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Mirrors > Home > MPE Home > Th. List > ltmul1dd | Structured version Visualization version GIF version |
Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.) |
Ref | Expression |
---|---|
ltmul1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltmul1d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltmul1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
ltdiv1dd.4 | ⊢ (𝜑 → 𝐴 < 𝐵) |
Ref | Expression |
---|---|
ltmul1dd | ⊢ (𝜑 → (𝐴 · 𝐶) < (𝐵 · 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltdiv1dd.4 | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
2 | ltmul1d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | ltmul1d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | ltmul1d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
5 | 2, 3, 4 | ltmul1d 13039 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶))) |
6 | 1, 5 | mpbid 231 | 1 ⊢ (𝜑 → (𝐴 · 𝐶) < (𝐵 · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5141 (class class class)co 7393 ℝcr 11091 · cmul 11097 < clt 11230 ℝ+crp 12956 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-ltxr 11235 df-sub 11428 df-neg 11429 df-rp 12957 |
This theorem is referenced by: mul2lt0bi 13062 o1rlimmul 15545 2expltfac 17008 lhop1lem 25459 ftalem5 26508 chtppilimlem1 26903 pntibndlem2 27021 pntlemb 27027 pntlemr 27032 ostth2lem1 27048 ostth2lem3 27065 tgoldbachgtde 33503 fltnltalem 41186 ioodvbdlimc1lem1 44420 stoweidlem11 44500 stoweidlem13 44502 stoweidlem26 44515 wallispi 44559 stirlinglem1 44563 dirkercncflem4 44595 fourierdlem4 44600 |
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