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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 15940 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 11995 | . 2 ⊢ 0 ∈ ℤ | |
2 | lcmval 15939 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑀 lcm 0) = if((𝑀 = 0 ∨ 0 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 0 ∥ 𝑛)}, ℝ, < ))) | |
3 | eqid 2824 | . . . . 5 ⊢ 0 = 0 | |
4 | 3 | olci 862 | . . . 4 ⊢ (𝑀 = 0 ∨ 0 = 0) |
5 | 4 | iftruei 4477 | . . 3 ⊢ if((𝑀 = 0 ∨ 0 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 0 ∥ 𝑛)}, ℝ, < )) = 0 |
6 | 2, 5 | syl6eq 2875 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑀 lcm 0) = 0) |
7 | 1, 6 | mpan2 689 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀 lcm 0) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 {crab 3145 ifcif 4470 class class class wbr 5069 (class class class)co 7159 infcinf 8908 ℝcr 10539 0cc0 10540 < clt 10678 ℕcn 11641 ℤcz 11984 ∥ cdvds 15610 lcm clcm 15935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-i2m1 10608 ax-rnegex 10611 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-ltxr 10683 df-neg 10876 df-z 11985 df-lcm 15937 |
This theorem is referenced by: dvdslcm 15945 lcmeq0 15947 lcmcl 15948 lcmneg 15950 lcmgcd 15954 lcmdvds 15955 lcmid 15956 lcmftp 15983 lcmfunsnlem2 15987 |
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