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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 11203. (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 11154 | . . . 4 class ℝ | |
| 3 | 1, 2 | wcel 2108 | . . 3 wff 𝐴 ∈ ℝ |
| 4 | cc0 11155 | . . . 4 class 0 | |
| 5 | 1, 4 | wne 2940 | . . 3 wff 𝐴 ≠ 0 |
| 6 | 3, 5 | wa 395 | . 2 wff (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) |
| 7 | vx | . . . . . 6 setvar 𝑥 | |
| 8 | 7 | cv 1539 | . . . . 5 class 𝑥 |
| 9 | cmul 11160 | . . . . 5 class · | |
| 10 | 1, 8, 9 | co 7431 | . . . 4 class (𝐴 · 𝑥) |
| 11 | c1 11156 | . . . 4 class 1 | |
| 12 | 10, 11 | wceq 1540 | . . 3 wff (𝐴 · 𝑥) = 1 |
| 13 | 12, 7, 2 | wrex 3070 | . 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 11261 00id 11436 mul02lem1 11437 addrid 11441 recex 11895 rereccl 11985 xrecex 32902 remulcan2d 42298 remul02 42435 remul01 42437 remulinvcom 42462 remullid 42463 remulcand 42468 sn-0tie0 42469 sn-itrere 42498 sn-retire 42499 |
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