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 12674 | . . . . . . 7 ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℝ) |
4 | 1, 3 | resubcld 10458 | . . . . . 6 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ) |
5 | 1 | rexrd 10089 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
6 | icossre 12254 | . . . . . 6 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ ∧ 𝐴 ∈ ℝ*) → ((𝐴 − (𝐴 mod 𝐵))[,)𝐴) ⊆ ℝ) | |
7 | 4, 5, 6 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵))[,)𝐴) ⊆ ℝ) |
8 | ltmod.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴)) | |
9 | 7, 8 | sseldd 3604 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
10 | 2 | rpred 11872 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
11 | 9, 2 | rerpdivcld 11903 | . . . . . . 7 ⊢ (𝜑 → (𝐶 / 𝐵) ∈ ℝ) |
12 | 11 | flcld 12599 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℤ) |
13 | 12 | zred 11482 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) |
14 | 10, 13 | remulcld 10070 | . . . 4 ⊢ (𝜑 → (𝐵 · (⌊‘(𝐶 / 𝐵))) ∈ ℝ) |
15 | 4 | rexrd 10089 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ*) |
16 | icoltub 39732 | . . . . 5 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴)) → 𝐶 < 𝐴) | |
17 | 15, 5, 8, 16 | syl3anc 1326 | . . . 4 ⊢ (𝜑 → 𝐶 < 𝐴) |
18 | 9, 1, 14, 17 | ltsub1dd 10639 | . . 3 ⊢ (𝜑 → (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) < (𝐴 − (𝐵 · (⌊‘(𝐶 / 𝐵))))) |
19 | icossicc 12260 | . . . . . . . 8 ⊢ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴) ⊆ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) | |
20 | 19, 8 | sseldi 3601 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) |
21 | 1, 2, 20 | lefldiveq 39505 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵))) |
22 | 21 | eqcomd 2628 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) = (⌊‘(𝐴 / 𝐵))) |
23 | 22 | oveq2d 6666 | . . . 4 ⊢ (𝜑 → (𝐵 · (⌊‘(𝐶 / 𝐵))) = (𝐵 · (⌊‘(𝐴 / 𝐵)))) |
24 | 23 | oveq2d 6666 | . . 3 ⊢ (𝜑 → (𝐴 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
25 | 18, 24 | breqtrd 4679 | . 2 ⊢ (𝜑 → (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) < (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
26 | modval 12670 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐶 mod 𝐵) = (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵))))) | |
27 | 9, 2, 26 | syl2anc 693 | . 2 ⊢ (𝜑 → (𝐶 mod 𝐵) = (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵))))) |
28 | modval 12670 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) | |
29 | 1, 2, 28 | syl2anc 693 | . 2 ⊢ (𝜑 → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
30 | 25, 27, 29 | 3brtr4d 4685 | 1 ⊢ (𝜑 → (𝐶 mod 𝐵) < (𝐴 mod 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 · cmul 9941 ℝ*cxr 10073 < clt 10074 − cmin 10266 / cdiv 10684 ℝ+crp 11832 [,)cico 12177 [,]cicc 12178 ⌊cfl 12591 mod cmo 12668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-icc 12182 df-fl 12593 df-mod 12669 |
This theorem is referenced by: fouriersw 40448 |
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