Axiom ax-rrecex 10680

Description: Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by Theorem axrrecex 10656. (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 10607	. . . 4 class ℝ
3	1, 2	wcel 2113	. . 3 wff 𝐴 ∈ ℝ
4		cc0 10608	. . . 4 class 0
5	1, 4	wne 2934	. . 3 wff 𝐴 ≠ 0
6	3, 5	wa 399	. 2 wff (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)
7		vx	. . . . . 6 setvar 𝑥
8	7	cv 1541	. . . . 5 class 𝑥
9		cmul 10613	. . . . 5 class ·
10	1, 8, 9	co 7164	. . . 4 class (𝐴 · 𝑥)
11		c1 10609	. . . 4 class 1
12	10, 11	wceq 1542	. . 3 wff (𝐴 · 𝑥) = 1
13	12, 7, 2	wrex 3054	. 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  10712  00id  10886  mul02lem1  10887  addid1  10891  recex  11343  rereccl  11429  xrecex  30761  remulcan2d  39853  remul02  39949  remul01  39951  remulinvcom  39975  remulid2  39976  remulcand  39981  sn-0tie0  39982  itrere  39997  retire  39998