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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 10766 | . 2 ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)) |
6 | 1, 2, 5 | mp2an 690 | 1 ⊢ 0 < (𝐴 · 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 class class class wbr 5059 (class class class)co 7150 ℝcr 10530 0cc0 10531 · cmul 10536 < clt 10669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-addrcl 10592 ax-mulrcl 10594 ax-rnegex 10602 ax-cnre 10604 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 |
This theorem is referenced by: ef01bndlem 15531 efif1olem2 25121 efif1olem4 25123 ang180lem1 25381 ang180lem2 25382 chebbnd1lem3 26041 chebbnd1 26042 sinaover2ne0 42141 dirkercncflem4 42384 fourierdlem24 42409 fourierswlem 42508 fouriersw 42509 |
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