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Mirrors > Home > ILE Home > Th. List > dvdstr | GIF version |
Description: The divides relation is transitive. Theorem 1.1(b) in [ApostolNT] p. 14 (transitive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvdstr | ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ 𝑀 ∧ 𝑀 ∥ 𝑁) → 𝐾 ∥ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 994 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) | |
2 | 3simpc 996 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) | |
3 | 3simpb 995 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ)) | |
4 | zmulcl 9282 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 · 𝑦) ∈ ℤ) | |
5 | 4 | adantl 277 | . 2 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑥 · 𝑦) ∈ ℤ) |
6 | oveq2 5876 | . . . . 5 ⊢ ((𝑥 · 𝐾) = 𝑀 → (𝑦 · (𝑥 · 𝐾)) = (𝑦 · 𝑀)) | |
7 | 6 | adantr 276 | . . . 4 ⊢ (((𝑥 · 𝐾) = 𝑀 ∧ (𝑦 · 𝑀) = 𝑁) → (𝑦 · (𝑥 · 𝐾)) = (𝑦 · 𝑀)) |
8 | eqeq2 2187 | . . . . 5 ⊢ ((𝑦 · 𝑀) = 𝑁 → ((𝑦 · (𝑥 · 𝐾)) = (𝑦 · 𝑀) ↔ (𝑦 · (𝑥 · 𝐾)) = 𝑁)) | |
9 | 8 | adantl 277 | . . . 4 ⊢ (((𝑥 · 𝐾) = 𝑀 ∧ (𝑦 · 𝑀) = 𝑁) → ((𝑦 · (𝑥 · 𝐾)) = (𝑦 · 𝑀) ↔ (𝑦 · (𝑥 · 𝐾)) = 𝑁)) |
10 | 7, 9 | mpbid 147 | . . 3 ⊢ (((𝑥 · 𝐾) = 𝑀 ∧ (𝑦 · 𝑀) = 𝑁) → (𝑦 · (𝑥 · 𝐾)) = 𝑁) |
11 | zcn 9234 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
12 | zcn 9234 | . . . . . . . 8 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
13 | zcn 9234 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ) | |
14 | mulass 7920 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑥 · 𝑦) · 𝐾) = (𝑥 · (𝑦 · 𝐾))) | |
15 | mul12 8063 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝐾 ∈ ℂ) → (𝑥 · (𝑦 · 𝐾)) = (𝑦 · (𝑥 · 𝐾))) | |
16 | 14, 15 | eqtrd 2210 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑥 · 𝑦) · 𝐾) = (𝑦 · (𝑥 · 𝐾))) |
17 | 11, 12, 13, 16 | syl3an 1280 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑥 · 𝑦) · 𝐾) = (𝑦 · (𝑥 · 𝐾))) |
18 | 17 | 3comr 1211 | . . . . . 6 ⊢ ((𝐾 ∈ ℤ ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 · 𝑦) · 𝐾) = (𝑦 · (𝑥 · 𝐾))) |
19 | 18 | 3expb 1204 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑥 · 𝑦) · 𝐾) = (𝑦 · (𝑥 · 𝐾))) |
20 | 19 | 3ad2antl1 1159 | . . . 4 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑥 · 𝑦) · 𝐾) = (𝑦 · (𝑥 · 𝐾))) |
21 | 20 | eqeq1d 2186 | . . 3 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 · 𝑦) · 𝐾) = 𝑁 ↔ (𝑦 · (𝑥 · 𝐾)) = 𝑁)) |
22 | 10, 21 | syl5ibr 156 | . 2 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 · 𝐾) = 𝑀 ∧ (𝑦 · 𝑀) = 𝑁) → ((𝑥 · 𝑦) · 𝐾) = 𝑁)) |
23 | 1, 2, 3, 5, 22 | dvds2lem 11781 | 1 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ 𝑀 ∧ 𝑀 ∥ 𝑁) → 𝐾 ∥ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 (class class class)co 5868 ℂcc 7787 · cmul 7794 ℤcz 9229 ∥ cdvds 11765 |
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-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 |
This theorem depends on definitions: df-bi 117 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-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-iota 5173 df-fun 5213 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-sub 8107 df-neg 8108 df-inn 8896 df-n0 9153 df-z 9230 df-dvds 11766 |
This theorem is referenced by: dvdstrd 11808 dvdsmultr1 11809 dvdsmultr2 11811 4dvdseven 11892 dvdsgcdb 11984 dvdsmulgcd 11996 gcddvdslcm 12043 lcmgcdeq 12053 lcmdvdsb 12054 mulgcddvds 12064 rpmulgcd2 12065 rpdvds 12069 exprmfct 12108 rpexp 12123 phimullem 12195 pcpremul 12263 pcdvdsb 12289 pcprmpw2 12302 |
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