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Mirrors > Home > ILE Home > Th. List > lcmeq0 | GIF version |
Description: The lcm of two integers is zero iff either is zero. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
Ref | Expression |
---|---|
lcmeq0 | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = 0 ↔ (𝑀 = 0 ∨ 𝑁 = 0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcmmndc 11973 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∨ 𝑁 = 0)) | |
2 | lcmn0cl 11979 | . . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ ℕ) | |
3 | 2 | nnne0d 8893 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ≠ 0) |
4 | 3 | neneqd 2355 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 lcm 𝑁) = 0) |
5 | 4 | ex 114 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∨ 𝑁 = 0) → ¬ (𝑀 lcm 𝑁) = 0)) |
6 | condc 843 | . . 3 ⊢ (DECID (𝑀 = 0 ∨ 𝑁 = 0) → ((¬ (𝑀 = 0 ∨ 𝑁 = 0) → ¬ (𝑀 lcm 𝑁) = 0) → ((𝑀 lcm 𝑁) = 0 → (𝑀 = 0 ∨ 𝑁 = 0)))) | |
7 | 1, 5, 6 | sylc 62 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = 0 → (𝑀 = 0 ∨ 𝑁 = 0))) |
8 | oveq1 5843 | . . . . . 6 ⊢ (𝑀 = 0 → (𝑀 lcm 𝑁) = (0 lcm 𝑁)) | |
9 | 0z 9193 | . . . . . . . 8 ⊢ 0 ∈ ℤ | |
10 | lcmcom 11975 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑁 lcm 0) = (0 lcm 𝑁)) | |
11 | 9, 10 | mpan2 422 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = (0 lcm 𝑁)) |
12 | lcm0val 11976 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = 0) | |
13 | 11, 12 | eqtr3d 2199 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (0 lcm 𝑁) = 0) |
14 | 8, 13 | sylan9eqr 2219 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = 0) |
15 | 14 | adantll 468 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = 0) |
16 | oveq2 5844 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑀 lcm 𝑁) = (𝑀 lcm 0)) | |
17 | lcm0val 11976 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0) | |
18 | 16, 17 | sylan9eqr 2219 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = 0) |
19 | 18 | adantlr 469 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = 0) |
20 | 15, 19 | jaodan 787 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) = 0) |
21 | 20 | ex 114 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 = 0 ∨ 𝑁 = 0) → (𝑀 lcm 𝑁) = 0)) |
22 | 7, 21 | impbid 128 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = 0 ↔ (𝑀 = 0 ∨ 𝑁 = 0))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 DECID wdc 824 = wceq 1342 ∈ wcel 2135 (class class class)co 5836 0cc0 7744 ℤcz 9182 lcm clcm 11971 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-sup 6940 df-inf 6941 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fzo 10068 df-fl 10195 df-mod 10248 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-dvds 11714 df-lcm 11972 |
This theorem is referenced by: lcmass 11996 |
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