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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 13919 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵))) | |
4 | 1, 2, 3 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵))) |
5 | 1, 2 | rerpdivcld 13106 | . . . . . . 7 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) |
6 | 5 | flcld 13835 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ∈ ℤ) |
7 | 4, 6 | eqeltrd 2839 | . . . . 5 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℤ) |
8 | flid 13845 | . . . . 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 2776 | . . 3 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘((𝐴 − (𝐴 mod 𝐵)) / 𝐵))) |
11 | 1, 2 | modcld 13912 | . . . . . 6 ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℝ) |
12 | 1, 11 | resubcld 11689 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ) |
13 | 12, 2 | rerpdivcld 13106 | . . . 4 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℝ) |
14 | iccssre 13466 | . . . . . . 7 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) ⊆ ℝ) | |
15 | 12, 1, 14 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) ⊆ ℝ) |
16 | lefldiveq.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) | |
17 | 15, 16 | sseldd 3996 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
18 | 17, 2 | rerpdivcld 13106 | . . . 4 ⊢ (𝜑 → (𝐶 / 𝐵) ∈ ℝ) |
19 | 12 | rexrd 11309 | . . . . . 6 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ*) |
20 | 1 | rexrd 11309 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
21 | iccgelb 13440 | . . . . . 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 1370 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ≤ 𝐶) |
23 | 12, 17, 2, 22 | lediv1dd 13133 | . . . 4 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ≤ (𝐶 / 𝐵)) |
24 | flwordi 13849 | . . . 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 1370 | . . 3 ⊢ (𝜑 → (⌊‘((𝐴 − (𝐴 mod 𝐵)) / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵))) |
26 | 10, 25 | eqbrtrd 5170 | . 2 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵))) |
27 | iccleub 13439 | . . . . 5 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) → 𝐶 ≤ 𝐴) | |
28 | 19, 20, 16, 27 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → 𝐶 ≤ 𝐴) |
29 | 17, 1, 2, 28 | lediv1dd 13133 | . . 3 ⊢ (𝜑 → (𝐶 / 𝐵) ≤ (𝐴 / 𝐵)) |
30 | flwordi 13849 | . . 3 ⊢ (((𝐶 / 𝐵) ∈ ℝ ∧ (𝐴 / 𝐵) ∈ ℝ ∧ (𝐶 / 𝐵) ≤ (𝐴 / 𝐵)) → (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))) | |
31 | 18, 5, 29, 30 | syl3anc 1370 | . 2 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))) |
32 | reflcl 13833 | . . . 4 ⊢ ((𝐴 / 𝐵) ∈ ℝ → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) | |
33 | 5, 32 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) |
34 | reflcl 13833 | . . . 4 ⊢ ((𝐶 / 𝐵) ∈ ℝ → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) | |
35 | 18, 34 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) |
36 | 33, 35 | letri3d 11401 | . 2 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵)) ↔ ((⌊‘(𝐴 / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵)) ∧ (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))))) |
37 | 26, 31, 36 | mpbir2and 713 | 1 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 ℝ*cxr 11292 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℤcz 12611 ℝ+crp 13032 [,]cicc 13387 ⌊cfl 13827 mod cmo 13906 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-icc 13391 df-fl 13829 df-mod 13907 |
This theorem is referenced by: ltmod 45594 |
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