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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 12543 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵))) | |
4 | 1, 2, 3 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵))) |
5 | 1, 2 | rerpdivcld 11779 | . . . . . . 7 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ) |
6 | 5 | flcld 12461 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ∈ ℤ) |
7 | 4, 6 | eqeltrd 2688 | . . . . 5 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℤ) |
8 | flid 12471 | . . . . 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 2645 | . . 3 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘((𝐴 − (𝐴 mod 𝐵)) / 𝐵))) |
11 | 1, 2 | modcld 12536 | . . . . . 6 ⊢ (𝜑 → (𝐴 mod 𝐵) ∈ ℝ) |
12 | 1, 11 | resubcld 10337 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ) |
13 | 12, 2 | rerpdivcld 11779 | . . . 4 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℝ) |
14 | iccssre 12126 | . . . . . . 7 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) ⊆ ℝ) | |
15 | 12, 1, 14 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵))[,]𝐴) ⊆ ℝ) |
16 | lefldiveq.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) | |
17 | 15, 16 | sseldd 3569 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
18 | 17, 2 | rerpdivcld 11779 | . . . 4 ⊢ (𝜑 → (𝐶 / 𝐵) ∈ ℝ) |
19 | 12 | rexrd 9968 | . . . . . 6 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ∈ ℝ*) |
20 | 1 | rexrd 9968 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
21 | iccgelb 12101 | . . . . . 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 1318 | . . . . 5 ⊢ (𝜑 → (𝐴 − (𝐴 mod 𝐵)) ≤ 𝐶) |
23 | 12, 17, 2, 22 | lediv1dd 11806 | . . . 4 ⊢ (𝜑 → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ≤ (𝐶 / 𝐵)) |
24 | flwordi 12475 | . . . 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 1318 | . . 3 ⊢ (𝜑 → (⌊‘((𝐴 − (𝐴 mod 𝐵)) / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵))) |
26 | 10, 25 | eqbrtrd 4605 | . 2 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵))) |
27 | iccleub 12100 | . . . . 5 ⊢ (((𝐴 − (𝐴 mod 𝐵)) ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,]𝐴)) → 𝐶 ≤ 𝐴) | |
28 | 19, 20, 16, 27 | syl3anc 1318 | . . . 4 ⊢ (𝜑 → 𝐶 ≤ 𝐴) |
29 | 17, 1, 2, 28 | lediv1dd 11806 | . . 3 ⊢ (𝜑 → (𝐶 / 𝐵) ≤ (𝐴 / 𝐵)) |
30 | flwordi 12475 | . . 3 ⊢ (((𝐶 / 𝐵) ∈ ℝ ∧ (𝐴 / 𝐵) ∈ ℝ ∧ (𝐶 / 𝐵) ≤ (𝐴 / 𝐵)) → (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))) | |
31 | 18, 5, 29, 30 | syl3anc 1318 | . 2 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))) |
32 | reflcl 12459 | . . . 4 ⊢ ((𝐴 / 𝐵) ∈ ℝ → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) | |
33 | 5, 32 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) ∈ ℝ) |
34 | reflcl 12459 | . . . 4 ⊢ ((𝐶 / 𝐵) ∈ ℝ → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) | |
35 | 18, 34 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐶 / 𝐵)) ∈ ℝ) |
36 | 33, 35 | letri3d 10058 | . 2 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵)) ↔ ((⌊‘(𝐴 / 𝐵)) ≤ (⌊‘(𝐶 / 𝐵)) ∧ (⌊‘(𝐶 / 𝐵)) ≤ (⌊‘(𝐴 / 𝐵))))) |
37 | 26, 31, 36 | mpbir2and 959 | 1 ⊢ (𝜑 → (⌊‘(𝐴 / 𝐵)) = (⌊‘(𝐶 / 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 ℝ*cxr 9952 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℤcz 11254 ℝ+crp 11708 [,]cicc 12049 ⌊cfl 12453 mod cmo 12530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-icc 12053 df-fl 12455 df-mod 12531 |
This theorem is referenced by: ltmod 38705 |
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