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Mirrors > Home > ILE Home > Th. List > adddivflid | GIF version |
Description: The floor of a sum of an integer and a fraction is equal to the integer iff the denominator of the fraction is less than the numerator. (Contributed by AV, 14-Jul-2021.) |
Ref | Expression |
---|---|
adddivflid | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → (𝐵 < 𝐶 ↔ (⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 981 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℤ) | |
2 | nn0z 9074 | . . . . . 6 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℤ) | |
3 | znq 9416 | . . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (𝐵 / 𝐶) ∈ ℚ) | |
4 | 2, 3 | sylan 281 | . . . . 5 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → (𝐵 / 𝐶) ∈ ℚ) |
5 | 4 | 3adant1 999 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → (𝐵 / 𝐶) ∈ ℚ) |
6 | 1, 5 | jca 304 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → (𝐴 ∈ ℤ ∧ (𝐵 / 𝐶) ∈ ℚ)) |
7 | flqbi2 10064 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐵 / 𝐶) ∈ ℚ) → ((⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴 ↔ (0 ≤ (𝐵 / 𝐶) ∧ (𝐵 / 𝐶) < 1))) | |
8 | 6, 7 | syl 14 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → ((⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴 ↔ (0 ≤ (𝐵 / 𝐶) ∧ (𝐵 / 𝐶) < 1))) |
9 | nn0re 8986 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ) | |
10 | nn0ge0 9002 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ0 → 0 ≤ 𝐵) | |
11 | 9, 10 | jca 304 | . . . . . 6 ⊢ (𝐵 ∈ ℕ0 → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) |
12 | nnre 8727 | . . . . . . 7 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℝ) | |
13 | nngt0 8745 | . . . . . . 7 ⊢ (𝐶 ∈ ℕ → 0 < 𝐶) | |
14 | 12, 13 | jca 304 | . . . . . 6 ⊢ (𝐶 ∈ ℕ → (𝐶 ∈ ℝ ∧ 0 < 𝐶)) |
15 | 11, 14 | anim12i 336 | . . . . 5 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → ((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶))) |
16 | 15 | 3adant1 999 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → ((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶))) |
17 | divge0 8631 | . . . 4 ⊢ (((𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → 0 ≤ (𝐵 / 𝐶)) | |
18 | 16, 17 | syl 14 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → 0 ≤ (𝐵 / 𝐶)) |
19 | 18 | biantrurd 303 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → ((𝐵 / 𝐶) < 1 ↔ (0 ≤ (𝐵 / 𝐶) ∧ (𝐵 / 𝐶) < 1))) |
20 | nnrp 9451 | . . . . 5 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℝ+) | |
21 | 9, 20 | anim12i 336 | . . . 4 ⊢ ((𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ+)) |
22 | 21 | 3adant1 999 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ+)) |
23 | divlt1lt 9511 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ+) → ((𝐵 / 𝐶) < 1 ↔ 𝐵 < 𝐶)) | |
24 | 22, 23 | syl 14 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → ((𝐵 / 𝐶) < 1 ↔ 𝐵 < 𝐶)) |
25 | 8, 19, 24 | 3bitr2rd 216 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → (𝐵 < 𝐶 ↔ (⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ℝcr 7619 0cc0 7620 1c1 7621 + caddc 7623 < clt 7800 ≤ cle 7801 / cdiv 8432 ℕcn 8720 ℕ0cn0 8977 ℤcz 9054 ℚcq 9411 ℝ+crp 9441 ⌊cfl 10041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-n0 8978 df-z 9055 df-q 9412 df-rp 9442 df-fl 10043 |
This theorem is referenced by: (None) |
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