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 13246 | . . . . . . 7 ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℝ) |
4 | 1, 3 | resubcld 11070 | . . . . . 6 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ) |
5 | 1 | rexrd 10693 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
6 | icossre 12820 | . . . . . 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 3970 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
10 | 2 | rpred 12434 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
11 | 9, 2 | rerpdivcld 12465 | . . . . . . 7 ⊢ (𝜑 → (𝐶 / 𝐵) ∈ ℝ) |
12 | 11 | flcld 13171 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℤ) |
13 | 12 | zred 12090 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) |
14 | 10, 13 | remulcld 10673 | . . . 4 ⊢ (𝜑 → (𝐵 · (⌊‘(𝐶 / 𝐵))) ∈ ℝ) |
15 | 4 | rexrd 10693 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ*) |
16 | icoltub 41791 | . . . . 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 11254 | . . 3 ⊢ (𝜑 → (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) < (𝐴 − (𝐵 · (⌊‘(𝐶 / 𝐵))))) |
19 | icossicc 12827 | . . . . . . . 8 ⊢ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴) ⊆ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) | |
20 | 19, 8 | sseldi 3967 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) |
21 | 1, 2, 20 | lefldiveq 41566 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵))) |
22 | 21 | eqcomd 2829 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) = (⌊‘(𝐴 / 𝐵))) |
23 | 22 | oveq2d 7174 | . . . 4 ⊢ (𝜑 → (𝐵 · (⌊‘(𝐶 / 𝐵))) = (𝐵 · (⌊‘(𝐴 / 𝐵)))) |
24 | 23 | oveq2d 7174 | . . 3 ⊢ (𝜑 → (𝐴 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
25 | 18, 24 | breqtrd 5094 | . 2 ⊢ (𝜑 → (𝐶 − (𝐵 · (⌊‘(𝐶 / 𝐵)))) < (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵))))) |
26 | modval 13242 | . . 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 13242 | . . 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 5100 | 1 ⊢ (𝜑 → (𝐶 mod 𝐵) < (𝐴 mod 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 · cmul 10544 ℝ*cxr 10676 < clt 10677 − cmin 10872 / cdiv 11299 ℝ+crp 12392 [,)cico 12743 [,]cicc 12744 ⌊cfl 13163 mod cmo 13240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-icc 12748 df-fl 13165 df-mod 13241 |
This theorem is referenced by: fouriersw 42523 |
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