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| Mirrors > Home > MPE Home > Th. List > mulgt0d | Structured version Visualization version GIF version | ||
| Description: The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.) | 
| Ref | Expression | 
|---|---|
| ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) | 
| ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) | 
| mulgt0d.3 | ⊢ (𝜑 → 0 < 𝐴) | 
| mulgt0d.4 | ⊢ (𝜑 → 0 < 𝐵) | 
| Ref | Expression | 
|---|---|
| mulgt0d | ⊢ (𝜑 → 0 < (𝐴 · 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ltd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 2 | mulgt0d.3 | . 2 ⊢ (𝜑 → 0 < 𝐴) | |
| 3 | ltd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 4 | mulgt0d.4 | . 2 ⊢ (𝜑 → 0 < 𝐵) | |
| 5 | mulgt0 11339 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵)) | |
| 6 | 1, 2, 3, 4, 5 | syl22anc 838 | 1 ⊢ (𝜑 → 0 < (𝐴 · 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 5142 (class class class)co 7432 ℝcr 11155 0cc0 11156 · cmul 11161 < clt 11296 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-addrcl 11217 ax-mulrcl 11219 ax-rnegex 11227 ax-cnre 11229 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-ltxr 11301 | 
| This theorem is referenced by: recgt0 12114 prodgt0 12115 ltmul1a 12117 prodge0rd 13143 expmulnbnd 14275 itg2monolem3 25788 tangtx 26548 tanregt0 26582 asinsinlem 26935 asinsin 26936 ostth2lem3 27680 xrge0iifhom 33937 unbdqndv2lem2 36512 knoppndvlem14 36527 knoppndvlem18 36531 knoppndvlem19 36532 knoppndvlem21 36534 itg2gt0cn 37683 lcmineqlem15 42045 posbezout 42102 mulgt0con1d 42493 mulgt0con2d 42494 mulgt0b2d 42495 sn-0lt1 42498 pell14qrmulcl 42879 rmxypos 42964 jm2.27a 43022 stoweidlem1 46021 stoweidlem26 46046 stoweidlem44 46064 stoweidlem49 46069 wallispilem4 46088 stirlinglem6 46099 itscnhlinecirc02plem1 48708 | 
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