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Mirrors > Home > ILE Home > Th. List > lcmabs | GIF version |
Description: The lcm of two integers is the same as that of their absolute values. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
Ref | Expression |
---|---|
lcmabs | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zq 9418 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℚ) | |
2 | zq 9418 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℚ) | |
3 | qabsor 10847 | . . . 4 ⊢ (𝑀 ∈ ℚ → ((abs‘𝑀) = 𝑀 ∨ (abs‘𝑀) = -𝑀)) | |
4 | qabsor 10847 | . . . 4 ⊢ (𝑁 ∈ ℚ → ((abs‘𝑁) = 𝑁 ∨ (abs‘𝑁) = -𝑁)) | |
5 | 3, 4 | anim12i 336 | . . 3 ⊢ ((𝑀 ∈ ℚ ∧ 𝑁 ∈ ℚ) → (((abs‘𝑀) = 𝑀 ∨ (abs‘𝑀) = -𝑀) ∧ ((abs‘𝑁) = 𝑁 ∨ (abs‘𝑁) = -𝑁))) |
6 | 1, 2, 5 | syl2an 287 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) = 𝑀 ∨ (abs‘𝑀) = -𝑀) ∧ ((abs‘𝑁) = 𝑁 ∨ (abs‘𝑁) = -𝑁))) |
7 | oveq12 5783 | . . . 4 ⊢ (((abs‘𝑀) = 𝑀 ∧ (abs‘𝑁) = 𝑁) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁)) | |
8 | 7 | a1i 9 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) = 𝑀 ∧ (abs‘𝑁) = 𝑁) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁))) |
9 | oveq12 5783 | . . . . 5 ⊢ (((abs‘𝑀) = -𝑀 ∧ (abs‘𝑁) = 𝑁) → ((abs‘𝑀) lcm (abs‘𝑁)) = (-𝑀 lcm 𝑁)) | |
10 | neglcm 11756 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-𝑀 lcm 𝑁) = (𝑀 lcm 𝑁)) | |
11 | 9, 10 | sylan9eqr 2194 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((abs‘𝑀) = -𝑀 ∧ (abs‘𝑁) = 𝑁)) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁)) |
12 | 11 | ex 114 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) = -𝑀 ∧ (abs‘𝑁) = 𝑁) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁))) |
13 | oveq12 5783 | . . . . 5 ⊢ (((abs‘𝑀) = 𝑀 ∧ (abs‘𝑁) = -𝑁) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm -𝑁)) | |
14 | lcmneg 11755 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (𝑀 lcm 𝑁)) | |
15 | 13, 14 | sylan9eqr 2194 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((abs‘𝑀) = 𝑀 ∧ (abs‘𝑁) = -𝑁)) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁)) |
16 | 15 | ex 114 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) = 𝑀 ∧ (abs‘𝑁) = -𝑁) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁))) |
17 | oveq12 5783 | . . . . 5 ⊢ (((abs‘𝑀) = -𝑀 ∧ (abs‘𝑁) = -𝑁) → ((abs‘𝑀) lcm (abs‘𝑁)) = (-𝑀 lcm -𝑁)) | |
18 | znegcl 9085 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → -𝑀 ∈ ℤ) | |
19 | lcmneg 11755 | . . . . . . 7 ⊢ ((-𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-𝑀 lcm -𝑁) = (-𝑀 lcm 𝑁)) | |
20 | 18, 19 | sylan 281 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-𝑀 lcm -𝑁) = (-𝑀 lcm 𝑁)) |
21 | 20, 10 | eqtrd 2172 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-𝑀 lcm -𝑁) = (𝑀 lcm 𝑁)) |
22 | 17, 21 | sylan9eqr 2194 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((abs‘𝑀) = -𝑀 ∧ (abs‘𝑁) = -𝑁)) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁)) |
23 | 22 | ex 114 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) = -𝑀 ∧ (abs‘𝑁) = -𝑁) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁))) |
24 | 8, 12, 16, 23 | ccased 949 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((((abs‘𝑀) = 𝑀 ∨ (abs‘𝑀) = -𝑀) ∧ ((abs‘𝑁) = 𝑁 ∨ (abs‘𝑁) = -𝑁)) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁))) |
25 | 6, 24 | mpd 13 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) lcm (abs‘𝑁)) = (𝑀 lcm 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ‘cfv 5123 (class class class)co 5774 -cneg 7934 ℤcz 9054 ℚcq 9411 abscabs 10769 lcm clcm 11741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-fz 9791 df-fzo 9920 df-fl 10043 df-mod 10096 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-dvds 11494 df-lcm 11742 |
This theorem is referenced by: lcmgcd 11759 lcmdvds 11760 lcmgcdeq 11764 |
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