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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 9610 | . . . . . 6 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) | |
2 | 1 | 3ad2ant1 1018 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐴 ∈ ℂ) |
3 | qcn 9610 | . . . . . 6 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ) | |
4 | 3 | 3ad2ant2 1019 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ∈ ℂ) |
5 | qre 9601 | . . . . . . 7 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ) | |
6 | 5 | 3ad2ant2 1019 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ∈ ℝ) |
7 | simp3 999 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 < 𝐵) | |
8 | 6, 7 | gt0ap0d 8563 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 # 0) |
9 | 2, 4, 8 | divcanap2d 8725 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐵 · (𝐴 / 𝐵)) = 𝐴) |
10 | 9 | oveq1d 5883 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐵 · (𝐴 / 𝐵)) − (𝐵 · (⌊‘(𝐴 / 𝐵)))) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
11 | 7 | gt0ne0d 8446 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ≠ 0) |
12 | qdivcl 9619 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ) | |
13 | 11, 12 | syld3an3 1283 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 / 𝐵) ∈ ℚ) |
14 | qcn 9610 | . . . . 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 10250 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (⌊‘(𝐴 / 𝐵)) ∈ ℤ) |
17 | 16 | zcnd 9352 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (⌊‘(𝐴 / 𝐵)) ∈ ℂ) |
18 | 4, 15, 17 | subdid 8348 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵)))) = ((𝐵 · (𝐴 / 𝐵)) − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
19 | modqval 10297 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) | |
20 | 10, 18, 19 | 3eqtr4rd 2221 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))))) |
21 | qfraclt1 10253 | . . . . 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 8719 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐵 / 𝐵) = 1) |
24 | 22, 23 | breqtrrd 4028 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < (𝐵 / 𝐵)) |
25 | qre 9601 | . . . . . 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 9351 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) |
28 | 26, 27 | resubcld 8315 | . . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) ∈ ℝ) |
29 | ltmuldiv2 8808 | . . . 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 1242 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵)))) < 𝐵 ↔ ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < (𝐵 / 𝐵))) |
31 | 24, 30 | mpbird 167 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵)))) < 𝐵) |
32 | 20, 31 | eqbrtrd 4022 | 1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) < 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 978 ∈ wcel 2148 ≠ wne 2347 class class class wbr 4000 ‘cfv 5211 (class class class)co 5868 ℂcc 7787 ℝcr 7788 0cc0 7789 1c1 7790 · cmul 7794 < clt 7969 − cmin 8105 / cdiv 8605 ℚcq 9595 ⌊cfl 10241 mod cmo 10295 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 ax-pre-mulext 7907 ax-arch 7908 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-po 4292 df-iso 4293 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 df-div 8606 df-inn 8896 df-n0 9153 df-z 9230 df-q 9596 df-rp 9628 df-fl 10243 df-mod 10296 |
This theorem is referenced by: modqelico 10307 zmodfz 10319 modqid2 10324 modqabs 10330 modqmuladdim 10340 modaddmodup 10360 modqsubdir 10366 divalglemnn 11893 divalgmod 11902 bezoutlemnewy 11967 bezoutlemstep 11968 eucalglt 12027 odzdvds 12215 fldivp1 12316 4sqlem6 12351 |
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