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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 11245 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
| 2 | rpcnne0 13053 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) | |
| 3 | divcan2 11930 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐵 · (𝐴 / 𝐵)) = 𝐴) | |
| 4 | 3 | 3expb 1121 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐵 · (𝐴 / 𝐵)) = 𝐴) |
| 5 | 1, 2, 4 | syl2an 596 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐵 · (𝐴 / 𝐵)) = 𝐴) |
| 6 | 5 | oveq1d 7446 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐵 · (𝐴 / 𝐵)) − (𝐵 · (⌊‘(𝐴 / 𝐵)))) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
| 7 | rpcn 13045 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℂ) | |
| 8 | 7 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℂ) |
| 9 | rerpdivcl 13065 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ) | |
| 10 | 9 | recnd 11289 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℂ) |
| 11 | refldivcl 13863 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) | |
| 12 | 11 | recnd 11289 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (⌊‘(𝐴 / 𝐵)) ∈ ℂ) |
| 13 | 8, 10, 12 | subdid 11719 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵)))) = ((𝐵 · (𝐴 / 𝐵)) − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
| 14 | modval 13911 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) | |
| 15 | 6, 13, 14 | 3eqtr4rd 2788 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))))) |
| 16 | fraclt1 13842 | . . . . 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 11953 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐵 / 𝐵) = 1) | |
| 19 | 2, 18 | syl 17 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → (𝐵 / 𝐵) = 1) |
| 20 | 19 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐵 / 𝐵) = 1) |
| 21 | 17, 20 | breqtrrd 5171 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < (𝐵 / 𝐵)) |
| 22 | 9, 11 | resubcld 11691 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) ∈ ℝ) |
| 23 | rpre 13043 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
| 24 | 23 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℝ) |
| 25 | rpregt0 13049 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → (𝐵 ∈ ℝ ∧ 0 < 𝐵)) | |
| 26 | 25 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐵 ∈ ℝ ∧ 0 < 𝐵)) |
| 27 | ltmuldiv2 12142 | . . . 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 1373 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵)))) < 𝐵 ↔ ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < (𝐵 / 𝐵))) |
| 29 | 21, 28 | mpbird 257 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐵 · ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵)))) < 𝐵) |
| 30 | 15, 29 | eqbrtrd 5165 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) < 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 < clt 11295 − cmin 11492 / cdiv 11920 ℝ+crp 13034 ⌊cfl 13830 mod cmo 13909 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fl 13832 df-mod 13910 |
| This theorem is referenced by: modelico 13921 zmodfz 13933 modid2 13938 modabs 13944 modaddmodup 13975 modsubdir 13981 digit1 14276 cshwidxmod 14841 repswcshw 14850 divalgmod 16443 bitsmod 16473 bitsinv1lem 16478 bezoutlem3 16578 eucalglt 16622 odzdvds 16833 fldivp1 16935 4sqlem6 16981 4sqlem12 16994 mndodcong 19560 oddvds 19565 gexdvds 19602 zringlpirlem3 21475 sineq0 26566 efif1olem2 26585 lgseisenlem1 27419 irrapxlem1 42833 pellfund14 42909 jm2.19 43005 sineq0ALT 44957 fourierswlem 46245 fouriersw 46246 gpg3nbgrvtx0 48032 |
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