**Description:** A number is equal to the reciprocal of its reciprocal. (Contributed by Mario Carneiro, 27-May-2016)

Ref | Expression | ||
---|---|---|---|

Hypotheses | div1d.1 | $${\u22a2}{\phi}\to {A}\in \u2102$$ | |

reccld.2 | $${\u22a2}{\phi}\to {A}\ne 0$$ | ||

Assertion | recrecd | $${\u22a2}{\phi}\to \frac{1}{\frac{1}{{A}}}={A}$$ |

Step | Hyp | Ref | Expression |
---|---|---|---|

1 | div1d.1 | $${\u22a2}{\phi}\to {A}\in \u2102$$ | |

2 | reccld.2 | $${\u22a2}{\phi}\to {A}\ne 0$$ | |

3 | recrec | $${\u22a2}\left({A}\in \u2102\wedge {A}\ne 0\right)\to \frac{1}{\frac{1}{{A}}}={A}$$ | |

4 | 1 2 3 | syl2anc | $${\u22a2}{\phi}\to \frac{1}{\frac{1}{{A}}}={A}$$ |