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Mirrors > Home > MPE Home > Th. List > zdiv | Structured version Visualization version GIF version |
Description: Two ways to express "𝑀 divides 𝑁. (Contributed by NM, 3-Oct-2008.) |
Ref | Expression |
---|---|
zdiv | ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnne0 11385 | . . 3 ⊢ (𝑀 ∈ ℕ → 𝑀 ≠ 0) | |
2 | 1 | adantr 474 | . 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → 𝑀 ≠ 0) |
3 | nncn 11358 | . . 3 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ) | |
4 | zcn 11708 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
5 | zcn 11708 | . . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℂ) | |
6 | divcan3 11035 | . . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑀 ≠ 0) → ((𝑀 · 𝑘) / 𝑀) = 𝑘) | |
7 | 6 | 3coml 1163 | . . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℂ ∧ 𝑀 ≠ 0 ∧ 𝑘 ∈ ℂ) → ((𝑀 · 𝑘) / 𝑀) = 𝑘) |
8 | 7 | 3expa 1153 | . . . . . . . . . 10 ⊢ (((𝑀 ∈ ℂ ∧ 𝑀 ≠ 0) ∧ 𝑘 ∈ ℂ) → ((𝑀 · 𝑘) / 𝑀) = 𝑘) |
9 | 5, 8 | sylan2 588 | . . . . . . . . 9 ⊢ (((𝑀 ∈ ℂ ∧ 𝑀 ≠ 0) ∧ 𝑘 ∈ ℤ) → ((𝑀 · 𝑘) / 𝑀) = 𝑘) |
10 | 9 | 3adantl2 1214 | . . . . . . . 8 ⊢ (((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝑀 ≠ 0) ∧ 𝑘 ∈ ℤ) → ((𝑀 · 𝑘) / 𝑀) = 𝑘) |
11 | oveq1 6911 | . . . . . . . 8 ⊢ ((𝑀 · 𝑘) = 𝑁 → ((𝑀 · 𝑘) / 𝑀) = (𝑁 / 𝑀)) | |
12 | 10, 11 | sylan9req 2881 | . . . . . . 7 ⊢ ((((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝑀 ≠ 0) ∧ 𝑘 ∈ ℤ) ∧ (𝑀 · 𝑘) = 𝑁) → 𝑘 = (𝑁 / 𝑀)) |
13 | simplr 787 | . . . . . . 7 ⊢ ((((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝑀 ≠ 0) ∧ 𝑘 ∈ ℤ) ∧ (𝑀 · 𝑘) = 𝑁) → 𝑘 ∈ ℤ) | |
14 | 12, 13 | eqeltrrd 2906 | . . . . . 6 ⊢ ((((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝑀 ≠ 0) ∧ 𝑘 ∈ ℤ) ∧ (𝑀 · 𝑘) = 𝑁) → (𝑁 / 𝑀) ∈ ℤ) |
15 | 14 | rexlimdva2 3242 | . . . . 5 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝑀 ≠ 0) → (∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁 → (𝑁 / 𝑀) ∈ ℤ)) |
16 | divcan2 11017 | . . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑀 ≠ 0) → (𝑀 · (𝑁 / 𝑀)) = 𝑁) | |
17 | 16 | 3com12 1159 | . . . . . 6 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝑀 ≠ 0) → (𝑀 · (𝑁 / 𝑀)) = 𝑁) |
18 | oveq2 6912 | . . . . . . . . 9 ⊢ (𝑘 = (𝑁 / 𝑀) → (𝑀 · 𝑘) = (𝑀 · (𝑁 / 𝑀))) | |
19 | 18 | eqeq1d 2826 | . . . . . . . 8 ⊢ (𝑘 = (𝑁 / 𝑀) → ((𝑀 · 𝑘) = 𝑁 ↔ (𝑀 · (𝑁 / 𝑀)) = 𝑁)) |
20 | 19 | rspcev 3525 | . . . . . . 7 ⊢ (((𝑁 / 𝑀) ∈ ℤ ∧ (𝑀 · (𝑁 / 𝑀)) = 𝑁) → ∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁) |
21 | 20 | expcom 404 | . . . . . 6 ⊢ ((𝑀 · (𝑁 / 𝑀)) = 𝑁 → ((𝑁 / 𝑀) ∈ ℤ → ∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁)) |
22 | 17, 21 | syl 17 | . . . . 5 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝑀 ≠ 0) → ((𝑁 / 𝑀) ∈ ℤ → ∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁)) |
23 | 15, 22 | impbid 204 | . . . 4 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝑀 ≠ 0) → (∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ)) |
24 | 23 | 3expia 1156 | . . 3 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑀 ≠ 0 → (∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ))) |
25 | 3, 4, 24 | syl2an 591 | . 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 ≠ 0 → (∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ))) |
26 | 2, 25 | mpd 15 | 1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ≠ wne 2998 ∃wrex 3117 (class class class)co 6904 ℂcc 10249 0cc0 10251 · cmul 10256 / cdiv 11008 ℕcn 11349 ℤcz 11703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-z 11704 |
This theorem is referenced by: addmodlteq 13039 fmtnoprmfac2lem1 42307 |
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