Axiom ax-rrecex 10008

Description: Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 9984. (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

Step	Hyp	Ref	Expression
1		cA	. . . 4 class A
2		cr 9935	. . . 4 class RR
3	1, 2	wcel 1990	. . 3 wff A e. RR
4		cc0 9936	. . . 4 class 0
5	1, 4	wne 2794	. . 3 wff A =/= 0
6	3, 5	wa 384	. 2 wff ( A e. RR /\ A =/= 0 )
7		vx	. . . . . 6 setvar x
8	7	cv 1482	. . . . 5 class x
9		cmul 9941	. . . . 5 class x.
10	1, 8, 9	co 6650	. . . 4 class ( A x. x )
11		c1 9937	. . . 4 class 1
12	10, 11	wceq 1483	. . 3 wff ( A x. x ) = 1
13	12, 7, 2	wrex 2913	. 2 wff E. x e. RR ( A x. x ) = 1
14	6, 13	wi 4	1 wff ( ( A e. RR /\ A =/= 0 ) -> E. x e. RR ( A x. x ) = 1 )

This axiom is referenced by:  1re  10039  00id  10211  mul02lem1  10212  addid1  10216  recex  10659  rereccl  10743  xrecex  29628