**Description:** A nonzero square is positive. Theorem I.20 of Apostol p. 20.
(Contributed by Mario Carneiro, 27-May-2016)

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

Hypotheses | leidd.1 | $${\u22a2}{\phi}\to {A}\in \mathbb{R}$$ | |

msqgt0d.2 | $${\u22a2}{\phi}\to {A}\ne 0$$ | ||

Assertion | msqgt0d | $${\u22a2}{\phi}\to 0<{A}{A}$$ |

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

1 | leidd.1 | $${\u22a2}{\phi}\to {A}\in \mathbb{R}$$ | |

2 | msqgt0d.2 | $${\u22a2}{\phi}\to {A}\ne 0$$ | |

3 | msqgt0 | $${\u22a2}\left({A}\in \mathbb{R}\wedge {A}\ne 0\right)\to 0<{A}{A}$$ | |

4 | 1 2 3 | syl2anc | $${\u22a2}{\phi}\to 0<{A}{A}$$ |