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Mirrors > Home > ILE Home > Th. List > coprmdvds | GIF version |
Description: Euclid's Lemma (see ProofWiki "Euclid's Lemma", 10-Jul-2021, https://proofwiki.org/wiki/Euclid's_Lemma): If an integer divides the product of two integers and is coprime to one of them, then it divides the other. See also theorem 1.5 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by AV, 10-Jul-2021.) |
Ref | Expression |
---|---|
coprmdvds | ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ (𝑀 · 𝑁) ∧ (𝐾 gcd 𝑀) = 1) → 𝐾 ∥ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 8745 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
2 | zcn 8745 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
3 | mulcom 7461 | . . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑀 · 𝑁) = (𝑁 · 𝑀)) | |
4 | 1, 2, 3 | syl2an 283 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) = (𝑁 · 𝑀)) |
5 | 4 | breq2d 3855 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ (𝑀 · 𝑁) ↔ 𝐾 ∥ (𝑁 · 𝑀))) |
6 | dvdsmulgcd 11279 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾 ∥ (𝑁 · 𝑀) ↔ 𝐾 ∥ (𝑁 · (𝑀 gcd 𝐾)))) | |
7 | 6 | ancoms 264 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ (𝑁 · 𝑀) ↔ 𝐾 ∥ (𝑁 · (𝑀 gcd 𝐾)))) |
8 | 5, 7 | bitrd 186 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ (𝑀 · 𝑁) ↔ 𝐾 ∥ (𝑁 · (𝑀 gcd 𝐾)))) |
9 | 8 | 3adant1 961 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ (𝑀 · 𝑁) ↔ 𝐾 ∥ (𝑁 · (𝑀 gcd 𝐾)))) |
10 | 9 | adantr 270 | . . . . . 6 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → (𝐾 ∥ (𝑀 · 𝑁) ↔ 𝐾 ∥ (𝑁 · (𝑀 gcd 𝐾)))) |
11 | gcdcom 11230 | . . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾 gcd 𝑀) = (𝑀 gcd 𝐾)) | |
12 | 11 | 3adant3 963 | . . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 gcd 𝑀) = (𝑀 gcd 𝐾)) |
13 | 12 | eqeq1d 2096 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 gcd 𝑀) = 1 ↔ (𝑀 gcd 𝐾) = 1)) |
14 | oveq2 5652 | . . . . . . . . . 10 ⊢ ((𝑀 gcd 𝐾) = 1 → (𝑁 · (𝑀 gcd 𝐾)) = (𝑁 · 1)) | |
15 | 13, 14 | syl6bi 161 | . . . . . . . . 9 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 gcd 𝑀) = 1 → (𝑁 · (𝑀 gcd 𝐾)) = (𝑁 · 1))) |
16 | 15 | imp 122 | . . . . . . . 8 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → (𝑁 · (𝑀 gcd 𝐾)) = (𝑁 · 1)) |
17 | 2 | mulid1d 7495 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (𝑁 · 1) = 𝑁) |
18 | 17 | 3ad2ant3 966 | . . . . . . . . 9 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 · 1) = 𝑁) |
19 | 18 | adantr 270 | . . . . . . . 8 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → (𝑁 · 1) = 𝑁) |
20 | 16, 19 | eqtrd 2120 | . . . . . . 7 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → (𝑁 · (𝑀 gcd 𝐾)) = 𝑁) |
21 | 20 | breq2d 3855 | . . . . . 6 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → (𝐾 ∥ (𝑁 · (𝑀 gcd 𝐾)) ↔ 𝐾 ∥ 𝑁)) |
22 | 10, 21 | bitrd 186 | . . . . 5 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → (𝐾 ∥ (𝑀 · 𝑁) ↔ 𝐾 ∥ 𝑁)) |
23 | 22 | biimpd 142 | . . . 4 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → (𝐾 ∥ (𝑀 · 𝑁) → 𝐾 ∥ 𝑁)) |
24 | 23 | ex 113 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 gcd 𝑀) = 1 → (𝐾 ∥ (𝑀 · 𝑁) → 𝐾 ∥ 𝑁))) |
25 | 24 | com23 77 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ (𝑀 · 𝑁) → ((𝐾 gcd 𝑀) = 1 → 𝐾 ∥ 𝑁))) |
26 | 25 | impd 251 | 1 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ (𝑀 · 𝑁) ∧ (𝐾 gcd 𝑀) = 1) → 𝐾 ∥ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∧ w3a 924 = wceq 1289 ∈ wcel 1438 class class class wbr 3843 (class class class)co 5644 ℂcc 7338 1c1 7341 · cmul 7345 ℤcz 8740 ∥ cdvds 11061 gcd cgcd 11203 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3952 ax-sep 3955 ax-nul 3963 ax-pow 4007 ax-pr 4034 ax-un 4258 ax-setind 4351 ax-iinf 4401 ax-cnex 7426 ax-resscn 7427 ax-1cn 7428 ax-1re 7429 ax-icn 7430 ax-addcl 7431 ax-addrcl 7432 ax-mulcl 7433 ax-mulrcl 7434 ax-addcom 7435 ax-mulcom 7436 ax-addass 7437 ax-mulass 7438 ax-distr 7439 ax-i2m1 7440 ax-0lt1 7441 ax-1rid 7442 ax-0id 7443 ax-rnegex 7444 ax-precex 7445 ax-cnre 7446 ax-pre-ltirr 7447 ax-pre-ltwlin 7448 ax-pre-lttrn 7449 ax-pre-apti 7450 ax-pre-ltadd 7451 ax-pre-mulgt0 7452 ax-pre-mulext 7453 ax-arch 7454 ax-caucvg 7455 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-if 3392 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-int 3687 df-iun 3730 df-br 3844 df-opab 3898 df-mpt 3899 df-tr 3935 df-id 4118 df-po 4121 df-iso 4122 df-iord 4191 df-on 4193 df-ilim 4194 df-suc 4196 df-iom 4404 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-rn 4447 df-res 4448 df-ima 4449 df-iota 4975 df-fun 5012 df-fn 5013 df-f 5014 df-f1 5015 df-fo 5016 df-f1o 5017 df-fv 5018 df-riota 5600 df-ov 5647 df-oprab 5648 df-mpt2 5649 df-1st 5903 df-2nd 5904 df-recs 6062 df-frec 6148 df-sup 6669 df-pnf 7514 df-mnf 7515 df-xr 7516 df-ltxr 7517 df-le 7518 df-sub 7645 df-neg 7646 df-reap 8042 df-ap 8049 df-div 8130 df-inn 8413 df-2 8471 df-3 8472 df-4 8473 df-n0 8664 df-z 8741 df-uz 9010 df-q 9095 df-rp 9125 df-fz 9415 df-fzo 9542 df-fl 9665 df-mod 9718 df-iseq 9841 df-seq3 9842 df-exp 9943 df-cj 10264 df-re 10265 df-im 10266 df-rsqrt 10419 df-abs 10420 df-dvds 11062 df-gcd 11204 |
This theorem is referenced by: coprmdvds2 11340 qredeq 11343 cncongr1 11350 euclemma 11390 |
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