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Mirrors > Home > MPE Home > Th. List > mulgt0ii | Structured version Visualization version GIF version |
Description: The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
Ref | Expression |
---|---|
lt.1 | ⊢ 𝐴 ∈ ℝ |
lt.2 | ⊢ 𝐵 ∈ ℝ |
mulgt0i.3 | ⊢ 0 < 𝐴 |
mulgt0i.4 | ⊢ 0 < 𝐵 |
Ref | Expression |
---|---|
mulgt0ii | ⊢ 0 < (𝐴 · 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulgt0i.3 | . 2 ⊢ 0 < 𝐴 | |
2 | mulgt0i.4 | . 2 ⊢ 0 < 𝐵 | |
3 | lt.1 | . . 3 ⊢ 𝐴 ∈ ℝ | |
4 | lt.2 | . . 3 ⊢ 𝐵 ∈ ℝ | |
5 | 3, 4 | mulgt0i 10761 | . 2 ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)) |
6 | 1, 2, 5 | mp2an 691 | 1 ⊢ 0 < (𝐴 · 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 · cmul 10531 < clt 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-addrcl 10587 ax-mulrcl 10589 ax-rnegex 10597 ax-cnre 10599 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 |
This theorem is referenced by: ef01bndlem 15529 efif1olem2 25135 efif1olem4 25137 ang180lem1 25395 ang180lem2 25396 chebbnd1lem3 26055 chebbnd1 26056 sinaover2ne0 42510 dirkercncflem4 42748 fourierdlem24 42773 fourierswlem 42872 fouriersw 42873 |
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