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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltmod | Structured version Visualization version GIF version |
Description: A sufficient condition for a "less than" relationship for the mod operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
ltmod.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltmod.b | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
ltmod.c | ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴)) |
Ref | Expression |
---|---|
ltmod | ⊢ (𝜑 → (𝐶 mod 𝐵) < (𝐴 mod 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmod.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltmod.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
3 | 1, 2 | modcld 13734 | . . . . . . 7 ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℝ) |
4 | 1, 3 | resubcld 11541 | . . . . . 6 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ) |
5 | 1 | rexrd 11163 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
6 | icossre 13299 | . . . . . 6 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ ∧ 𝐴 ∈ ℝ*) → ((𝐴 − (𝐴 mod 𝐵))[,)𝐴) ⊆ ℝ) | |
7 | 4, 5, 6 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵))[,)𝐴) ⊆ ℝ) |
8 | ltmod.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴)) | |
9 | 7, 8 | sseldd 3943 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
10 | 2 | rpred 12911 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
11 | 9, 2 | rerpdivcld 12942 | . . . . . . 7 ⊢ (𝜑 → (𝐶 / 𝐵) ∈ ℝ) |
12 | 11 | flcld 13657 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℤ) |
13 | 12 | zred 12565 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) |
14 | 10, 13 | remulcld 11143 | . . . 4 ⊢ (𝜑 → (𝐵 · (⌊‘(𝐶 / 𝐵))) ∈ ℝ) |
15 | 4 | rexrd 11163 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ*) |
16 | icoltub 43640 | . . . . 5 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴)) → 𝐶 < 𝐴) | |
17 | 15, 5, 8, 16 | syl3anc 1371 | . . . 4 ⊢ (𝜑 → 𝐶 < 𝐴) |
18 | 9, 1, 14, 17 | ltsub1dd 11725 | . . 3 ⊢ (𝜑 → (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) < (𝐴 − (𝐵 · (⌊‘(𝐶 / 𝐵))))) |
19 | icossicc 13307 | . . . . . . . 8 ⊢ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴) ⊆ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) | |
20 | 19, 8 | sselid 3940 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) |
21 | 1, 2, 20 | lefldiveq 43424 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵))) |
22 | 21 | eqcomd 2743 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) = (⌊‘(𝐴 / 𝐵))) |
23 | 22 | oveq2d 7367 | . . . 4 ⊢ (𝜑 → (𝐵 · (⌊‘(𝐶 / 𝐵))) = (𝐵 · (⌊‘(𝐴 / 𝐵)))) |
24 | 23 | oveq2d 7367 | . . 3 ⊢ (𝜑 → (𝐴 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
25 | 18, 24 | breqtrd 5129 | . 2 ⊢ (𝜑 → (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) < (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
26 | modval 13730 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐶 mod 𝐵) = (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵))))) | |
27 | 9, 2, 26 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐶 mod 𝐵) = (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵))))) |
28 | modval 13730 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) | |
29 | 1, 2, 28 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
30 | 25, 27, 29 | 3brtr4d 5135 | 1 ⊢ (𝜑 → (𝐶 mod 𝐵) < (𝐴 mod 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 class class class wbr 5103 ‘cfv 6493 (class class class)co 7351 ℝcr 11008 · cmul 11014 ℝ*cxr 11146 < clt 11147 − cmin 11343 / cdiv 11770 ℝ+crp 12869 [,)cico 13220 [,]cicc 13221 ⌊cfl 13649 mod cmo 13728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-rp 12870 df-ico 13224 df-icc 13225 df-fl 13651 df-mod 13729 |
This theorem is referenced by: fouriersw 44366 |
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