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Mirrors > Home > ILE Home > Th. List > zmodfz | GIF version |
Description: An integer mod 𝐵 lies in the first 𝐵 nonnegative integers. (Contributed by Jeff Madsen, 17-Jun-2010.) |
Ref | Expression |
---|---|
zmodfz | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ (0...(𝐵 − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zmodcl 9805 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ ℕ0) | |
2 | 1 | nn0zd 8920 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ ℤ) |
3 | 1 | nn0ge0d 8783 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 0 ≤ (𝐴 mod 𝐵)) |
4 | zq 9165 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) | |
5 | 4 | adantr 271 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℚ) |
6 | nnq 9172 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℚ) | |
7 | 6 | adantl 272 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℚ) |
8 | nngt0 8501 | . . . 4 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵) | |
9 | 8 | adantl 272 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 0 < 𝐵) |
10 | modqlt 9794 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) < 𝐵) | |
11 | 5, 7, 9, 10 | syl3anc 1175 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) < 𝐵) |
12 | 0z 8815 | . . 3 ⊢ 0 ∈ ℤ | |
13 | nnz 8823 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ) | |
14 | 13 | adantl 272 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℤ) |
15 | elfzm11 9559 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 mod 𝐵) ∈ (0...(𝐵 − 1)) ↔ ((𝐴 mod 𝐵) ∈ ℤ ∧ 0 ≤ (𝐴 mod 𝐵) ∧ (𝐴 mod 𝐵) < 𝐵))) | |
16 | 12, 14, 15 | sylancr 406 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴 mod 𝐵) ∈ (0...(𝐵 − 1)) ↔ ((𝐴 mod 𝐵) ∈ ℤ ∧ 0 ≤ (𝐴 mod 𝐵) ∧ (𝐴 mod 𝐵) < 𝐵))) |
17 | 2, 3, 11, 16 | mpbir3and 1127 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 mod 𝐵) ∈ (0...(𝐵 − 1))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 925 ∈ wcel 1439 class class class wbr 3851 (class class class)co 5666 0cc0 7404 1c1 7405 < clt 7576 ≤ cle 7577 − cmin 7707 ℕcn 8476 ℤcz 8804 ℚcq 9158 ...cfz 9478 mod cmo 9783 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-mulrcl 7498 ax-addcom 7499 ax-mulcom 7500 ax-addass 7501 ax-mulass 7502 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-1rid 7506 ax-0id 7507 ax-rnegex 7508 ax-precex 7509 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-apti 7514 ax-pre-ltadd 7515 ax-pre-mulgt0 7516 ax-pre-mulext 7517 ax-arch 7518 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-po 4132 df-iso 4133 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-reap 8106 df-ap 8113 df-div 8194 df-inn 8477 df-n0 8728 df-z 8805 df-q 9159 df-rp 9189 df-fz 9479 df-fl 9731 df-mod 9784 |
This theorem is referenced by: zmodfzo 9808 mod2eq1n2dvds 11211 bezoutlemmain 11319 |
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