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Theorem recexgt0 7645
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 7052 . 2  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. x  e.  RR  ( 0  <RR  x  /\  ( A  x.  x
)  =  1 ) )
2 0re 7085 . . . 4  |-  0  e.  RR
3 ltxrlt 7144 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  <->  0 
<RR  A ) )
42, 3mpan 408 . . 3  |-  ( A  e.  RR  ->  (
0  <  A  <->  0  <RR  A ) )
54pm5.32i 435 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  <->  ( A  e.  RR  /\  0  <RR  A ) )
6 ltxrlt 7144 . . . . 5  |-  ( ( 0  e.  RR  /\  x  e.  RR )  ->  ( 0  <  x  <->  0 
<RR  x ) )
72, 6mpan 408 . . . 4  |-  ( x  e.  RR  ->  (
0  <  x  <->  0  <RR  x ) )
87anbi1d 446 . . 3  |-  ( x  e.  RR  ->  (
( 0  <  x  /\  ( A  x.  x
)  =  1 )  <-> 
( 0  <RR  x  /\  ( A  x.  x
)  =  1 ) ) )
98rexbiia 2356 . 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 194 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 101    <-> wb 102    = wceq 1259    e. wcel 1409   E.wrex 2324   class class class wbr 3792  (class class class)co 5540   RRcr 6946   0cc0 6947   1c1 6948    <RR cltrr 6951    x. cmul 6952    < clt 7119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-1re 7036  ax-addrcl 7039  ax-rnegex 7051  ax-precex 7052
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-xp 4379  df-pnf 7121  df-mnf 7122  df-ltxr 7124
This theorem is referenced by:  ltmul1  7657
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