![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > modqelico | GIF version |
Description: Modular reduction produces a half-open interval. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Ref | Expression |
---|---|
modqelico | ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ (0[,)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modqcl 9622 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℚ) | |
2 | qre 9005 | . . 3 ⊢ ((𝐴 mod 𝐵) ∈ ℚ → (𝐴 mod 𝐵) ∈ ℝ) | |
3 | 1, 2 | syl 14 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℝ) |
4 | modqge0 9628 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵)) | |
5 | modqlt 9629 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) < 𝐵) | |
6 | 0re 7391 | . . 3 ⊢ 0 ∈ ℝ | |
7 | qre 9005 | . . . . 5 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ) | |
8 | 7 | rexrd 7440 | . . . 4 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ*) |
9 | 8 | 3ad2ant2 961 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 𝐵 ∈ ℝ*) |
10 | elico2 9250 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → ((𝐴 mod 𝐵) ∈ (0[,)𝐵) ↔ ((𝐴 mod 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 mod 𝐵) ∧ (𝐴 mod 𝐵) < 𝐵))) | |
11 | 6, 9, 10 | sylancr 405 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) ∈ (0[,)𝐵) ↔ ((𝐴 mod 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 mod 𝐵) ∧ (𝐴 mod 𝐵) < 𝐵))) |
12 | 3, 4, 5, 11 | mpbir3and 1122 | 1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ (0[,)𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∧ w3a 920 ∈ wcel 1434 class class class wbr 3811 (class class class)co 5591 ℝcr 7252 0cc0 7253 ℝ*cxr 7424 < clt 7425 ≤ cle 7426 ℚcq 8999 [,)cico 9203 mod cmo 9618 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-mulrcl 7347 ax-addcom 7348 ax-mulcom 7349 ax-addass 7350 ax-mulass 7351 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-1rid 7355 ax-0id 7356 ax-rnegex 7357 ax-precex 7358 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-apti 7363 ax-pre-ltadd 7364 ax-pre-mulgt0 7365 ax-pre-mulext 7366 ax-arch 7367 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-id 4084 df-po 4087 df-iso 4088 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-1st 5846 df-2nd 5847 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-reap 7952 df-ap 7959 df-div 8038 df-inn 8317 df-n0 8566 df-z 8647 df-q 9000 df-rp 9030 df-ico 9207 df-fl 9566 df-mod 9619 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |