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Theorem recexgt0 8455
Description: Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.)
Assertion
Ref Expression
recexgt0  |-  ( ( A  e.  RR  /\  0  <  A )  ->  E. x  e.  RR  ( 0  <  x  /\  ( A  x.  x
)  =  1 ) )
Distinct variable group:    x, A

Proof of Theorem recexgt0
StepHypRef Expression
1 ax-precex 7842 . 2  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. x  e.  RR  ( 0  <RR  x  /\  ( A  x.  x
)  =  1 ) )
2 0re 7878 . . . 4  |-  0  e.  RR
3 ltxrlt 7943 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  <->  0 
<RR  A ) )
42, 3mpan 421 . . 3  |-  ( A  e.  RR  ->  (
0  <  A  <->  0  <RR  A ) )
54pm5.32i 450 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  <->  ( A  e.  RR  /\  0  <RR  A ) )
6 ltxrlt 7943 . . . . 5  |-  ( ( 0  e.  RR  /\  x  e.  RR )  ->  ( 0  <  x  <->  0 
<RR  x ) )
72, 6mpan 421 . . . 4  |-  ( x  e.  RR  ->  (
0  <  x  <->  0  <RR  x ) )
87anbi1d 461 . . 3  |-  ( x  e.  RR  ->  (
( 0  <  x  /\  ( A  x.  x
)  =  1 )  <-> 
( 0  <RR  x  /\  ( A  x.  x
)  =  1 ) ) )
98rexbiia 2472 . 2  |-  ( E. x  e.  RR  (
0  <  x  /\  ( A  x.  x
)  =  1 )  <->  E. x  e.  RR  ( 0  <RR  x  /\  ( A  x.  x
)  =  1 ) )
101, 5, 93imtr4i 200 1  |-  ( ( A  e.  RR  /\  0  <  A )  ->  E. x  e.  RR  ( 0  <  x  /\  ( A  x.  x
)  =  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   E.wrex 2436   class class class wbr 3965  (class class class)co 5824   RRcr 7731   0cc0 7732   1c1 7733    <RR cltrr 7736    x. cmul 7737    < clt 7912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1re 7826  ax-addrcl 7829  ax-rnegex 7841  ax-precex 7842
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4592  df-pnf 7914  df-mnf 7915  df-ltxr 7917
This theorem is referenced by:  ltmul1  8467
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