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Mirrors > Home > ILE Home > Th. List > modqge0 | GIF version |
Description: The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Ref | Expression |
---|---|
modqge0 | ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 968 | . . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 < 𝐵) | |
2 | 1 | gt0ne0d 8242 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ≠ 0) |
3 | qdivcl 9403 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ) | |
4 | 2, 3 | syld3an3 1246 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 / 𝐵) ∈ ℚ) |
5 | flqle 10019 | . . . . 5 ⊢ ((𝐴 / 𝐵) ∈ ℚ → (⌊‘(𝐴 / 𝐵)) ≤ (𝐴 / 𝐵)) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (⌊‘(𝐴 / 𝐵)) ≤ (𝐴 / 𝐵)) |
7 | 4 | flqcld 10018 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (⌊‘(𝐴 / 𝐵)) ∈ ℤ) |
8 | 7 | zred 9141 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) |
9 | qre 9385 | . . . . . 6 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) | |
10 | 9 | 3ad2ant1 987 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐴 ∈ ℝ) |
11 | qre 9385 | . . . . . 6 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ) | |
12 | 11 | 3ad2ant2 988 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ∈ ℝ) |
13 | lemuldiv2 8608 | . . . . 5 ⊢ (((⌊‘(𝐴 / 𝐵)) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐵 · (⌊‘(𝐴 / 𝐵))) ≤ 𝐴 ↔ (⌊‘(𝐴 / 𝐵)) ≤ (𝐴 / 𝐵))) | |
14 | 8, 10, 12, 1, 13 | syl112anc 1205 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐵 · (⌊‘(𝐴 / 𝐵))) ≤ 𝐴 ↔ (⌊‘(𝐴 / 𝐵)) ≤ (𝐴 / 𝐵))) |
15 | 6, 14 | mpbird 166 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐵 · (⌊‘(𝐴 / 𝐵))) ≤ 𝐴) |
16 | 12, 8 | remulcld 7764 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐵 · (⌊‘(𝐴 / 𝐵))) ∈ ℝ) |
17 | 10, 16 | subge0d 8265 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (0 ≤ (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))) ↔ (𝐵 · (⌊‘(𝐴 / 𝐵))) ≤ 𝐴)) |
18 | 15, 17 | mpbird 166 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
19 | modqval 10065 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) | |
20 | 18, 19 | breqtrrd 3926 | 1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 947 ∈ wcel 1465 ≠ wne 2285 class class class wbr 3899 ‘cfv 5093 (class class class)co 5742 ℝcr 7587 0cc0 7588 · cmul 7593 < clt 7768 ≤ cle 7769 − cmin 7901 / cdiv 8400 ℚcq 9379 ⌊cfl 10009 mod cmo 10063 |
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 ax-arch 7707 |
This theorem depends on definitions: df-bi 116 df-3or 948 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-csb 2976 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-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 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-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 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 df-inn 8689 df-n0 8946 df-z 9023 df-q 9380 df-rp 9410 df-fl 10011 df-mod 10064 |
This theorem is referenced by: modqelico 10075 zmodcl 10085 modqid2 10092 modqabs 10098 modqmuladdim 10108 modqltm1p1mod 10117 modqsubdir 10134 modqeqmodmin 10135 |
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