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Definition df-xrs 13726
 Description: The extended real number structure. Unlike df-cnfld 16704, 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 16704. 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 -ball around it). All components of this structure agree with ℂfld when restricted to . (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
df-xrs TopSet ordTop
Distinct variable group:   ,

Detailed syntax breakdown of Definition df-xrs
StepHypRef Expression
1 cxrs 13722 . 2
2 cnx 13466 . . . . . 6
3 cbs 13469 . . . . . 6
42, 3cfv 5454 . . . . 5
5 cxr 9119 . . . . 5
64, 5cop 3817 . . . 4
7 cplusg 13529 . . . . . 6
82, 7cfv 5454 . . . . 5
9 cxad 10708 . . . . 5
108, 9cop 3817 . . . 4
11 cmulr 13530 . . . . . 6
122, 11cfv 5454 . . . . 5
13 cxmu 10709 . . . . 5
1412, 13cop 3817 . . . 4
156, 10, 14ctp 3816 . . 3
16 cts 13535 . . . . . 6 TopSet
172, 16cfv 5454 . . . . 5 TopSet
18 cle 9121 . . . . . 6
19 cordt 13721 . . . . . 6 ordTop
2018, 19cfv 5454 . . . . 5 ordTop
2117, 20cop 3817 . . . 4 TopSet ordTop
22 cple 13536 . . . . . 6
232, 22cfv 5454 . . . . 5
2423, 18cop 3817 . . . 4
25 cds 13538 . . . . . 6
262, 25cfv 5454 . . . . 5
27 vx . . . . . 6
28 vy . . . . . 6
2927cv 1651 . . . . . . . 8
3028cv 1651 . . . . . . . 8
3129, 30, 18wbr 4212 . . . . . . 7
3229cxne 10707 . . . . . . . 8
3330, 32, 9co 6081 . . . . . . 7
3430cxne 10707 . . . . . . . 8
3529, 34, 9co 6081 . . . . . . 7
3631, 33, 35cif 3739 . . . . . 6
3727, 28, 5, 5, 36cmpt2 6083 . . . . 5
3826, 37cop 3817 . . . 4
3921, 24, 38ctp 3816 . . 3 TopSet ordTop
4015, 39cun 3318 . 2 TopSet ordTop
411, 40wceq 1652 1 TopSet ordTop
 Colors of variables: wff set class This definition is referenced by:  xrsstr  16715  xrsex  16716  xrsbas  16717  xrsadd  16718  xrsmul  16719  xrstset  16720  xrsle  16721  xrsds  16741
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