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Mirrors > Home > ILE Home > Th. List > ltmul1dd | 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 9493 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶))) |
6 | 1, 5 | mpbid 146 | 1 ⊢ (𝜑 → (𝐴 · 𝐶) < (𝐵 · 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 class class class wbr 3899 (class class class)co 5742 ℝcr 7587 · cmul 7593 < clt 7768 ℝ+crp 9409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-ltxr 7773 df-sub 7903 df-neg 7904 df-rp 9410 |
This theorem is referenced by: mul2lt0np 9518 cvg1nlemcau 10724 resqrexlemnm 10758 cvgcmp2nlemabs 13154 |
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