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Mirrors > Home > ILE Home > Th. List > modqlt | GIF version |
Description: The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Ref | Expression |
---|---|
modqlt | ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) < 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qcn 9563 | . . . . . 6 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) | |
2 | 1 | 3ad2ant1 1007 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐴 ∈ ℂ) |
3 | qcn 9563 | . . . . . 6 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ) | |
4 | 3 | 3ad2ant2 1008 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ∈ ℂ) |
5 | qre 9554 | . . . . . . 7 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ) | |
6 | 5 | 3ad2ant2 1008 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ∈ ℝ) |
7 | simp3 988 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 < 𝐵) | |
8 | 6, 7 | gt0ap0d 8518 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 # 0) |
9 | 2, 4, 8 | divcanap2d 8679 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐵 · (𝐴 / 𝐵)) = 𝐴) |
10 | 9 | oveq1d 5851 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐵 · (𝐴 / 𝐵)) − (𝐵 · (⌊‘(𝐴 / 𝐵)))) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
11 | 7 | gt0ne0d 8401 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ≠ 0) |
12 | qdivcl 9572 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ) | |
13 | 11, 12 | syld3an3 1272 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 / 𝐵) ∈ ℚ) |
14 | qcn 9563 | . . . . 5 ⊢ ((𝐴 / 𝐵) ∈ ℚ → (𝐴 / 𝐵) ∈ ℂ) | |
15 | 13, 14 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 / 𝐵) ∈ ℂ) |
16 | 13 | flqcld 10202 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (⌊‘(𝐴 / 𝐵)) ∈ ℤ) |
17 | 16 | zcnd 9305 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (⌊‘(𝐴 / 𝐵)) ∈ ℂ) |
18 | 4, 15, 17 | subdid 8303 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵)))) = ((𝐵 · (𝐴 / 𝐵)) − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
19 | modqval 10249 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) | |
20 | 10, 18, 19 | 3eqtr4rd 2208 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))))) |
21 | qfraclt1 10205 | . . . . 5 ⊢ ((𝐴 / 𝐵) ∈ ℚ → ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < 1) | |
22 | 13, 21 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < 1) |
23 | 4, 8 | dividapd 8673 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐵 / 𝐵) = 1) |
24 | 22, 23 | breqtrrd 4004 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < (𝐵 / 𝐵)) |
25 | qre 9554 | . . . . . 6 ⊢ ((𝐴 / 𝐵) ∈ ℚ → (𝐴 / 𝐵) ∈ ℝ) | |
26 | 13, 25 | syl 14 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 / 𝐵) ∈ ℝ) |
27 | 16 | zred 9304 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) |
28 | 26, 27 | resubcld 8270 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) ∈ ℝ) |
29 | ltmuldiv2 8761 | . . . 4 ⊢ ((((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵)))) < 𝐵 ↔ ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < (𝐵 / 𝐵))) | |
30 | 28, 6, 6, 7, 29 | syl112anc 1231 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵)))) < 𝐵 ↔ ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < (𝐵 / 𝐵))) |
31 | 24, 30 | mpbird 166 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵)))) < 𝐵) |
32 | 20, 31 | eqbrtrd 3998 | 1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) < 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 967 ∈ wcel 2135 ≠ wne 2334 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℂcc 7742 ℝcr 7743 0cc0 7744 1c1 7745 · cmul 7749 < clt 7924 − cmin 8060 / cdiv 8559 ℚcq 9548 ⌊cfl 10193 mod cmo 10247 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-n0 9106 df-z 9183 df-q 9549 df-rp 9581 df-fl 10195 df-mod 10248 |
This theorem is referenced by: modqelico 10259 zmodfz 10271 modqid2 10276 modqabs 10282 modqmuladdim 10292 modaddmodup 10312 modqsubdir 10318 divalglemnn 11840 divalgmod 11849 bezoutlemnewy 11914 bezoutlemstep 11915 eucalglt 11968 odzdvds 12156 fldivp1 12257 |
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