**Description:** The ratio of nonzero numbers is nonzero. (Contributed by Mario
Carneiro, 27-May-2016)

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

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

divcld.2 | $${\u22a2}{\phi}\to {B}\in \u2102$$ | ||

divne0d.3 | $${\u22a2}{\phi}\to {A}\ne 0$$ | ||

divne0d.4 | $${\u22a2}{\phi}\to {B}\ne 0$$ | ||

Assertion | divne0d | $${\u22a2}{\phi}\to \frac{{A}}{{B}}\ne 0$$ |

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

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

2 | divcld.2 | $${\u22a2}{\phi}\to {B}\in \u2102$$ | |

3 | divne0d.3 | $${\u22a2}{\phi}\to {A}\ne 0$$ | |

4 | divne0d.4 | $${\u22a2}{\phi}\to {B}\ne 0$$ | |

5 | divne0 | $${\u22a2}\left(\left({A}\in \u2102\wedge {A}\ne 0\right)\wedge \left({B}\in \u2102\wedge {B}\ne 0\right)\right)\to \frac{{A}}{{B}}\ne 0$$ | |

6 | 1 3 2 4 5 | syl22anc | $${\u22a2}{\phi}\to \frac{{A}}{{B}}\ne 0$$ |