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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
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cr 10870 . . . 4 class
31, 2wcel 2106 . . 3 wff 𝐴 ∈ ℝ
4 cc0 10871 . . . 4 class 0
51, 4wne 2943 . . 3 wff 𝐴 ≠ 0
63, 5wa 396 . 2 wff (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)
7 vx . . . . . 6 setvar 𝑥
87cv 1538 . . . . 5 class 𝑥
9 cmul 10876 . . . . 5 class ·
101, 8, 9co 7275 . . . 4 class (𝐴 · 𝑥)
11 c1 10872 . . . 4 class 1
1210, 11wceq 1539 . . 3 wff (𝐴 · 𝑥) = 1
1312, 7, 2wrex 3065 . 2 wff 𝑥 ∈ ℝ (𝐴 · 𝑥) = 1
146, 13wi 4 1 wff ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
Colors of variables: wff setvar class
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
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