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Axiom ax-precex 7399
Description: Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by theorem axprecex 7359. (Contributed by Jim Kingdon, 6-Feb-2020.)
Assertion
Ref Expression
ax-precex  |-  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. x  e.  RR  ( 0  <RR  x  /\  ( A  x.  x
)  =  1 ) )
Distinct variable group:    x, A

Detailed syntax breakdown of Axiom ax-precex
StepHypRef Expression
1 cA . . . 4  class  A
2 cr 7293 . . . 4  class  RR
31, 2wcel 1436 . . 3  wff  A  e.  RR
4 cc0 7294 . . . 4  class  0
5 cltrr 7298 . . . 4  class  <RR
64, 1, 5wbr 3820 . . 3  wff  0  <RR  A
73, 6wa 102 . 2  wff  ( A  e.  RR  /\  0  <RR  A )
8 vx . . . . . 6  setvar  x
98cv 1286 . . . . 5  class  x
104, 9, 5wbr 3820 . . . 4  wff  0  <RR  x
11 cmul 7299 . . . . . 6  class  x.
121, 9, 11co 5613 . . . . 5  class  ( A  x.  x )
13 c1 7295 . . . . 5  class  1
1412, 13wceq 1287 . . . 4  wff  ( A  x.  x )  =  1
1510, 14wa 102 . . 3  wff  ( 0 
<RR  x  /\  ( A  x.  x )  =  1 )
1615, 8, 2wrex 2356 . 2  wff  E. x  e.  RR  ( 0  <RR  x  /\  ( A  x.  x )  =  1 )
177, 16wi 4 1  wff  ( ( A  e.  RR  /\  0  <RR  A )  ->  E. x  e.  RR  ( 0  <RR  x  /\  ( A  x.  x
)  =  1 ) )
Colors of variables: wff set class
This axiom is referenced by:  recexre  7996  recexgt0  7998
  Copyright terms: Public domain W3C validator