Axiom ax-rrecex 11178

Description: Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by Theorem axrrecex 11154. (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 11105	. . . 4 class ℝ
3	1, 2	wcel 2142	. . 3 wff 𝐴 ∈ ℝ
4		cc0 11106	. . . 4 class 0
5	1, 4	wne 2957	. . 3 wff 𝐴 ≠ 0
6	3, 5	wa 400	. 2 wff (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)
7		vx	. . . . . 6 setvar 𝑥
8	7	cv 1568	. . . . 5 class 𝑥
9		cmul 11111	. . . . 5 class ·
10	1, 8, 9	co 7412	. . . 4 class (𝐴 · 𝑥)
11		c1 11107	. . . 4 class 1
12	10, 11	wceq 1569	. . 3 wff (𝐴 · 𝑥) = 1
13	12, 7, 2	wrex 3088	. 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 used by:  1re  11214  00id  11391  mul02lem1  11392  addrid  11396  recex  11852  rereccl  11939  xrecex  33250  remulcan2d  43052  remul02  43194  remul01  43196  remulinvcom  43222  remullid  43223  remulcand  43228  rediveud  43232  sn-0tie0  43253