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Mirrors > Home > ILE Home > Th. List > lcmid | GIF version |
Description: The lcm of an integer and itself is its absolute value. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
Ref | Expression |
---|---|
lcmid | ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 𝑀) = (abs‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcm0val 12048 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0) | |
2 | 1 | adantr 276 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 = 0) → (𝑀 lcm 0) = 0) |
3 | oveq2 5877 | . . . . 5 ⊢ (𝑀 = 0 → (𝑀 lcm 𝑀) = (𝑀 lcm 0)) | |
4 | fveq2 5511 | . . . . . 6 ⊢ (𝑀 = 0 → (abs‘𝑀) = (abs‘0)) | |
5 | abs0 11051 | . . . . . 6 ⊢ (abs‘0) = 0 | |
6 | 4, 5 | eqtrdi 2226 | . . . . 5 ⊢ (𝑀 = 0 → (abs‘𝑀) = 0) |
7 | 3, 6 | eqeq12d 2192 | . . . 4 ⊢ (𝑀 = 0 → ((𝑀 lcm 𝑀) = (abs‘𝑀) ↔ (𝑀 lcm 0) = 0)) |
8 | 7 | adantl 277 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 = 0) → ((𝑀 lcm 𝑀) = (abs‘𝑀) ↔ (𝑀 lcm 0) = 0)) |
9 | 2, 8 | mpbird 167 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 = 0) → (𝑀 lcm 𝑀) = (abs‘𝑀)) |
10 | df-ne 2348 | . . 3 ⊢ (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0) | |
11 | lcmcl 12055 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 lcm 𝑀) ∈ ℕ0) | |
12 | 11 | nn0cnd 9220 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 lcm 𝑀) ∈ ℂ) |
13 | 12 | anidms 397 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 𝑀) ∈ ℂ) |
14 | 13 | adantr 276 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀 lcm 𝑀) ∈ ℂ) |
15 | zabscl 11079 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℤ) | |
16 | 15 | zcnd 9365 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℂ) |
17 | 16 | adantr 276 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℂ) |
18 | zcn 9247 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
19 | 18 | adantr 276 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 𝑀 ∈ ℂ) |
20 | simpr 110 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 𝑀 ≠ 0) | |
21 | 19, 20 | absne0d 11180 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ≠ 0) |
22 | 0zd 9254 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 0 ∈ ℤ) | |
23 | zapne 9316 | . . . . . 6 ⊢ (((abs‘𝑀) ∈ ℤ ∧ 0 ∈ ℤ) → ((abs‘𝑀) # 0 ↔ (abs‘𝑀) ≠ 0)) | |
24 | 15, 22, 23 | syl2an2r 595 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → ((abs‘𝑀) # 0 ↔ (abs‘𝑀) ≠ 0)) |
25 | 21, 24 | mpbird 167 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) # 0) |
26 | lcmgcd 12061 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑀 lcm 𝑀) · (𝑀 gcd 𝑀)) = (abs‘(𝑀 · 𝑀))) | |
27 | 26 | anidms 397 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → ((𝑀 lcm 𝑀) · (𝑀 gcd 𝑀)) = (abs‘(𝑀 · 𝑀))) |
28 | gcdid 11970 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 𝑀) = (abs‘𝑀)) | |
29 | 28 | oveq2d 5885 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → ((𝑀 lcm 𝑀) · (𝑀 gcd 𝑀)) = ((𝑀 lcm 𝑀) · (abs‘𝑀))) |
30 | 18, 18 | absmuld 11187 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (abs‘(𝑀 · 𝑀)) = ((abs‘𝑀) · (abs‘𝑀))) |
31 | 27, 29, 30 | 3eqtr3d 2218 | . . . . 5 ⊢ (𝑀 ∈ ℤ → ((𝑀 lcm 𝑀) · (abs‘𝑀)) = ((abs‘𝑀) · (abs‘𝑀))) |
32 | 31 | adantr 276 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → ((𝑀 lcm 𝑀) · (abs‘𝑀)) = ((abs‘𝑀) · (abs‘𝑀))) |
33 | 14, 17, 17, 25, 32 | mulcanap2ad 8610 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀 lcm 𝑀) = (abs‘𝑀)) |
34 | 10, 33 | sylan2br 288 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ ¬ 𝑀 = 0) → (𝑀 lcm 𝑀) = (abs‘𝑀)) |
35 | 0z 9253 | . . . 4 ⊢ 0 ∈ ℤ | |
36 | zdceq 9317 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑀 = 0) | |
37 | 35, 36 | mpan2 425 | . . 3 ⊢ (𝑀 ∈ ℤ → DECID 𝑀 = 0) |
38 | exmiddc 836 | . . 3 ⊢ (DECID 𝑀 = 0 → (𝑀 = 0 ∨ ¬ 𝑀 = 0)) | |
39 | 37, 38 | syl 14 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑀 = 0 ∨ ¬ 𝑀 = 0)) |
40 | 9, 34, 39 | mpjaodan 798 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 𝑀) = (abs‘𝑀)) |
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 ≠ wne 2347 class class class wbr 4000 ‘cfv 5212 (class class class)co 5869 ℂcc 7800 0cc0 7802 · cmul 7807 # cap 8528 ℤcz 9242 abscabs 10990 gcd cgcd 11926 lcm clcm 12043 |
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 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922 |
This theorem depends on definitions: df-bi 117 df-stab 831 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-isom 5221 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-inf 6978 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-fz 9996 df-fzo 10129 df-fl 10256 df-mod 10309 df-seqfrec 10432 df-exp 10506 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-dvds 11779 df-gcd 11927 df-lcm 12044 |
This theorem is referenced by: lcmgcdeq 12066 |
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