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Axiom ax-precex 6975
Description: Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by theorem axprecex 6935. (Contributed by Jim Kingdon, 6-Feb-2020.)
Assertion
Ref Expression
ax-precex ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1))
Distinct variable group:   𝑥,𝐴

Detailed syntax breakdown of Axiom ax-precex
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cr 6869 . . . 4 class
31, 2wcel 1393 . . 3 wff 𝐴 ∈ ℝ
4 cc0 6870 . . . 4 class 0
5 cltrr 6874 . . . 4 class <
64, 1, 5wbr 3761 . . 3 wff 0 < 𝐴
73, 6wa 97 . 2 wff (𝐴 ∈ ℝ ∧ 0 < 𝐴)
8 vx . . . . . 6 setvar 𝑥
98cv 1242 . . . . 5 class 𝑥
104, 9, 5wbr 3761 . . . 4 wff 0 < 𝑥
11 cmul 6875 . . . . . 6 class ·
121, 9, 11co 5499 . . . . 5 class (𝐴 · 𝑥)
13 c1 6871 . . . . 5 class 1
1412, 13wceq 1243 . . . 4 wff (𝐴 · 𝑥) = 1
1510, 14wa 97 . . 3 wff (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)
1615, 8, 2wrex 2304 . 2 wff 𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)
177, 16wi 4 1 wff ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1))
Colors of variables: wff set class
This axiom is referenced by:  recexre  7545  recexgt0  7547
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