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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 1001 | . . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 < 𝐵) | |
2 | 1 | gt0ne0d 8498 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ≠ 0) |
3 | qdivcl 9672 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ) | |
4 | 2, 3 | syld3an3 1294 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 / 𝐵) ∈ ℚ) |
5 | flqle 10308 | . . . . 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 10307 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (⌊‘(𝐴 / 𝐵)) ∈ ℤ) |
8 | 7 | zred 9404 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) |
9 | qre 9654 | . . . . . 6 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) | |
10 | 9 | 3ad2ant1 1020 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐴 ∈ ℝ) |
11 | qre 9654 | . . . . . 6 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ) | |
12 | 11 | 3ad2ant2 1021 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ∈ ℝ) |
13 | lemuldiv2 8868 | . . . . 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 1253 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐵 · (⌊‘(𝐴 / 𝐵))) ≤ 𝐴 ↔ (⌊‘(𝐴 / 𝐵)) ≤ (𝐴 / 𝐵))) |
15 | 6, 14 | mpbird 167 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐵 · (⌊‘(𝐴 / 𝐵))) ≤ 𝐴) |
16 | 12, 8 | remulcld 8017 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐵 · (⌊‘(𝐴 / 𝐵))) ∈ ℝ) |
17 | 10, 16 | subge0d 8521 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (0 ≤ (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))) ↔ (𝐵 · (⌊‘(𝐴 / 𝐵))) ≤ 𝐴)) |
18 | 15, 17 | mpbird 167 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
19 | modqval 10354 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) | |
20 | 18, 19 | breqtrrd 4046 | 1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 980 ∈ wcel 2160 ≠ wne 2360 class class class wbr 4018 ‘cfv 5235 (class class class)co 5895 ℝcr 7839 0cc0 7840 · cmul 7845 < clt 8021 ≤ cle 8022 − cmin 8157 / cdiv 8658 ℚcq 9648 ⌊cfl 10298 mod cmo 10352 |
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 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 ax-pre-mulext 7958 ax-arch 7959 |
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-csb 3073 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-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-reap 8561 df-ap 8568 df-div 8659 df-inn 8949 df-n0 9206 df-z 9283 df-q 9649 df-rp 9683 df-fl 10300 df-mod 10353 |
This theorem is referenced by: modqelico 10364 zmodcl 10374 modqid2 10381 modqabs 10387 modqmuladdim 10397 modqltm1p1mod 10406 modqsubdir 10423 modqeqmodmin 10424 4sqlem6 12414 |
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