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 13242 | . . . . . . 7 ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℝ) |
4 | 1, 3 | resubcld 11067 | . . . . . 6 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ) |
5 | 1 | rexrd 10690 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
6 | icossre 12816 | . . . . . 6 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ ∧ 𝐴 ∈ ℝ*) → ((𝐴 − (𝐴 mod 𝐵))[,)𝐴) ⊆ ℝ) | |
7 | 4, 5, 6 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵))[,)𝐴) ⊆ ℝ) |
8 | ltmod.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴)) | |
9 | 7, 8 | sseldd 3967 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
10 | 2 | rpred 12430 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
11 | 9, 2 | rerpdivcld 12461 | . . . . . . 7 ⊢ (𝜑 → (𝐶 / 𝐵) ∈ ℝ) |
12 | 11 | flcld 13167 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℤ) |
13 | 12 | zred 12086 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) |
14 | 10, 13 | remulcld 10670 | . . . 4 ⊢ (𝜑 → (𝐵 · (⌊‘(𝐶 / 𝐵))) ∈ ℝ) |
15 | 4 | rexrd 10690 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ*) |
16 | icoltub 41782 | . . . . 5 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴)) → 𝐶 < 𝐴) | |
17 | 15, 5, 8, 16 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → 𝐶 < 𝐴) |
18 | 9, 1, 14, 17 | ltsub1dd 11251 | . . 3 ⊢ (𝜑 → (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) < (𝐴 − (𝐵 · (⌊‘(𝐶 / 𝐵))))) |
19 | icossicc 12823 | . . . . . . . 8 ⊢ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴) ⊆ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) | |
20 | 19, 8 | sseldi 3964 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) |
21 | 1, 2, 20 | lefldiveq 41557 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵))) |
22 | 21 | eqcomd 2827 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) = (⌊‘(𝐴 / 𝐵))) |
23 | 22 | oveq2d 7171 | . . . 4 ⊢ (𝜑 → (𝐵 · (⌊‘(𝐶 / 𝐵))) = (𝐵 · (⌊‘(𝐴 / 𝐵)))) |
24 | 23 | oveq2d 7171 | . . 3 ⊢ (𝜑 → (𝐴 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
25 | 18, 24 | breqtrd 5091 | . 2 ⊢ (𝜑 → (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) < (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
26 | modval 13238 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐶 mod 𝐵) = (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵))))) | |
27 | 9, 2, 26 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐶 mod 𝐵) = (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵))))) |
28 | modval 13238 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) | |
29 | 1, 2, 28 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
30 | 25, 27, 29 | 3brtr4d 5097 | 1 ⊢ (𝜑 → (𝐶 mod 𝐵) < (𝐴 mod 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 · cmul 10541 ℝ*cxr 10673 < clt 10674 − cmin 10869 / cdiv 11296 ℝ+crp 12388 [,)cico 12739 [,]cicc 12740 ⌊cfl 13159 mod cmo 13236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-ico 12743 df-icc 12744 df-fl 13161 df-mod 13237 |
This theorem is referenced by: fouriersw 42515 |
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