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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lefldiveq | Structured version Visualization version GIF version |
Description: A closed enough, smaller real 𝐶 has the same floor of 𝐴 when both are divided by 𝐵. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
lefldiveq.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
lefldiveq.b | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
lefldiveq.c | ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) |
Ref | Expression |
---|---|
lefldiveq | ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lefldiveq.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | lefldiveq.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
3 | moddiffl 12983 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵))) | |
4 | 1, 2, 3 | syl2anc 579 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵))) |
5 | 1, 2 | rerpdivcld 12194 | . . . . . . 7 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) |
6 | 5 | flcld 12901 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ∈ ℤ) |
7 | 4, 6 | eqeltrd 2906 | . . . . 5 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℤ) |
8 | flid 12911 | . . . . 5 ⊢ (((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℤ → (⌊‘((𝐴 − (𝐴 mod 𝐵)) / 𝐵)) = ((𝐴 − (𝐴 mod 𝐵)) / 𝐵)) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (⌊‘((𝐴 − (𝐴 mod 𝐵)) / 𝐵)) = ((𝐴 − (𝐴 mod 𝐵)) / 𝐵)) |
10 | 9, 4 | eqtr2d 2862 | . . 3 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘((𝐴 − (𝐴 mod 𝐵)) / 𝐵))) |
11 | 1, 2 | modcld 12976 | . . . . . 6 ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℝ) |
12 | 1, 11 | resubcld 10789 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ) |
13 | 12, 2 | rerpdivcld 12194 | . . . 4 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℝ) |
14 | iccssre 12550 | . . . . . . 7 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) ⊆ ℝ) | |
15 | 12, 1, 14 | syl2anc 579 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) ⊆ ℝ) |
16 | lefldiveq.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) | |
17 | 15, 16 | sseldd 3828 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
18 | 17, 2 | rerpdivcld 12194 | . . . 4 ⊢ (𝜑 → (𝐶 / 𝐵) ∈ ℝ) |
19 | 12 | rexrd 10413 | . . . . . 6 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ*) |
20 | 1 | rexrd 10413 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
21 | iccgelb 12525 | . . . . . 6 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) → (𝐴 − (𝐴 mod 𝐵)) ≤ 𝐶) | |
22 | 19, 20, 16, 21 | syl3anc 1494 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ≤ 𝐶) |
23 | 12, 17, 2, 22 | lediv1dd 12221 | . . . 4 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ≤ (𝐶 / 𝐵)) |
24 | flwordi 12915 | . . . 4 ⊢ ((((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℝ ∧ (𝐶 / 𝐵) ∈ ℝ ∧ ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ≤ (𝐶 / 𝐵)) → (⌊‘((𝐴 − (𝐴 mod 𝐵)) / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵))) | |
25 | 13, 18, 23, 24 | syl3anc 1494 | . . 3 ⊢ (𝜑 → (⌊‘((𝐴 − (𝐴 mod 𝐵)) / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵))) |
26 | 10, 25 | eqbrtrd 4897 | . 2 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵))) |
27 | iccleub 12524 | . . . . 5 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) → 𝐶 ≤ 𝐴) | |
28 | 19, 20, 16, 27 | syl3anc 1494 | . . . 4 ⊢ (𝜑 → 𝐶 ≤ 𝐴) |
29 | 17, 1, 2, 28 | lediv1dd 12221 | . . 3 ⊢ (𝜑 → (𝐶 / 𝐵) ≤ (𝐴 / 𝐵)) |
30 | flwordi 12915 | . . 3 ⊢ (((𝐶 / 𝐵) ∈ ℝ ∧ (𝐴 / 𝐵) ∈ ℝ ∧ (𝐶 / 𝐵) ≤ (𝐴 / 𝐵)) → (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))) | |
31 | 18, 5, 29, 30 | syl3anc 1494 | . 2 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))) |
32 | reflcl 12899 | . . . 4 ⊢ ((𝐴 / 𝐵) ∈ ℝ → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) | |
33 | 5, 32 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) |
34 | reflcl 12899 | . . . 4 ⊢ ((𝐶 / 𝐵) ∈ ℝ → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) | |
35 | 18, 34 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) |
36 | 33, 35 | letri3d 10505 | . 2 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵)) ↔ ((⌊‘(𝐴 / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵)) ∧ (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))))) |
37 | 26, 31, 36 | mpbir2and 704 | 1 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 ℝcr 10258 ℝ*cxr 10397 ≤ cle 10399 − cmin 10592 / cdiv 11016 ℤcz 11711 ℝ+crp 12119 [,]cicc 12473 ⌊cfl 12893 mod cmo 12970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 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 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-sup 8623 df-inf 8624 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-n0 11626 df-z 11712 df-uz 11976 df-rp 12120 df-icc 12477 df-fl 12895 df-mod 12971 |
This theorem is referenced by: ltmod 40659 |
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