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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 9994 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵)) | |
6 | 1, 2, 3, 4, 5 | syl22anc 1319 | 1 ⊢ (𝜑 → 0 < (𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 0cc0 9815 · cmul 9820 < clt 9953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 |
This theorem is referenced by: recgt0 10746 prodgt0 10747 prodge0 10749 ltmul1a 10751 expmulnbnd 12858 itg2monolem3 23325 tangtx 24061 tanregt0 24089 asinsinlem 24418 asinsin 24419 ostth2lem3 25124 xrge0iifhom 29311 unbdqndv2lem2 31671 knoppndvlem14 31686 knoppndvlem18 31690 knoppndvlem19 31691 knoppndvlem21 31693 itg2gt0cn 32635 pell14qrmulcl 36445 rmxypos 36532 jm2.27a 36590 stoweidlem1 38894 stoweidlem26 38919 stoweidlem44 38937 stoweidlem49 38942 wallispilem4 38961 stirlinglem6 38972 |
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