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Theorem lcmval 15685
 Description: Value of the lcm operator. (𝑀 lcm 𝑁) is the least common multiple of 𝑀 and 𝑁. If either 𝑀 or 𝑁 is 0, the result is defined conventionally as 0. Contrast with df-gcd 15597 and gcdval 15598. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
Assertion
Ref Expression
lcmval ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )))
Distinct variable groups:   𝑛,𝑀   𝑛,𝑁

Proof of Theorem lcmval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2829 . . . 4 (𝑥 = 𝑀 → (𝑥 = 0 ↔ 𝑀 = 0))
21orbi1d 945 . . 3 (𝑥 = 𝑀 → ((𝑥 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑦 = 0)))
3 breq1 4878 . . . . . 6 (𝑥 = 𝑀 → (𝑥𝑛𝑀𝑛))
43anbi1d 623 . . . . 5 (𝑥 = 𝑀 → ((𝑥𝑛𝑦𝑛) ↔ (𝑀𝑛𝑦𝑛)))
54rabbidv 3402 . . . 4 (𝑥 = 𝑀 → {𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)})
65infeq1d 8658 . . 3 (𝑥 = 𝑀 → inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)}, ℝ, < ))
72, 6ifbieq2d 4333 . 2 (𝑥 = 𝑀 → if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)}, ℝ, < )))
8 eqeq1 2829 . . . 4 (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
98orbi2d 944 . . 3 (𝑦 = 𝑁 → ((𝑀 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑁 = 0)))
10 breq1 4878 . . . . . 6 (𝑦 = 𝑁 → (𝑦𝑛𝑁𝑛))
1110anbi2d 622 . . . . 5 (𝑦 = 𝑁 → ((𝑀𝑛𝑦𝑛) ↔ (𝑀𝑛𝑁𝑛)))
1211rabbidv 3402 . . . 4 (𝑦 = 𝑁 → {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)})
1312infeq1d 8658 . . 3 (𝑦 = 𝑁 → inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < ))
149, 13ifbieq2d 4333 . 2 (𝑦 = 𝑁 → if((𝑀 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )))
15 df-lcm 15683 . 2 lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )))
16 c0ex 10357 . . 3 0 ∈ V
17 ltso 10444 . . . 4 < Or ℝ
1817infex 8674 . . 3 inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < ) ∈ V
1916, 18ifex 4356 . 2 if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )) ∈ V
207, 14, 15, 19ovmpt2 7061 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∨ wo 878   = wceq 1656   ∈ wcel 2164  {crab 3121  ifcif 4308   class class class wbr 4875  (class class class)co 6910  infcinf 8622  ℝcr 10258  0cc0 10259   < clt 10398  ℕcn 11357  ℤcz 11711   ∥ cdvds 15364   lcm clcm 15681 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-mulcl 10321  ax-i2m1 10327  ax-pre-lttri 10333  ax-pre-lttrn 10334 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-po 5265  df-so 5266  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-sup 8623  df-inf 8624  df-pnf 10400  df-mnf 10401  df-ltxr 10403  df-lcm 15683 This theorem is referenced by:  lcmcom  15686  lcm0val  15687  lcmn0val  15688  lcmass  15707  lcmfpr  15720
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