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Mirrors > Home > MPE Home > Th. List > modlt | Structured version Visualization version GIF version |
Description: The modulo operation is less than its second argument. (Contributed by NM, 10-Nov-2008.) |
Ref | Expression |
---|---|
modlt | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) < 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 10615 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | rpcnne0 12395 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) | |
3 | divcan2 11294 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐵 · (𝐴 / 𝐵)) = 𝐴) | |
4 | 3 | 3expb 1112 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐵 · (𝐴 / 𝐵)) = 𝐴) |
5 | 1, 2, 4 | syl2an 595 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐵 · (𝐴 / 𝐵)) = 𝐴) |
6 | 5 | oveq1d 7160 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐵 · (𝐴 / 𝐵)) − (𝐵 · (⌊‘(𝐴 / 𝐵)))) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
7 | rpcn 12387 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℂ) | |
8 | 7 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℂ) |
9 | rerpdivcl 12407 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ) | |
10 | 9 | recnd 10657 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℂ) |
11 | refldivcl 13181 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) | |
12 | 11 | recnd 10657 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (⌊‘(𝐴 / 𝐵)) ∈ ℂ) |
13 | 8, 10, 12 | subdid 11084 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵)))) = ((𝐵 · (𝐴 / 𝐵)) − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
14 | modval 13227 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) | |
15 | 6, 13, 14 | 3eqtr4rd 2864 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))))) |
16 | fraclt1 13160 | . . . . 5 ⊢ ((𝐴 / 𝐵) ∈ ℝ → ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < 1) | |
17 | 9, 16 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < 1) |
18 | divid 11315 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐵 / 𝐵) = 1) | |
19 | 2, 18 | syl 17 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → (𝐵 / 𝐵) = 1) |
20 | 19 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐵 / 𝐵) = 1) |
21 | 17, 20 | breqtrrd 5085 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < (𝐵 / 𝐵)) |
22 | 9, 11 | resubcld 11056 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) ∈ ℝ) |
23 | rpre 12385 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
24 | 23 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℝ) |
25 | rpregt0 12391 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → (𝐵 ∈ ℝ ∧ 0 < 𝐵)) | |
26 | 25 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐵 ∈ ℝ ∧ 0 < 𝐵)) |
27 | ltmuldiv2 11502 | . . . 4 ⊢ ((((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵)))) < 𝐵 ↔ ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < (𝐵 / 𝐵))) | |
28 | 22, 24, 26, 27 | syl3anc 1363 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵)))) < 𝐵 ↔ ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < (𝐵 / 𝐵))) |
29 | 21, 28 | mpbird 258 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵)))) < 𝐵) |
30 | 15, 29 | eqbrtrd 5079 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 0cc0 10525 1c1 10526 · cmul 10530 < clt 10663 − cmin 10858 / cdiv 11285 ℝ+crp 12377 ⌊cfl 13148 mod cmo 13225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13150 df-mod 13226 |
This theorem is referenced by: modelico 13237 zmodfz 13249 modid2 13254 modabs 13260 modaddmodup 13290 modsubdir 13296 digit1 13586 cshwidxmod 14153 repswcshw 14162 divalgmod 15745 bitsmod 15773 bitsinv1lem 15778 bezoutlem3 15877 eucalglt 15917 odzdvds 16120 fldivp1 16221 4sqlem6 16267 4sqlem12 16280 mndodcong 18599 oddvds 18604 gexdvds 18638 zringlpirlem3 20561 sineq0 25036 efif1olem2 25054 lgseisenlem1 25878 irrapxlem1 39297 pellfund14 39373 jm2.19 39468 sineq0ALT 41148 fourierswlem 42392 fouriersw 42393 |
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