadd  19974  xrsmul  19975  xrstset  19976  xrsle  19977  xrsds  20000
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