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Axiom ax-rrecex 10603
Description: Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 10579. (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 10530 . . . 4 class
31, 2wcel 2107 . . 3 wff 𝐴 ∈ ℝ
4 cc0 10531 . . . 4 class 0
51, 4wne 3021 . . 3 wff 𝐴 ≠ 0
63, 5wa 396 . 2 wff (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)
7 vx . . . . . 6 setvar 𝑥
87cv 1529 . . . . 5 class 𝑥
9 cmul 10536 . . . . 5 class ·
101, 8, 9co 7150 . . . 4 class (𝐴 · 𝑥)
11 c1 10532 . . . 4 class 1
1210, 11wceq 1530 . . 3 wff (𝐴 · 𝑥) = 1
1312, 7, 2wrex 3144 . 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  10635  00id  10809  mul02lem1  10810  addid1  10814  recex  11266  rereccl  11352  xrecex  30529  remulcan2d  39040  remul02  39119  remul01  39121  remulinvcom  39132  remulid2  39133  remulcand  39134
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