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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 12030 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0) | |
2 | 1 | adantr 276 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 = 0) → (𝑀 lcm 0) = 0) |
3 | oveq2 5873 | . . . . 5 ⊢ (𝑀 = 0 → (𝑀 lcm 𝑀) = (𝑀 lcm 0)) | |
4 | fveq2 5507 | . . . . . 6 ⊢ (𝑀 = 0 → (abs‘𝑀) = (abs‘0)) | |
5 | abs0 11033 | . . . . . 6 ⊢ (abs‘0) = 0 | |
6 | 4, 5 | eqtrdi 2224 | . . . . 5 ⊢ (𝑀 = 0 → (abs‘𝑀) = 0) |
7 | 3, 6 | eqeq12d 2190 | . . . 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 2346 | . . 3 ⊢ (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0) | |
11 | lcmcl 12037 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀 lcm 𝑀) ∈ ℕ0) | |
12 | 11 | nn0cnd 9202 | . . . . . 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 11061 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℤ) | |
16 | 15 | zcnd 9347 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℂ) |
17 | 16 | adantr 276 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℂ) |
18 | zcn 9229 | . . . . . . 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 11162 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ≠ 0) |
22 | 0zd 9236 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 0 ∈ ℤ) | |
23 | zapne 9298 | . . . . . 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 12043 | . . . . . . 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 11952 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 𝑀) = (abs‘𝑀)) | |
29 | 28 | oveq2d 5881 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → ((𝑀 lcm 𝑀) · (𝑀 gcd 𝑀)) = ((𝑀 lcm 𝑀) · (abs‘𝑀))) |
30 | 18, 18 | absmuld 11169 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (abs‘(𝑀 · 𝑀)) = ((abs‘𝑀) · (abs‘𝑀))) |
31 | 27, 29, 30 | 3eqtr3d 2216 | . . . . 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 8594 | . . 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 9235 | . . . 4 ⊢ 0 ∈ ℤ | |
36 | zdceq 9299 | . . . 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 2146 ≠ wne 2345 class class class wbr 3998 ‘cfv 5208 (class class class)co 5865 ℂcc 7784 0cc0 7786 · cmul 7791 # cap 8512 ℤcz 9224 abscabs 10972 gcd cgcd 11908 lcm clcm 12025 |
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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906 |
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 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-isom 5217 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-sup 6973 df-inf 6974 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8602 df-inn 8891 df-2 8949 df-3 8950 df-4 8951 df-n0 9148 df-z 9225 df-uz 9500 df-q 9591 df-rp 9623 df-fz 9978 df-fzo 10111 df-fl 10238 df-mod 10291 df-seqfrec 10414 df-exp 10488 df-cj 10817 df-re 10818 df-im 10819 df-rsqrt 10973 df-abs 10974 df-dvds 11761 df-gcd 11909 df-lcm 12026 |
This theorem is referenced by: lcmgcdeq 12048 |
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