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Mirrors > Home > MPE Home > Th. List > lcm0val | Structured version Visualization version GIF version |
Description: The value, by convention, of the lcm operator when either operand is 0. (Use lcmcom 15925 for a left-hand 0.) (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
Ref | Expression |
---|---|
lcm0val | ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 11980 | . 2 ⊢ 0 ∈ ℤ | |
2 | lcmval 15924 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑀 lcm 0) = if((𝑀 = 0 ∨ 0 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 0 ∥ 𝑛)}, ℝ, < ))) | |
3 | eqid 2818 | . . . . 5 ⊢ 0 = 0 | |
4 | 3 | olci 860 | . . . 4 ⊢ (𝑀 = 0 ∨ 0 = 0) |
5 | 4 | iftruei 4470 | . . 3 ⊢ if((𝑀 = 0 ∨ 0 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 0 ∥ 𝑛)}, ℝ, < )) = 0 |
6 | 2, 5 | syl6eq 2869 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑀 lcm 0) = 0) |
7 | 1, 6 | mpan2 687 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 {crab 3139 ifcif 4463 class class class wbr 5057 (class class class)co 7145 infcinf 8893 ℝcr 10524 0cc0 10525 < clt 10663 ℕcn 11626 ℤcz 11969 ∥ cdvds 15595 lcm clcm 15920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-i2m1 10593 ax-rnegex 10596 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-neg 10861 df-z 11970 df-lcm 15922 |
This theorem is referenced by: dvdslcm 15930 lcmeq0 15932 lcmcl 15933 lcmneg 15935 lcmgcd 15939 lcmdvds 15940 lcmid 15941 lcmftp 15968 lcmfunsnlem2 15972 |
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