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Mirrors > Home > MPE Home > Th. List > ax-rrecex | Structured version Visualization version GIF version |
Description: Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by Theorem axrrecex 11200. (Contributed by Eric Schmidt, 11-Apr-2007.) |
Ref | Expression |
---|---|
ax-rrecex | ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | cr 11151 | . . . 4 class ℝ | |
3 | 1, 2 | wcel 2105 | . . 3 wff 𝐴 ∈ ℝ |
4 | cc0 11152 | . . . 4 class 0 | |
5 | 1, 4 | wne 2937 | . . 3 wff 𝐴 ≠ 0 |
6 | 3, 5 | wa 395 | . 2 wff (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) |
7 | vx | . . . . . 6 setvar 𝑥 | |
8 | 7 | cv 1535 | . . . . 5 class 𝑥 |
9 | cmul 11157 | . . . . 5 class · | |
10 | 1, 8, 9 | co 7430 | . . . 4 class (𝐴 · 𝑥) |
11 | c1 11153 | . . . 4 class 1 | |
12 | 10, 11 | wceq 1536 | . . 3 wff (𝐴 · 𝑥) = 1 |
13 | 12, 7, 2 | wrex 3067 | . 2 wff ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1 |
14 | 6, 13 | wi 4 | 1 wff ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) |
Colors of variables: wff setvar class |
This axiom is referenced by: 1re 11258 00id 11433 mul02lem1 11434 addrid 11438 recex 11892 rereccl 11982 xrecex 32886 remulcan2d 42276 remul02 42411 remul01 42413 remulinvcom 42438 remullid 42439 remulcand 42444 sn-0tie0 42445 sn-itrere 42474 sn-retire 42475 |
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