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Mirrors > Home > ILE Home > Th. List > coprmdvds2 | GIF version |
Description: If an integer is divisible by two coprime integers, then it is divisible by their product. (Contributed by Mario Carneiro, 24-Feb-2014.) |
Ref | Expression |
---|---|
coprmdvds2 | ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 gcd 𝑁) = 1) → ((𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾) → (𝑀 · 𝑁) ∥ 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divides 11777 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 ∥ 𝐾 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝑁) = 𝐾)) | |
2 | 1 | 3adant1 1015 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 ∥ 𝐾 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝑁) = 𝐾)) |
3 | 2 | adantr 276 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 ∥ 𝐾 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝑁) = 𝐾)) |
4 | simprr 531 | . . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ ((𝑀 gcd 𝑁) = 1 ∧ 𝑥 ∈ ℤ)) → 𝑥 ∈ ℤ) | |
5 | simpl2 1001 | . . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ ((𝑀 gcd 𝑁) = 1 ∧ 𝑥 ∈ ℤ)) → 𝑁 ∈ ℤ) | |
6 | zcn 9244 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
7 | zcn 9244 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
8 | mulcom 7928 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑥 · 𝑁) = (𝑁 · 𝑥)) | |
9 | 6, 7, 8 | syl2an 289 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥 · 𝑁) = (𝑁 · 𝑥)) |
10 | 4, 5, 9 | syl2anc 411 | . . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ ((𝑀 gcd 𝑁) = 1 ∧ 𝑥 ∈ ℤ)) → (𝑥 · 𝑁) = (𝑁 · 𝑥)) |
11 | 10 | breq2d 4012 | . . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ ((𝑀 gcd 𝑁) = 1 ∧ 𝑥 ∈ ℤ)) → (𝑀 ∥ (𝑥 · 𝑁) ↔ 𝑀 ∥ (𝑁 · 𝑥))) |
12 | simprl 529 | . . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ ((𝑀 gcd 𝑁) = 1 ∧ 𝑥 ∈ ℤ)) → (𝑀 gcd 𝑁) = 1) | |
13 | simpl1 1000 | . . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ ((𝑀 gcd 𝑁) = 1 ∧ 𝑥 ∈ ℤ)) → 𝑀 ∈ ℤ) | |
14 | coprmdvds 12072 | . . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑥 ∈ ℤ) → ((𝑀 ∥ (𝑁 · 𝑥) ∧ (𝑀 gcd 𝑁) = 1) → 𝑀 ∥ 𝑥)) | |
15 | 13, 5, 4, 14 | syl3anc 1238 | . . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ ((𝑀 gcd 𝑁) = 1 ∧ 𝑥 ∈ ℤ)) → ((𝑀 ∥ (𝑁 · 𝑥) ∧ (𝑀 gcd 𝑁) = 1) → 𝑀 ∥ 𝑥)) |
16 | 12, 15 | mpan2d 428 | . . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ ((𝑀 gcd 𝑁) = 1 ∧ 𝑥 ∈ ℤ)) → (𝑀 ∥ (𝑁 · 𝑥) → 𝑀 ∥ 𝑥)) |
17 | 11, 16 | sylbid 150 | . . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ ((𝑀 gcd 𝑁) = 1 ∧ 𝑥 ∈ ℤ)) → (𝑀 ∥ (𝑥 · 𝑁) → 𝑀 ∥ 𝑥)) |
18 | dvdsmulc 11807 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑥 → (𝑀 · 𝑁) ∥ (𝑥 · 𝑁))) | |
19 | 13, 4, 5, 18 | syl3anc 1238 | . . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ ((𝑀 gcd 𝑁) = 1 ∧ 𝑥 ∈ ℤ)) → (𝑀 ∥ 𝑥 → (𝑀 · 𝑁) ∥ (𝑥 · 𝑁))) |
20 | 17, 19 | syld 45 | . . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ ((𝑀 gcd 𝑁) = 1 ∧ 𝑥 ∈ ℤ)) → (𝑀 ∥ (𝑥 · 𝑁) → (𝑀 · 𝑁) ∥ (𝑥 · 𝑁))) |
21 | breq2 4004 | . . . . . . . 8 ⊢ ((𝑥 · 𝑁) = 𝐾 → (𝑀 ∥ (𝑥 · 𝑁) ↔ 𝑀 ∥ 𝐾)) | |
22 | breq2 4004 | . . . . . . . 8 ⊢ ((𝑥 · 𝑁) = 𝐾 → ((𝑀 · 𝑁) ∥ (𝑥 · 𝑁) ↔ (𝑀 · 𝑁) ∥ 𝐾)) | |
23 | 21, 22 | imbi12d 234 | . . . . . . 7 ⊢ ((𝑥 · 𝑁) = 𝐾 → ((𝑀 ∥ (𝑥 · 𝑁) → (𝑀 · 𝑁) ∥ (𝑥 · 𝑁)) ↔ (𝑀 ∥ 𝐾 → (𝑀 · 𝑁) ∥ 𝐾))) |
24 | 20, 23 | syl5ibcom 155 | . . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ ((𝑀 gcd 𝑁) = 1 ∧ 𝑥 ∈ ℤ)) → ((𝑥 · 𝑁) = 𝐾 → (𝑀 ∥ 𝐾 → (𝑀 · 𝑁) ∥ 𝐾))) |
25 | 24 | anassrs 400 | . . . . 5 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 gcd 𝑁) = 1) ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝑁) = 𝐾 → (𝑀 ∥ 𝐾 → (𝑀 · 𝑁) ∥ 𝐾))) |
26 | 25 | rexlimdva 2594 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 gcd 𝑁) = 1) → (∃𝑥 ∈ ℤ (𝑥 · 𝑁) = 𝐾 → (𝑀 ∥ 𝐾 → (𝑀 · 𝑁) ∥ 𝐾))) |
27 | 3, 26 | sylbid 150 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 gcd 𝑁) = 1) → (𝑁 ∥ 𝐾 → (𝑀 ∥ 𝐾 → (𝑀 · 𝑁) ∥ 𝐾))) |
28 | 27 | com23 78 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 gcd 𝑁) = 1) → (𝑀 ∥ 𝐾 → (𝑁 ∥ 𝐾 → (𝑀 · 𝑁) ∥ 𝐾))) |
29 | 28 | impd 254 | 1 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 gcd 𝑁) = 1) → ((𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾) → (𝑀 · 𝑁) ∥ 𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 class class class wbr 4000 (class class class)co 5869 ℂcc 7797 1c1 7800 · cmul 7804 ℤcz 9239 ∥ cdvds 11775 gcd cgcd 11923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-mulrcl 7898 ax-addcom 7899 ax-mulcom 7900 ax-addass 7901 ax-mulass 7902 ax-distr 7903 ax-i2m1 7904 ax-0lt1 7905 ax-1rid 7906 ax-0id 7907 ax-rnegex 7908 ax-precex 7909 ax-cnre 7910 ax-pre-ltirr 7911 ax-pre-ltwlin 7912 ax-pre-lttrn 7913 ax-pre-apti 7914 ax-pre-ltadd 7915 ax-pre-mulgt0 7916 ax-pre-mulext 7917 ax-arch 7918 ax-caucvg 7919 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-sup 6977 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-sub 8117 df-neg 8118 df-reap 8519 df-ap 8526 df-div 8616 df-inn 8906 df-2 8964 df-3 8965 df-4 8966 df-n0 9163 df-z 9240 df-uz 9515 df-q 9606 df-rp 9638 df-fz 9993 df-fzo 10126 df-fl 10253 df-mod 10306 df-seqfrec 10429 df-exp 10503 df-cj 10832 df-re 10833 df-im 10834 df-rsqrt 10988 df-abs 10989 df-dvds 11776 df-gcd 11924 |
This theorem is referenced by: rpmulgcd2 12075 crth 12204 |
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