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Theorem lcmval 15914
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 15822 and gcdval 15823. (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 2824 . . . 4 (𝑥 = 𝑀 → (𝑥 = 0 ↔ 𝑀 = 0))
21orbi1d 913 . . 3 (𝑥 = 𝑀 → ((𝑥 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑦 = 0)))
3 breq1 5045 . . . . . 6 (𝑥 = 𝑀 → (𝑥𝑛𝑀𝑛))
43anbi1d 631 . . . . 5 (𝑥 = 𝑀 → ((𝑥𝑛𝑦𝑛) ↔ (𝑀𝑛𝑦𝑛)))
54rabbidv 3459 . . . 4 (𝑥 = 𝑀 → {𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)})
65infeq1d 8919 . . 3 (𝑥 = 𝑀 → inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)}, ℝ, < ))
72, 6ifbieq2d 4468 . 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 2824 . . . 4 (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
98orbi2d 912 . . 3 (𝑦 = 𝑁 → ((𝑀 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑁 = 0)))
10 breq1 5045 . . . . . 6 (𝑦 = 𝑁 → (𝑦𝑛𝑁𝑛))
1110anbi2d 630 . . . . 5 (𝑦 = 𝑁 → ((𝑀𝑛𝑦𝑛) ↔ (𝑀𝑛𝑁𝑛)))
1211rabbidv 3459 . . . 4 (𝑦 = 𝑁 → {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)})
1312infeq1d 8919 . . 3 (𝑦 = 𝑁 → inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < ))
149, 13ifbieq2d 4468 . 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 15912 . 2 lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )))
16 c0ex 10613 . . 3 0 ∈ V
17 ltso 10699 . . . 4 < Or ℝ
1817infex 8935 . . 3 inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < ) ∈ V
1916, 18ifex 4491 . 2 if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )) ∈ V
207, 14, 15, 19ovmpo 7287 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 398  wo 843   = wceq 1537  wcel 2114  {crab 3129  ifcif 4443   class class class wbr 5042  (class class class)co 7133  infcinf 8883  cr 10514  0cc0 10515   < clt 10653  cn 11616  cz 11960  cdvds 15587   lcm clcm 15910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-mulcl 10577  ax-i2m1 10583  ax-pre-lttri 10589  ax-pre-lttrn 10590
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-po 5450  df-so 5451  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-er 8267  df-en 8488  df-dom 8489  df-sdom 8490  df-sup 8884  df-inf 8885  df-pnf 10655  df-mnf 10656  df-ltxr 10658  df-lcm 15912
This theorem is referenced by:  lcmcom  15915  lcm0val  15916  lcmn0val  15917  lcmass  15936  lcmfpr  15949
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