MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-rrecex Structured version   Visualization version   GIF version

Axiom ax-rrecex 10046
Description: Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 10022. (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 9973 . . . 4 class
31, 2wcel 2030 . . 3 wff 𝐴 ∈ ℝ
4 cc0 9974 . . . 4 class 0
51, 4wne 2823 . . 3 wff 𝐴 ≠ 0
63, 5wa 383 . 2 wff (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)
7 vx . . . . . 6 setvar 𝑥
87cv 1522 . . . . 5 class 𝑥
9 cmul 9979 . . . . 5 class ·
101, 8, 9co 6690 . . . 4 class (𝐴 · 𝑥)
11 c1 9975 . . . 4 class 1
1210, 11wceq 1523 . . 3 wff (𝐴 · 𝑥) = 1
1312, 7, 2wrex 2942 . 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  10077  00id  10249  mul02lem1  10250  addid1  10254  recex  10697  rereccl  10781  xrecex  29756
  Copyright terms: Public domain W3C validator