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Mirrors > Home > ILE Home > Th. List > divalg | GIF version |
Description: The division algorithm (theorem). Dividing an integer 𝑁 by a nonzero integer 𝐷 produces a (unique) quotient 𝑞 and a unique remainder 0 ≤ 𝑟 < (abs‘𝐷). Theorem 1.14 in [ApostolNT] p. 19. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
divalg | ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 942 | . . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) ∧ 𝐷 < 0) → 𝑁 ∈ ℤ) | |
2 | simpl2 943 | . . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) ∧ 𝐷 < 0) → 𝐷 ∈ ℤ) | |
3 | simpr 108 | . . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) ∧ 𝐷 < 0) → 𝐷 < 0) | |
4 | divalglemeuneg 10702 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) | |
5 | 1, 2, 3, 4 | syl3anc 1170 | . 2 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) ∧ 𝐷 < 0) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) |
6 | simpr 108 | . . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) ∧ 𝐷 = 0) → 𝐷 = 0) | |
7 | simpl3 944 | . . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) ∧ 𝐷 = 0) → 𝐷 ≠ 0) | |
8 | 6, 7 | pm2.21ddne 2332 | . 2 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) ∧ 𝐷 = 0) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) |
9 | simpl1 942 | . . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) ∧ 0 < 𝐷) → 𝑁 ∈ ℤ) | |
10 | simpl2 943 | . . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) ∧ 0 < 𝐷) → 𝐷 ∈ ℤ) | |
11 | simpr 108 | . . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) ∧ 0 < 𝐷) → 0 < 𝐷) | |
12 | elnnz 8655 | . . . 4 ⊢ (𝐷 ∈ ℕ ↔ (𝐷 ∈ ℤ ∧ 0 < 𝐷)) | |
13 | 10, 11, 12 | sylanbrc 408 | . . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) ∧ 0 < 𝐷) → 𝐷 ∈ ℕ) |
14 | divalglemeunn 10700 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) | |
15 | 9, 13, 14 | syl2anc 403 | . 2 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) ∧ 0 < 𝐷) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) |
16 | ztri3or0 8687 | . . 3 ⊢ (𝐷 ∈ ℤ → (𝐷 < 0 ∨ 𝐷 = 0 ∨ 0 < 𝐷)) | |
17 | 16 | 3ad2ant2 961 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (𝐷 < 0 ∨ 𝐷 = 0 ∨ 0 < 𝐷)) |
18 | 5, 8, 15, 17 | mpjao3dan 1239 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∨ w3o 919 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 ≠ wne 2249 ∃wrex 2354 ∃!wreu 2355 class class class wbr 3811 ‘cfv 4968 (class class class)co 5590 0cc0 7252 + caddc 7255 · cmul 7257 < clt 7424 ≤ cle 7425 ℕcn 8315 ℤcz 8645 abscabs 10256 |
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-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 ax-cnex 7338 ax-resscn 7339 ax-1cn 7340 ax-1re 7341 ax-icn 7342 ax-addcl 7343 ax-addrcl 7344 ax-mulcl 7345 ax-mulrcl 7346 ax-addcom 7347 ax-mulcom 7348 ax-addass 7349 ax-mulass 7350 ax-distr 7351 ax-i2m1 7352 ax-0lt1 7353 ax-1rid 7354 ax-0id 7355 ax-rnegex 7356 ax-precex 7357 ax-cnre 7358 ax-pre-ltirr 7359 ax-pre-ltwlin 7360 ax-pre-lttrn 7361 ax-pre-apti 7362 ax-pre-ltadd 7363 ax-pre-mulgt0 7364 ax-pre-mulext 7365 ax-arch 7366 |
This theorem depends on definitions: df-bi 115 df-dc 777 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-nul 3270 df-if 3374 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-tr 3902 df-id 4083 df-po 4086 df-iso 4087 df-iord 4156 df-on 4158 df-ilim 4159 df-suc 4161 df-iom 4368 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-rn 4411 df-res 4412 df-ima 4413 df-iota 4933 df-fun 4970 df-fn 4971 df-f 4972 df-f1 4973 df-fo 4974 df-f1o 4975 df-fv 4976 df-riota 5546 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-1st 5845 df-2nd 5846 df-recs 6001 df-frec 6087 df-pnf 7426 df-mnf 7427 df-xr 7428 df-ltxr 7429 df-le 7430 df-sub 7557 df-neg 7558 df-reap 7951 df-ap 7958 df-div 8037 df-inn 8316 df-2 8374 df-n0 8565 df-z 8646 df-uz 8914 df-q 8999 df-rp 9029 df-fl 9565 df-mod 9618 df-iseq 9740 df-iexp 9791 df-cj 10102 df-re 10103 df-im 10104 df-rsqrt 10257 df-abs 10258 |
This theorem is referenced by: divalg2 10705 |
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