**Description:** A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016)

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

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

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

divcld.3 | $${\u22a2}{\phi}\to {B}\ne 0$$ | ||

Assertion | divcan3d | $${\u22a2}{\phi}\to \frac{{B}{A}}{{B}}={A}$$ |

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

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

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

3 | divcld.3 | $${\u22a2}{\phi}\to {B}\ne 0$$ | |

4 | divcan3 | $${\u22a2}\left({A}\in \u2102\wedge {B}\in \u2102\wedge {B}\ne 0\right)\to \frac{{B}{A}}{{B}}={A}$$ | |

5 | 1 2 3 4 | syl3anc | $${\u22a2}{\phi}\to \frac{{B}{A}}{{B}}={A}$$ |