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Mirrors > Home > ILE Home > Th. List > lcmcl | GIF version |
Description: Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
Ref | Expression |
---|---|
lcmcl | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcmcom 12096 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = (𝑁 lcm 𝑀)) | |
2 | 1 | adantr 276 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = (𝑁 lcm 𝑀)) |
3 | oveq2 5904 | . . . . . . 7 ⊢ (𝑀 = 0 → (𝑁 lcm 𝑀) = (𝑁 lcm 0)) | |
4 | lcm0val 12097 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = 0) | |
5 | 3, 4 | sylan9eqr 2244 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 = 0) → (𝑁 lcm 𝑀) = 0) |
6 | 5 | adantll 476 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑁 lcm 𝑀) = 0) |
7 | 2, 6 | eqtrd 2222 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = 0) |
8 | oveq2 5904 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑀 lcm 𝑁) = (𝑀 lcm 0)) | |
9 | lcm0val 12097 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0) | |
10 | 8, 9 | sylan9eqr 2244 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = 0) |
11 | 10 | adantlr 477 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = 0) |
12 | 7, 11 | jaodan 798 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) = 0) |
13 | 0nn0 9221 | . . 3 ⊢ 0 ∈ ℕ0 | |
14 | 12, 13 | eqeltrdi 2280 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ ℕ0) |
15 | lcmn0cl 12100 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ ℕ) | |
16 | 15 | nnnn0d 9259 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ ℕ0) |
17 | lcmmndc 12094 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∨ 𝑁 = 0)) | |
18 | exmiddc 837 | . . 3 ⊢ (DECID (𝑀 = 0 ∨ 𝑁 = 0) → ((𝑀 = 0 ∨ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∨ 𝑁 = 0))) | |
19 | 17, 18 | syl 14 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 = 0 ∨ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∨ 𝑁 = 0))) |
20 | 14, 16, 19 | mpjaodan 799 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 709 DECID wdc 835 = wceq 1364 ∈ wcel 2160 (class class class)co 5896 0cc0 7841 ℕ0cn0 9206 ℤcz 9283 lcm clcm 12092 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 ax-pre-mulext 7959 ax-arch 7960 ax-caucvg 7961 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-frec 6416 df-sup 7013 df-inf 7014 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-reap 8562 df-ap 8569 df-div 8660 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-n0 9207 df-z 9284 df-uz 9559 df-q 9650 df-rp 9684 df-fz 10039 df-fzo 10173 df-fl 10301 df-mod 10354 df-seqfrec 10477 df-exp 10551 df-cj 10883 df-re 10884 df-im 10885 df-rsqrt 11039 df-abs 11040 df-dvds 11827 df-lcm 12093 |
This theorem is referenced by: gcddvdslcm 12105 lcmneg 12106 lcmdvds 12111 lcmid 12112 lcm1 12113 lcmgcdeq 12115 lcmdvdsb 12116 lcmass 12117 3lcm2e6woprm 12118 6lcm4e12 12119 3lcm2e6 12192 |
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