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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 10406 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵)) | |
6 | 1, 2, 3, 4, 5 | syl22anc 868 | 1 ⊢ (𝜑 → 0 < (𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2157 class class class wbr 4844 (class class class)co 6879 ℝcr 10224 0cc0 10225 · cmul 10230 < clt 10364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-resscn 10282 ax-1cn 10283 ax-addrcl 10286 ax-mulrcl 10288 ax-rnegex 10296 ax-cnre 10298 ax-pre-mulgt0 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-er 7983 df-en 8197 df-dom 8198 df-sdom 8199 df-pnf 10366 df-mnf 10367 df-ltxr 10369 |
This theorem is referenced by: recgt0 11160 prodgt0 11161 prodge0OLD 11163 ltmul1a 11165 prodge0rd 12181 expmulnbnd 13249 itg2monolem3 23859 tangtx 24598 tanregt0 24626 asinsinlem 24969 asinsin 24970 ostth2lem3 25675 xrge0iifhom 30498 unbdqndv2lem2 33008 knoppndvlem14 33023 knoppndvlem18 33027 knoppndvlem19 33028 knoppndvlem21 33030 itg2gt0cn 33952 pell14qrmulcl 38208 rmxypos 38294 jm2.27a 38352 stoweidlem1 40956 stoweidlem26 40981 stoweidlem44 40999 stoweidlem49 41004 wallispilem4 41023 stirlinglem6 41034 |
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