![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > lt2mul2divd | GIF version |
Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
lt2mul2divd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
lt2mul2divd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
lt2mul2divd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
lt2mul2divd.4 | ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
Ref | Expression |
---|---|
lt2mul2divd | ⊢ (𝜑 → ((𝐴 · 𝐵) < (𝐶 · 𝐷) ↔ (𝐴 / 𝐷) < (𝐶 / 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt2mul2divd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | lt2mul2divd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
3 | 2 | rpregt0d 9520 | . 2 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 0 < 𝐵)) |
4 | lt2mul2divd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | lt2mul2divd.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ+) | |
6 | 5 | rpregt0d 9520 | . 2 ⊢ (𝜑 → (𝐷 ∈ ℝ ∧ 0 < 𝐷)) |
7 | lt2mul2div 8661 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) ∧ (𝐶 ∈ ℝ ∧ (𝐷 ∈ ℝ ∧ 0 < 𝐷))) → ((𝐴 · 𝐵) < (𝐶 · 𝐷) ↔ (𝐴 / 𝐷) < (𝐶 / 𝐵))) | |
8 | 1, 3, 4, 6, 7 | syl22anc 1218 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵) < (𝐶 · 𝐷) ↔ (𝐴 / 𝐷) < (𝐶 / 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1481 class class class wbr 3937 (class class class)co 5782 ℝcr 7643 0cc0 7644 · cmul 7649 < clt 7824 / cdiv 8456 ℝ+crp 9470 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-rp 9471 |
This theorem is referenced by: qtri3or 10051 cvg1nlemcxze 10786 cvg1nlemcau 10788 efcllemp 11401 |
Copyright terms: Public domain | W3C validator |