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Mirrors > Home > ILE Home > Th. List > dvdslcm | GIF version |
Description: The lcm of two integers is divisible by each of them. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
Ref | Expression |
---|---|
dvdslcm | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvds0 11816 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∥ 0) | |
2 | 1 | ad2antrr 488 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∥ 0) |
3 | oveq1 5885 | . . . . . . 7 ⊢ (𝑀 = 0 → (𝑀 lcm 𝑁) = (0 lcm 𝑁)) | |
4 | 0z 9267 | . . . . . . . . 9 ⊢ 0 ∈ ℤ | |
5 | lcmcom 12067 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑁 lcm 0) = (0 lcm 𝑁)) | |
6 | 4, 5 | mpan2 425 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = (0 lcm 𝑁)) |
7 | lcm0val 12068 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = 0) | |
8 | 6, 7 | eqtr3d 2212 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (0 lcm 𝑁) = 0) |
9 | 3, 8 | sylan9eqr 2232 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = 0) |
10 | 9 | adantll 476 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = 0) |
11 | oveq2 5886 | . . . . . . 7 ⊢ (𝑁 = 0 → (𝑀 lcm 𝑁) = (𝑀 lcm 0)) | |
12 | lcm0val 12068 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0) | |
13 | 11, 12 | sylan9eqr 2232 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = 0) |
14 | 13 | adantlr 477 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = 0) |
15 | 10, 14 | jaodan 797 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) = 0) |
16 | 2, 15 | breqtrrd 4033 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∥ (𝑀 lcm 𝑁)) |
17 | dvds0 11816 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∥ 0) | |
18 | 17 | ad2antlr 489 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∥ 0) |
19 | 18, 15 | breqtrrd 4033 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∥ (𝑀 lcm 𝑁)) |
20 | 16, 19 | jca 306 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁))) |
21 | lcmcllem 12070 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}) | |
22 | lcmn0cl 12071 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ ℕ) | |
23 | breq2 4009 | . . . . . 6 ⊢ (𝑛 = (𝑀 lcm 𝑁) → (𝑀 ∥ 𝑛 ↔ 𝑀 ∥ (𝑀 lcm 𝑁))) | |
24 | breq2 4009 | . . . . . 6 ⊢ (𝑛 = (𝑀 lcm 𝑁) → (𝑁 ∥ 𝑛 ↔ 𝑁 ∥ (𝑀 lcm 𝑁))) | |
25 | 23, 24 | anbi12d 473 | . . . . 5 ⊢ (𝑛 = (𝑀 lcm 𝑁) → ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) ↔ (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))) |
26 | 25 | elrab3 2896 | . . . 4 ⊢ ((𝑀 lcm 𝑁) ∈ ℕ → ((𝑀 lcm 𝑁) ∈ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)} ↔ (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))) |
27 | 22, 26 | syl 14 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm 𝑁) ∈ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)} ↔ (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))) |
28 | 21, 27 | mpbid 147 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁))) |
29 | lcmmndc 12065 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∨ 𝑁 = 0)) | |
30 | exmiddc 836 | . . 3 ⊢ (DECID (𝑀 = 0 ∨ 𝑁 = 0) → ((𝑀 = 0 ∨ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∨ 𝑁 = 0))) | |
31 | 29, 30 | syl 14 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 = 0 ∨ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∨ 𝑁 = 0))) |
32 | 20, 28, 31 | mpjaodan 798 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 DECID wdc 834 = wceq 1353 ∈ wcel 2148 {crab 2459 class class class wbr 4005 (class class class)co 5878 0cc0 7814 ℕcn 8922 ℤcz 9256 ∥ cdvds 11797 lcm clcm 12063 |
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 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-mulrcl 7913 ax-addcom 7914 ax-mulcom 7915 ax-addass 7916 ax-mulass 7917 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-1rid 7921 ax-0id 7922 ax-rnegex 7923 ax-precex 7924 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 ax-pre-mulgt0 7931 ax-pre-mulext 7932 ax-arch 7933 ax-caucvg 7934 |
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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-recs 6309 df-frec 6395 df-sup 6986 df-inf 6987 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-reap 8535 df-ap 8542 df-div 8633 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-n0 9180 df-z 9257 df-uz 9532 df-q 9623 df-rp 9657 df-fz 10012 df-fzo 10146 df-fl 10273 df-mod 10326 df-seqfrec 10449 df-exp 10523 df-cj 10854 df-re 10855 df-im 10856 df-rsqrt 11010 df-abs 11011 df-dvds 11798 df-lcm 12064 |
This theorem is referenced by: gcddvdslcm 12076 lcmneg 12077 lcmgcdeq 12086 lcmdvdsb 12087 |
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