MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-rrecex Unicode version

Axiom ax-rrecex 8854
Description: Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 8830. (Contributed by Eric Schmidt, 11-Apr-2007.)
Assertion
Ref Expression
ax-rrecex  |-  ( ( A  e.  RR  /\  A  =/=  0 )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 )
Distinct variable group:    x, A

Detailed syntax breakdown of Axiom ax-rrecex
StepHypRef Expression
1 cA . . . 4  class  A
2 cr 8781 . . . 4  class  RR
31, 2wcel 1701 . . 3  wff  A  e.  RR
4 cc0 8782 . . . 4  class  0
51, 4wne 2479 . . 3  wff  A  =/=  0
63, 5wa 358 . 2  wff  ( A  e.  RR  /\  A  =/=  0 )
7 vx . . . . . 6  set  x
87cv 1632 . . . . 5  class  x
9 cmul 8787 . . . . 5  class  x.
101, 8, 9co 5900 . . . 4  class  ( A  x.  x )
11 c1 8783 . . . 4  class  1
1210, 11wceq 1633 . . 3  wff  ( A  x.  x )  =  1
1312, 7, 2wrex 2578 . 2  wff  E. x  e.  RR  ( A  x.  x )  =  1
146, 13wi 4 1  wff  ( ( A  e.  RR  /\  A  =/=  0 )  ->  E. x  e.  RR  ( A  x.  x
)  =  1 )
Colors of variables: wff set class
This axiom is referenced by:  1re  8882  00id  9032  mul02lem1  9033  addid1  9037  recex  9445  rereccl  9523  xrecex  23318
  Copyright terms: Public domain W3C validator