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Theorem lcmcom 15925
Description: The lcm operator is commutative. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
Assertion
Ref Expression
lcmcom ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = (𝑁 lcm 𝑀))

Proof of Theorem lcmcom
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 orcom 864 . . 3 ((𝑀 = 0 ∨ 𝑁 = 0) ↔ (𝑁 = 0 ∨ 𝑀 = 0))
2 ancom 461 . . . . 5 ((𝑀𝑛𝑁𝑛) ↔ (𝑁𝑛𝑀𝑛))
32rabbii 3471 . . . 4 {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)} = {𝑛 ∈ ℕ ∣ (𝑁𝑛𝑀𝑛)}
43infeq1i 8930 . . 3 inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑁𝑛𝑀𝑛)}, ℝ, < )
51, 4ifbieq2i 4487 . 2 if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )) = if((𝑁 = 0 ∨ 𝑀 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑁𝑛𝑀𝑛)}, ℝ, < ))
6 lcmval 15924 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )))
7 lcmval 15924 . . 3 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 lcm 𝑀) = if((𝑁 = 0 ∨ 𝑀 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑁𝑛𝑀𝑛)}, ℝ, < )))
87ancoms 459 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 lcm 𝑀) = if((𝑁 = 0 ∨ 𝑀 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑁𝑛𝑀𝑛)}, ℝ, < )))
95, 6, 83eqtr4a 2879 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = (𝑁 lcm 𝑀))
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  cr 10524  0cc0 10525   < clt 10663  cn 11626  cz 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-mulcl 10587  ax-i2m1 10593  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-lcm 15922
This theorem is referenced by:  dvdslcm  15930  lcmeq0  15932  lcmcl  15933  lcmneg  15935  neglcm  15936  lcmgcd  15939  lcmdvds  15940  lcmftp  15968  lcmfunsnlem2  15972  lcmfunsnlem  15973  lcmf2a3a4e12  15979
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