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Mirrors > Home > ILE Home > Th. List > nn0ge0div | GIF version |
Description: Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
Ref | Expression |
---|---|
nn0ge0div | ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → 0 ≤ (𝐾 / 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ge0 9220 | . . 3 ⊢ (𝐾 ∈ ℕ0 → 0 ≤ 𝐾) | |
2 | 1 | adantr 276 | . 2 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → 0 ≤ 𝐾) |
3 | elnnz 9282 | . . . 4 ⊢ (𝐿 ∈ ℕ ↔ (𝐿 ∈ ℤ ∧ 0 < 𝐿)) | |
4 | nn0re 9204 | . . . . . 6 ⊢ (𝐾 ∈ ℕ0 → 𝐾 ∈ ℝ) | |
5 | 4 | adantr 276 | . . . . 5 ⊢ ((𝐾 ∈ ℕ0 ∧ (𝐿 ∈ ℤ ∧ 0 < 𝐿)) → 𝐾 ∈ ℝ) |
6 | zre 9276 | . . . . . 6 ⊢ (𝐿 ∈ ℤ → 𝐿 ∈ ℝ) | |
7 | 6 | ad2antrl 490 | . . . . 5 ⊢ ((𝐾 ∈ ℕ0 ∧ (𝐿 ∈ ℤ ∧ 0 < 𝐿)) → 𝐿 ∈ ℝ) |
8 | simprr 531 | . . . . 5 ⊢ ((𝐾 ∈ ℕ0 ∧ (𝐿 ∈ ℤ ∧ 0 < 𝐿)) → 0 < 𝐿) | |
9 | 5, 7, 8 | 3jca 1179 | . . . 4 ⊢ ((𝐾 ∈ ℕ0 ∧ (𝐿 ∈ ℤ ∧ 0 < 𝐿)) → (𝐾 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 0 < 𝐿)) |
10 | 3, 9 | sylan2b 287 | . . 3 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝐾 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 0 < 𝐿)) |
11 | ge0div 8847 | . . 3 ⊢ ((𝐾 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 0 < 𝐿) → (0 ≤ 𝐾 ↔ 0 ≤ (𝐾 / 𝐿))) | |
12 | 10, 11 | syl 14 | . 2 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (0 ≤ 𝐾 ↔ 0 ≤ (𝐾 / 𝐿))) |
13 | 2, 12 | mpbid 147 | 1 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → 0 ≤ (𝐾 / 𝐿)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 ∈ wcel 2160 class class class wbr 4018 (class class class)co 5891 ℝcr 7829 0cc0 7830 < clt 8011 ≤ cle 8012 / cdiv 8648 ℕcn 8938 ℕ0cn0 9195 ℤcz 9272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-mulrcl 7929 ax-addcom 7930 ax-mulcom 7931 ax-addass 7932 ax-mulass 7933 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-1rid 7937 ax-0id 7938 ax-rnegex 7939 ax-precex 7940 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 ax-pre-mulgt0 7947 ax-pre-mulext 7948 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4308 df-po 4311 df-iso 4312 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5233 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-reap 8551 df-ap 8558 df-div 8649 df-inn 8939 df-n0 9196 df-z 9273 |
This theorem is referenced by: fldivnn0 10314 divfl0 10315 |
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