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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 13245 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵))) | |
4 | 1, 2, 3 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵))) |
5 | 1, 2 | rerpdivcld 12450 | . . . . . . 7 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) |
6 | 5 | flcld 13163 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ∈ ℤ) |
7 | 4, 6 | eqeltrd 2890 | . . . . 5 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℤ) |
8 | flid 13173 | . . . . 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 2834 | . . 3 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘((𝐴 − (𝐴 mod 𝐵)) / 𝐵))) |
11 | 1, 2 | modcld 13238 | . . . . . 6 ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℝ) |
12 | 1, 11 | resubcld 11057 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ) |
13 | 12, 2 | rerpdivcld 12450 | . . . 4 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℝ) |
14 | iccssre 12807 | . . . . . . 7 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) ⊆ ℝ) | |
15 | 12, 1, 14 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) ⊆ ℝ) |
16 | lefldiveq.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) | |
17 | 15, 16 | sseldd 3916 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
18 | 17, 2 | rerpdivcld 12450 | . . . 4 ⊢ (𝜑 → (𝐶 / 𝐵) ∈ ℝ) |
19 | 12 | rexrd 10680 | . . . . . 6 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ*) |
20 | 1 | rexrd 10680 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
21 | iccgelb 12781 | . . . . . 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 1368 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ≤ 𝐶) |
23 | 12, 17, 2, 22 | lediv1dd 12477 | . . . 4 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ≤ (𝐶 / 𝐵)) |
24 | flwordi 13177 | . . . 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 1368 | . . 3 ⊢ (𝜑 → (⌊‘((𝐴 − (𝐴 mod 𝐵)) / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵))) |
26 | 10, 25 | eqbrtrd 5052 | . 2 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵))) |
27 | iccleub 12780 | . . . . 5 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) → 𝐶 ≤ 𝐴) | |
28 | 19, 20, 16, 27 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → 𝐶 ≤ 𝐴) |
29 | 17, 1, 2, 28 | lediv1dd 12477 | . . 3 ⊢ (𝜑 → (𝐶 / 𝐵) ≤ (𝐴 / 𝐵)) |
30 | flwordi 13177 | . . 3 ⊢ (((𝐶 / 𝐵) ∈ ℝ ∧ (𝐴 / 𝐵) ∈ ℝ ∧ (𝐶 / 𝐵) ≤ (𝐴 / 𝐵)) → (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))) | |
31 | 18, 5, 29, 30 | syl3anc 1368 | . 2 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))) |
32 | reflcl 13161 | . . . 4 ⊢ ((𝐴 / 𝐵) ∈ ℝ → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) | |
33 | 5, 32 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) |
34 | reflcl 13161 | . . . 4 ⊢ ((𝐶 / 𝐵) ∈ ℝ → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) | |
35 | 18, 34 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) |
36 | 33, 35 | letri3d 10771 | . 2 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵)) ↔ ((⌊‘(𝐴 / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵)) ∧ (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))))) |
37 | 26, 31, 36 | mpbir2and 712 | 1 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 ℝ*cxr 10663 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℤcz 11969 ℝ+crp 12377 [,]cicc 12729 ⌊cfl 13155 mod cmo 13232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-icc 12733 df-fl 13157 df-mod 13233 |
This theorem is referenced by: ltmod 42280 |
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