Axiom ax-rrecex 10943

Description: Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by Theorem axrrecex 10919. (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 10870	. . . 4 class ℝ
3	1, 2	wcel 2106	. . 3 wff 𝐴 ∈ ℝ
4		cc0 10871	. . . 4 class 0
5	1, 4	wne 2943	. . 3 wff 𝐴 ≠ 0
6	3, 5	wa 396	. 2 wff (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)
7		vx	. . . . . 6 setvar 𝑥
8	7	cv 1538	. . . . 5 class 𝑥
9		cmul 10876	. . . . 5 class ·
10	1, 8, 9	co 7275	. . . 4 class (𝐴 · 𝑥)
11		c1 10872	. . . 4 class 1
12	10, 11	wceq 1539	. . 3 wff (𝐴 · 𝑥) = 1
13	12, 7, 2	wrex 3065	. 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  10975  00id  11150  mul02lem1  11151  addid1  11155  recex  11607  rereccl  11693  xrecex  31194  remulcan2d  40293  remul02  40388  remul01  40390  remulinvcom  40414  remulid2  40415  remulcand  40420  sn-0tie0  40421  itrere  40436  retire  40437