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Mirrors > Home > ILE Home > Th. List > ge0div | GIF version |
Description: Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.) |
Ref | Expression |
---|---|
ge0div | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7549 | . . . 4 ⊢ 0 ∈ ℝ | |
2 | lediv1 8391 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (0 ≤ 𝐴 ↔ (0 / 𝐵) ≤ (𝐴 / 𝐵))) | |
3 | 1, 2 | mp3an1 1261 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (0 ≤ 𝐴 ↔ (0 / 𝐵) ≤ (𝐴 / 𝐵))) |
4 | 3 | 3impb 1140 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 ≤ 𝐴 ↔ (0 / 𝐵) ≤ (𝐴 / 𝐵))) |
5 | gt0ap0 8163 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 0 < 𝐵) → 𝐵 # 0) | |
6 | recn 7536 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
7 | div0ap 8230 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 # 0) → (0 / 𝐵) = 0) | |
8 | 6, 7 | sylan 278 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 # 0) → (0 / 𝐵) = 0) |
9 | 5, 8 | syldan 277 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 / 𝐵) = 0) |
10 | 9 | breq1d 3861 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 0 < 𝐵) → ((0 / 𝐵) ≤ (𝐴 / 𝐵) ↔ 0 ≤ (𝐴 / 𝐵))) |
11 | 10 | 3adant1 962 | . 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 925 = wceq 1290 ∈ wcel 1439 class class class wbr 3851 (class class class)co 5666 ℂcc 7409 ℝcr 7410 0cc0 7411 < clt 7583 ≤ cle 7584 # cap 8119 / cdiv 8200 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-mulrcl 7505 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-precex 7516 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-apti 7521 ax-pre-ltadd 7522 ax-pre-mulgt0 7523 ax-pre-mulext 7524 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-po 4132 df-iso 4133 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-reap 8113 df-ap 8120 df-div 8201 |
This theorem is referenced by: divge0 8395 halfnneg2 8709 nn0ge0div 8894 ge0divd 9273 2tnp1ge0ge0 9769 nn0ehalf 11242 nn0oddm1d2 11248 |
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