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Theorem recexgt0 8310
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 7698 . 2  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. x  e.  RR  ( 0  <RR  x  /\  ( A  x.  x
)  =  1 ) )
2 0re 7734 . . . 4  |-  0  e.  RR
3 ltxrlt 7798 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  <->  0 
<RR  A ) )
42, 3mpan 420 . . 3  |-  ( A  e.  RR  ->  (
0  <  A  <->  0  <RR  A ) )
54pm5.32i 449 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  <->  ( A  e.  RR  /\  0  <RR  A ) )
6 ltxrlt 7798 . . . . 5  |-  ( ( 0  e.  RR  /\  x  e.  RR )  ->  ( 0  <  x  <->  0 
<RR  x ) )
72, 6mpan 420 . . . 4  |-  ( x  e.  RR  ->  (
0  <  x  <->  0  <RR  x ) )
87anbi1d 460 . . 3  |-  ( x  e.  RR  ->  (
( 0  <  x  /\  ( A  x.  x
)  =  1 )  <-> 
( 0  <RR  x  /\  ( A  x.  x
)  =  1 ) ) )
98rexbiia 2427 . 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 1316    e. wcel 1465   E.wrex 2394   class class class wbr 3899  (class class class)co 5742   RRcr 7587   0cc0 7588   1c1 7589    <RR cltrr 7592    x. cmul 7593    < clt 7768
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685  ax-rnegex 7697  ax-precex 7698
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-xp 4515  df-pnf 7770  df-mnf 7771  df-ltxr 7773
This theorem is referenced by:  ltmul1  8322
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