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Mirrors > Home > ILE Home > Th. List > modremain | GIF version |
Description: The result of the modulo operation is the remainder of the division algorithm. (Contributed by AV, 19-Aug-2021.) |
Ref | Expression |
---|---|
modremain | ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → ((𝑁 mod 𝐷) = 𝑅 ↔ ∃𝑧 ∈ ℤ ((𝑧 · 𝐷) + 𝑅) = 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2139 | . 2 ⊢ ((𝑁 mod 𝐷) = 𝑅 ↔ 𝑅 = (𝑁 mod 𝐷)) | |
2 | divalgmodcl 11614 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ 𝑅 ∈ ℕ0) → (𝑅 = (𝑁 mod 𝐷) ↔ (𝑅 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑅)))) | |
3 | 2 | 3adant3r 1213 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → (𝑅 = (𝑁 mod 𝐷) ↔ (𝑅 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑅)))) |
4 | ibar 299 | . . . . 5 ⊢ (𝑅 < 𝐷 → (𝐷 ∥ (𝑁 − 𝑅) ↔ (𝑅 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑅)))) | |
5 | 4 | adantl 275 | . . . 4 ⊢ ((𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷) → (𝐷 ∥ (𝑁 − 𝑅) ↔ (𝑅 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑅)))) |
6 | 5 | 3ad2ant3 1004 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → (𝐷 ∥ (𝑁 − 𝑅) ↔ (𝑅 < 𝐷 ∧ 𝐷 ∥ (𝑁 − 𝑅)))) |
7 | nnz 9066 | . . . . . 6 ⊢ (𝐷 ∈ ℕ → 𝐷 ∈ ℤ) | |
8 | 7 | 3ad2ant2 1003 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → 𝐷 ∈ ℤ) |
9 | simp1 981 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → 𝑁 ∈ ℤ) | |
10 | nn0z 9067 | . . . . . . . 8 ⊢ (𝑅 ∈ ℕ0 → 𝑅 ∈ ℤ) | |
11 | 10 | adantr 274 | . . . . . . 7 ⊢ ((𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷) → 𝑅 ∈ ℤ) |
12 | 11 | 3ad2ant3 1004 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → 𝑅 ∈ ℤ) |
13 | 9, 12 | zsubcld 9171 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → (𝑁 − 𝑅) ∈ ℤ) |
14 | divides 11484 | . . . . 5 ⊢ ((𝐷 ∈ ℤ ∧ (𝑁 − 𝑅) ∈ ℤ) → (𝐷 ∥ (𝑁 − 𝑅) ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝐷) = (𝑁 − 𝑅))) | |
15 | 8, 13, 14 | syl2anc 408 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → (𝐷 ∥ (𝑁 − 𝑅) ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝐷) = (𝑁 − 𝑅))) |
16 | eqcom 2139 | . . . . . 6 ⊢ ((𝑧 · 𝐷) = (𝑁 − 𝑅) ↔ (𝑁 − 𝑅) = (𝑧 · 𝐷)) | |
17 | zcn 9052 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
18 | 17 | 3ad2ant1 1002 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → 𝑁 ∈ ℂ) |
19 | 18 | adantr 274 | . . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) ∧ 𝑧 ∈ ℤ) → 𝑁 ∈ ℂ) |
20 | nn0cn 8980 | . . . . . . . . . 10 ⊢ (𝑅 ∈ ℕ0 → 𝑅 ∈ ℂ) | |
21 | 20 | adantr 274 | . . . . . . . . 9 ⊢ ((𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷) → 𝑅 ∈ ℂ) |
22 | 21 | 3ad2ant3 1004 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → 𝑅 ∈ ℂ) |
23 | 22 | adantr 274 | . . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) ∧ 𝑧 ∈ ℤ) → 𝑅 ∈ ℂ) |
24 | simpr 109 | . . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) ∧ 𝑧 ∈ ℤ) → 𝑧 ∈ ℤ) | |
25 | 8 | adantr 274 | . . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) ∧ 𝑧 ∈ ℤ) → 𝐷 ∈ ℤ) |
26 | 24, 25 | zmulcld 9172 | . . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) ∧ 𝑧 ∈ ℤ) → (𝑧 · 𝐷) ∈ ℤ) |
27 | 26 | zcnd 9167 | . . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) ∧ 𝑧 ∈ ℤ) → (𝑧 · 𝐷) ∈ ℂ) |
28 | 19, 23, 27 | subadd2d 8085 | . . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) ∧ 𝑧 ∈ ℤ) → ((𝑁 − 𝑅) = (𝑧 · 𝐷) ↔ ((𝑧 · 𝐷) + 𝑅) = 𝑁)) |
29 | 16, 28 | syl5bb 191 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) ∧ 𝑧 ∈ ℤ) → ((𝑧 · 𝐷) = (𝑁 − 𝑅) ↔ ((𝑧 · 𝐷) + 𝑅) = 𝑁)) |
30 | 29 | rexbidva 2432 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → (∃𝑧 ∈ ℤ (𝑧 · 𝐷) = (𝑁 − 𝑅) ↔ ∃𝑧 ∈ ℤ ((𝑧 · 𝐷) + 𝑅) = 𝑁)) |
31 | 15, 30 | bitrd 187 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → (𝐷 ∥ (𝑁 − 𝑅) ↔ ∃𝑧 ∈ ℤ ((𝑧 · 𝐷) + 𝑅) = 𝑁)) |
32 | 3, 6, 31 | 3bitr2d 215 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → (𝑅 = (𝑁 mod 𝐷) ↔ ∃𝑧 ∈ ℤ ((𝑧 · 𝐷) + 𝑅) = 𝑁)) |
33 | 1, 32 | syl5bb 191 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ ∧ (𝑅 ∈ ℕ0 ∧ 𝑅 < 𝐷)) → ((𝑁 mod 𝐷) = 𝑅 ↔ ∃𝑧 ∈ ℤ ((𝑧 · 𝐷) + 𝑅) = 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ∃wrex 2415 class class class wbr 3924 (class class class)co 5767 ℂcc 7611 + caddc 7616 · cmul 7618 < clt 7793 − cmin 7926 ℕcn 8713 ℕ0cn0 8970 ℤcz 9047 mod cmo 10088 ∥ cdvds 11482 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-fl 10036 df-mod 10089 df-seqfrec 10212 df-exp 10286 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-dvds 11483 |
This theorem is referenced by: bezoutlemnewy 11673 bezoutlemstep 11674 |
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