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 13244 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵))) | |
4 | 1, 2, 3 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵))) |
5 | 1, 2 | rerpdivcld 12456 | . . . . . . 7 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) |
6 | 5 | flcld 13162 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ∈ ℤ) |
7 | 4, 6 | eqeltrd 2913 | . . . . 5 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℤ) |
8 | flid 13172 | . . . . 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 2857 | . . 3 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘((𝐴 − (𝐴 mod 𝐵)) / 𝐵))) |
11 | 1, 2 | modcld 13237 | . . . . . 6 ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℝ) |
12 | 1, 11 | resubcld 11062 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ) |
13 | 12, 2 | rerpdivcld 12456 | . . . 4 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℝ) |
14 | iccssre 12812 | . . . . . . 7 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) ⊆ ℝ) | |
15 | 12, 1, 14 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) ⊆ ℝ) |
16 | lefldiveq.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) | |
17 | 15, 16 | sseldd 3967 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
18 | 17, 2 | rerpdivcld 12456 | . . . 4 ⊢ (𝜑 → (𝐶 / 𝐵) ∈ ℝ) |
19 | 12 | rexrd 10685 | . . . . . 6 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ*) |
20 | 1 | rexrd 10685 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
21 | iccgelb 12787 | . . . . . 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 1367 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ≤ 𝐶) |
23 | 12, 17, 2, 22 | lediv1dd 12483 | . . . 4 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ≤ (𝐶 / 𝐵)) |
24 | flwordi 13176 | . . . 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 1367 | . . 3 ⊢ (𝜑 → (⌊‘((𝐴 − (𝐴 mod 𝐵)) / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵))) |
26 | 10, 25 | eqbrtrd 5080 | . 2 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵))) |
27 | iccleub 12786 | . . . . 5 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) → 𝐶 ≤ 𝐴) | |
28 | 19, 20, 16, 27 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → 𝐶 ≤ 𝐴) |
29 | 17, 1, 2, 28 | lediv1dd 12483 | . . 3 ⊢ (𝜑 → (𝐶 / 𝐵) ≤ (𝐴 / 𝐵)) |
30 | flwordi 13176 | . . 3 ⊢ (((𝐶 / 𝐵) ∈ ℝ ∧ (𝐴 / 𝐵) ∈ ℝ ∧ (𝐶 / 𝐵) ≤ (𝐴 / 𝐵)) → (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))) | |
31 | 18, 5, 29, 30 | syl3anc 1367 | . 2 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))) |
32 | reflcl 13160 | . . . 4 ⊢ ((𝐴 / 𝐵) ∈ ℝ → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) | |
33 | 5, 32 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) |
34 | reflcl 13160 | . . . 4 ⊢ ((𝐶 / 𝐵) ∈ ℝ → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) | |
35 | 18, 34 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) |
36 | 33, 35 | letri3d 10776 | . 2 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵)) ↔ ((⌊‘(𝐴 / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵)) ∧ (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))))) |
37 | 26, 31, 36 | mpbir2and 711 | 1 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 ℝ*cxr 10668 ≤ cle 10670 − cmin 10864 / cdiv 11291 ℤcz 11975 ℝ+crp 12383 [,]cicc 12735 ⌊cfl 13154 mod cmo 13231 |
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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-icc 12739 df-fl 13156 df-mod 13232 |
This theorem is referenced by: ltmod 41912 |
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