Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 16404 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 12444 | . 2 ⊢ 0 ∈ ℤ | |
2 | lcmval 16403 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑀 lcm 0) = if((𝑀 = 0 ∨ 0 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 0 ∥ 𝑛)}, ℝ, < ))) | |
3 | eqid 2738 | . . . . 5 ⊢ 0 = 0 | |
4 | 3 | olci 865 | . . . 4 ⊢ (𝑀 = 0 ∨ 0 = 0) |
5 | 4 | iftruei 4492 | . . 3 ⊢ if((𝑀 = 0 ∨ 0 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 0 ∥ 𝑛)}, ℝ, < )) = 0 |
6 | 2, 5 | eqtrdi 2794 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑀 lcm 0) = 0) |
7 | 1, 6 | mpan2 690 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 {crab 3406 ifcif 4485 class class class wbr 5104 (class class class)co 7350 infcinf 9311 ℝcr 10984 0cc0 10985 < clt 11123 ℕcn 12087 ℤcz 12433 ∥ cdvds 16071 lcm clcm 16399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-i2m1 11053 ax-rnegex 11056 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7353 df-oprab 7354 df-mpo 7355 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-sup 9312 df-inf 9313 df-pnf 11125 df-mnf 11126 df-ltxr 11128 df-neg 11322 df-z 12434 df-lcm 16401 |
This theorem is referenced by: dvdslcm 16409 lcmeq0 16411 lcmcl 16412 lcmneg 16414 lcmgcd 16418 lcmdvds 16419 lcmid 16420 lcmftp 16447 lcmfunsnlem2 16451 |
Copyright terms: Public domain | W3C validator |