Axiom ax-rrecex 11184

Description: Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by Theorem axrrecex 11160. (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 11111	. . . 4 class ℝ
3	1, 2	wcel 2106	. . 3 wff 𝐴 ∈ ℝ
4		cc0 11112	. . . 4 class 0
5	1, 4	wne 2940	. . 3 wff 𝐴 ≠ 0
6	3, 5	wa 396	. 2 wff (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)
7		vx	. . . . . 6 setvar 𝑥
8	7	cv 1540	. . . . 5 class 𝑥
9		cmul 11117	. . . . 5 class ·
10	1, 8, 9	co 7411	. . . 4 class (𝐴 · 𝑥)
11		c1 11113	. . . 4 class 1
12	10, 11	wceq 1541	. . 3 wff (𝐴 · 𝑥) = 1
13	12, 7, 2	wrex 3070	. 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  11216  00id  11391  mul02lem1  11392  addrid  11396  recex  11848  rereccl  11934  xrecex  32124  remulcan2d  41259  remul02  41360  remul01  41362  remulinvcom  41387  remullid  41388  remulcand  41393  sn-0tie0  41394  itrere  41421  retire  41422