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Mirrors > Home > ILE Home > Th. List > gt0div | GIF version |
Description: Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.) |
Ref | Expression |
---|---|
gt0div | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 < 𝐴 ↔ 0 < (𝐴 / 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7734 | . . . 4 ⊢ 0 ∈ ℝ | |
2 | ltdiv1 8594 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (0 < 𝐴 ↔ (0 / 𝐵) < (𝐴 / 𝐵))) | |
3 | 1, 2 | mp3an1 1287 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (0 < 𝐴 ↔ (0 / 𝐵) < (𝐴 / 𝐵))) |
4 | 3 | 3impb 1162 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 < 𝐴 ↔ (0 / 𝐵) < (𝐴 / 𝐵))) |
5 | gt0ap0 8356 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 0 < 𝐵) → 𝐵 # 0) | |
6 | recn 7721 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
7 | div0ap 8430 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 # 0) → (0 / 𝐵) = 0) | |
8 | 6, 7 | sylan 281 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 # 0) → (0 / 𝐵) = 0) |
9 | 5, 8 | syldan 280 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 / 𝐵) = 0) |
10 | 9 | breq1d 3909 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 0 < 𝐵) → ((0 / 𝐵) < (𝐴 / 𝐵) ↔ 0 < (𝐴 / 𝐵))) |
11 | 10 | 3adant1 984 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → ((0 / 𝐵) < (𝐴 / 𝐵) ↔ 0 < (𝐴 / 𝐵))) |
12 | 4, 11 | bitrd 187 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 < 𝐴 ↔ 0 < (𝐴 / 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 947 = wceq 1316 ∈ wcel 1465 class class class wbr 3899 (class class class)co 5742 ℂcc 7586 ℝcr 7587 0cc0 7588 < clt 7768 # cap 8311 / cdiv 8400 |
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-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
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-rmo 2401 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-po 4188 df-iso 4189 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-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 |
This theorem is referenced by: divgt0 8598 halfpos2 8918 gt0divd 9489 |
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