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Mirrors > Home > MPE Home > Th. List > flltdivnn0lt | Structured version Visualization version GIF version |
Description: The floor function of a division of a nonnegative integer by a positive integer is less than the division of a greater dividend by the same positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
Ref | Expression |
---|---|
flltdivnn0lt | ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝐾 < 𝑁 → (⌊‘(𝐾 / 𝐿)) < (𝑁 / 𝐿))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0nndivcl 11954 | . . . . . . 7 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝐾 / 𝐿) ∈ ℝ) | |
2 | reflcl 13161 | . . . . . . 7 ⊢ ((𝐾 / 𝐿) ∈ ℝ → (⌊‘(𝐾 / 𝐿)) ∈ ℝ) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ∈ ℝ) |
4 | 3 | 3adant2 1128 | . . . . 5 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ∈ ℝ) |
5 | 1 | 3adant2 1128 | . . . . 5 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝐾 / 𝐿) ∈ ℝ) |
6 | nn0nndivcl 11954 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝑁 / 𝐿) ∈ ℝ) | |
7 | 6 | 3adant1 1127 | . . . . 5 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝑁 / 𝐿) ∈ ℝ) |
8 | 4, 5, 7 | 3jca 1125 | . . . 4 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → ((⌊‘(𝐾 / 𝐿)) ∈ ℝ ∧ (𝐾 / 𝐿) ∈ ℝ ∧ (𝑁 / 𝐿) ∈ ℝ)) |
9 | 8 | adantr 484 | . . 3 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) ∧ 𝐾 < 𝑁) → ((⌊‘(𝐾 / 𝐿)) ∈ ℝ ∧ (𝐾 / 𝐿) ∈ ℝ ∧ (𝑁 / 𝐿) ∈ ℝ)) |
10 | fldivnn0le 13197 | . . . . . 6 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿)) | |
11 | 10 | 3adant2 1128 | . . . . 5 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿)) |
12 | 11 | adantr 484 | . . . 4 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) ∧ 𝐾 < 𝑁) → (⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿)) |
13 | simpr 488 | . . . . 5 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) ∧ 𝐾 < 𝑁) → 𝐾 < 𝑁) | |
14 | nn0re 11894 | . . . . . . . 8 ⊢ (𝐾 ∈ ℕ0 → 𝐾 ∈ ℝ) | |
15 | nn0re 11894 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
16 | nnre 11632 | . . . . . . . . 9 ⊢ (𝐿 ∈ ℕ → 𝐿 ∈ ℝ) | |
17 | nngt0 11656 | . . . . . . . . 9 ⊢ (𝐿 ∈ ℕ → 0 < 𝐿) | |
18 | 16, 17 | jca 515 | . . . . . . . 8 ⊢ (𝐿 ∈ ℕ → (𝐿 ∈ ℝ ∧ 0 < 𝐿)) |
19 | 14, 15, 18 | 3anim123i 1148 | . . . . . . 7 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 0 < 𝐿))) |
20 | 19 | adantr 484 | . . . . . 6 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) ∧ 𝐾 < 𝑁) → (𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 0 < 𝐿))) |
21 | ltdiv1 11493 | . . . . . 6 ⊢ ((𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 0 < 𝐿)) → (𝐾 < 𝑁 ↔ (𝐾 / 𝐿) < (𝑁 / 𝐿))) | |
22 | 20, 21 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) ∧ 𝐾 < 𝑁) → (𝐾 < 𝑁 ↔ (𝐾 / 𝐿) < (𝑁 / 𝐿))) |
23 | 13, 22 | mpbid 235 | . . . 4 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) ∧ 𝐾 < 𝑁) → (𝐾 / 𝐿) < (𝑁 / 𝐿)) |
24 | 12, 23 | jca 515 | . . 3 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) ∧ 𝐾 < 𝑁) → ((⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿) ∧ (𝐾 / 𝐿) < (𝑁 / 𝐿))) |
25 | lelttr 10720 | . . 3 ⊢ (((⌊‘(𝐾 / 𝐿)) ∈ ℝ ∧ (𝐾 / 𝐿) ∈ ℝ ∧ (𝑁 / 𝐿) ∈ ℝ) → (((⌊‘(𝐾 / 𝐿)) ≤ (𝐾 / 𝐿) ∧ (𝐾 / 𝐿) < (𝑁 / 𝐿)) → (⌊‘(𝐾 / 𝐿)) < (𝑁 / 𝐿))) | |
26 | 9, 24, 25 | sylc 65 | . 2 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) ∧ 𝐾 < 𝑁) → (⌊‘(𝐾 / 𝐿)) < (𝑁 / 𝐿)) |
27 | 26 | ex 416 | 1 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ ℕ) → (𝐾 < 𝑁 → (⌊‘(𝐾 / 𝐿)) < (𝑁 / 𝐿))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 < clt 10664 ≤ cle 10665 / cdiv 11286 ℕcn 11625 ℕ0cn0 11885 ⌊cfl 13155 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13157 |
This theorem is referenced by: (None) |
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