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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 12969 | . . . . . . 7 ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℝ) |
4 | 1, 3 | resubcld 10782 | . . . . . 6 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ) |
5 | 1 | rexrd 10406 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
6 | icossre 12542 | . . . . . 6 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ ∧ 𝐴 ∈ ℝ*) → ((𝐴 − (𝐴 mod 𝐵))[,)𝐴) ⊆ ℝ) | |
7 | 4, 5, 6 | syl2anc 581 | . . . . 5 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵))[,)𝐴) ⊆ ℝ) |
8 | ltmod.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴)) | |
9 | 7, 8 | sseldd 3828 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
10 | 2 | rpred 12156 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
11 | 9, 2 | rerpdivcld 12187 | . . . . . . 7 ⊢ (𝜑 → (𝐶 / 𝐵) ∈ ℝ) |
12 | 11 | flcld 12894 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℤ) |
13 | 12 | zred 11810 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) |
14 | 10, 13 | remulcld 10387 | . . . 4 ⊢ (𝜑 → (𝐵 · (⌊‘(𝐶 / 𝐵))) ∈ ℝ) |
15 | 4 | rexrd 10406 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ*) |
16 | icoltub 40530 | . . . . 5 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴)) → 𝐶 < 𝐴) | |
17 | 15, 5, 8, 16 | syl3anc 1496 | . . . 4 ⊢ (𝜑 → 𝐶 < 𝐴) |
18 | 9, 1, 14, 17 | ltsub1dd 10964 | . . 3 ⊢ (𝜑 → (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) < (𝐴 − (𝐵 · (⌊‘(𝐶 / 𝐵))))) |
19 | icossicc 12549 | . . . . . . . 8 ⊢ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴) ⊆ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) | |
20 | 19, 8 | sseldi 3825 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) |
21 | 1, 2, 20 | lefldiveq 40304 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵))) |
22 | 21 | eqcomd 2831 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) = (⌊‘(𝐴 / 𝐵))) |
23 | 22 | oveq2d 6921 | . . . 4 ⊢ (𝜑 → (𝐵 · (⌊‘(𝐶 / 𝐵))) = (𝐵 · (⌊‘(𝐴 / 𝐵)))) |
24 | 23 | oveq2d 6921 | . . 3 ⊢ (𝜑 → (𝐴 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
25 | 18, 24 | breqtrd 4899 | . 2 ⊢ (𝜑 → (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) < (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
26 | modval 12965 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐶 mod 𝐵) = (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵))))) | |
27 | 9, 2, 26 | syl2anc 581 | . 2 ⊢ (𝜑 → (𝐶 mod 𝐵) = (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵))))) |
28 | modval 12965 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) | |
29 | 1, 2, 28 | syl2anc 581 | . 2 ⊢ (𝜑 → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
30 | 25, 27, 29 | 3brtr4d 4905 | 1 ⊢ (𝜑 → (𝐶 mod 𝐵) < (𝐴 mod 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ⊆ wss 3798 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 ℝcr 10251 · cmul 10257 ℝ*cxr 10390 < clt 10391 − cmin 10585 / cdiv 11009 ℝ+crp 12112 [,)cico 12465 [,]cicc 12466 ⌊cfl 12886 mod cmo 12963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-ico 12469 df-icc 12470 df-fl 12888 df-mod 12964 |
This theorem is referenced by: fouriersw 41242 |
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