Axiom ax-rrecex 11182

Description: Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by Theorem axrrecex 11158. (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 11109	. . . 4 class ℝ
3	1, 2	wcel 2107	. . 3 wff 𝐴 ∈ ℝ
4		cc0 11110	. . . 4 class 0
5	1, 4	wne 2941	. . 3 wff 𝐴 ≠ 0
6	3, 5	wa 397	. 2 wff (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)
7		vx	. . . . . 6 setvar 𝑥
8	7	cv 1541	. . . . 5 class 𝑥
9		cmul 11115	. . . . 5 class ·
10	1, 8, 9	co 7409	. . . 4 class (𝐴 · 𝑥)
11		c1 11111	. . . 4 class 1
12	10, 11	wceq 1542	. . 3 wff (𝐴 · 𝑥) = 1
13	12, 7, 2	wrex 3071	. 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  11214  00id  11389  mul02lem1  11390  addrid  11394  recex  11846  rereccl  11932  xrecex  32086  remulcan2d  41177  remul02  41278  remul01  41280  remulinvcom  41305  remullid  41306  remulcand  41311  sn-0tie0  41312  itrere  41339  retire  41340