Axiom ax-rrecex 11116

Description: Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by Theorem axrrecex 11092. (Contributed by Eric Schmidt, 11-Apr-2007.)

Assertion

Ref	Expression
ax-rrecex	⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 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 11043	. . . 4 class ℝ
3	1, 2	wcel 2109	. . 3 wff 𝐴 ∈ ℝ
4		cc0 11044	. . . 4 class 0
5	1, 4	wne 2925	. . 3 wff 𝐴 ≠ 0
6	3, 5	wa 395	. 2 wff (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)
7		vx	. . . . . 6 setvar 𝑥
8	7	cv 1539	. . . . 5 class 𝑥
9		cmul 11049	. . . . 5 class ·
10	1, 8, 9	co 7369	. . . 4 class (𝐴 · 𝑥)
11		c1 11045	. . . 4 class 1
12	10, 11	wceq 1540	. . 3 wff (𝐴 · 𝑥) = 1
13	12, 7, 2	wrex 3053	. 2 wff ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1
14	6, 13	wi 4	1 wff ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)

This axiom is referenced by:  1re  11150  00id  11325  mul02lem1  11326  addrid  11330  recex  11786  rereccl  11876  xrecex  32813  remulcan2d  42218  remul02  42366  remul01  42368  remulinvcom  42394  remullid  42395  remulcand  42400  rediveud  42404  sn-0tie0  42412