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Definition df-xrs 17509
Description: The extended real number structure. Unlike df-cnfld 21337, 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 21337. 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 17507 . 2 class *𝑠
2 cnx 17187 . . . . . 6 class ndx
3 cbs 17205 . . . . . 6 class Base
42, 3cfv 6543 . . . . 5 class (Base‘ndx)
5 cxr 11285 . . . . 5 class *
64, 5cop 4629 . . . 4 class ⟨(Base‘ndx), ℝ*
7 cplusg 17258 . . . . . 6 class +g
82, 7cfv 6543 . . . . 5 class (+g‘ndx)
9 cxad 13135 . . . . 5 class +𝑒
108, 9cop 4629 . . . 4 class ⟨(+g‘ndx), +𝑒
11 cmulr 17259 . . . . . 6 class .r
122, 11cfv 6543 . . . . 5 class (.r‘ndx)
13 cxmu 13136 . . . . 5 class ·e
1412, 13cop 4629 . . . 4 class ⟨(.r‘ndx), ·e
156, 10, 14ctp 4627 . . 3 class {⟨(Base‘ndx), ℝ*⟩, ⟨(+g‘ndx), +𝑒 ⟩, ⟨(.r‘ndx), ·e ⟩}
16 cts 17264 . . . . . 6 class TopSet
172, 16cfv 6543 . . . . 5 class (TopSet‘ndx)
18 cle 11287 . . . . . 6 class
19 cordt 17506 . . . . . 6 class ordTop
2018, 19cfv 6543 . . . . 5 class (ordTop‘ ≤ )
2117, 20cop 4629 . . . 4 class ⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩
22 cple 17265 . . . . . 6 class le
232, 22cfv 6543 . . . . 5 class (le‘ndx)
2423, 18cop 4629 . . . 4 class ⟨(le‘ndx), ≤ ⟩
25 cds 17267 . . . . . 6 class dist
262, 25cfv 6543 . . . . 5 class (dist‘ndx)
27 vx . . . . . 6 setvar 𝑥
28 vy . . . . . 6 setvar 𝑦
2927cv 1533 . . . . . . . 8 class 𝑥
3028cv 1533 . . . . . . . 8 class 𝑦
3129, 30, 18wbr 5143 . . . . . . 7 wff 𝑥𝑦
3229cxne 13134 . . . . . . . 8 class -𝑒𝑥
3330, 32, 9co 7413 . . . . . . 7 class (𝑦 +𝑒 -𝑒𝑥)
3430cxne 13134 . . . . . . . 8 class -𝑒𝑦
3529, 34, 9co 7413 . . . . . . 7 class (𝑥 +𝑒 -𝑒𝑦)
3631, 33, 35cif 4523 . . . . . 6 class if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))
3727, 28, 5, 5, 36cmpo 7415 . . . . 5 class (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))
3826, 37cop 4629 . . . 4 class ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩
3921, 24, 38ctp 4627 . . 3 class {⟨(TopSet‘ndx), (ordTop‘ ≤ )⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))⟩}
4015, 39cun 3944 . 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 1534 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  21366  xrsex  21367  xrsbas  21368  xrsadd  21369  xrsmul  21370  xrstset  21371  xrsle  21372  xrsds  21399
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