Theorem lcmval 16278

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 16183 and gcdval 16184. (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 ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, 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.

Step	Hyp	Ref	Expression
1		eqeq1 2743	. . . 4 ⊢ (𝑥 = 𝑀 → (𝑥 = 0 ↔ 𝑀 = 0))
2	1	orbi1d 913	. . 3 ⊢ (𝑥 = 𝑀 → ((𝑥 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑦 = 0)))
3		breq1 5081	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∥ 𝑛 ↔ 𝑀 ∥ 𝑛))
4	3	anbi1d 629	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛) ↔ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)))
5	4	rabbidv 3412	. . . 4 ⊢ (𝑥 = 𝑀 → {𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)})
6	5	infeq1d 9197	. . 3 ⊢ (𝑥 = 𝑀 → inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ))
7	2, 6	ifbieq2d 4490	. 2 ⊢ (𝑥 = 𝑀 → if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )))
8		eqeq1 2743	. . . 4 ⊢ (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
9	8	orbi2d 912	. . 3 ⊢ (𝑦 = 𝑁 → ((𝑀 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑁 = 0)))
10		breq1 5081	. . . . . 6 ⊢ (𝑦 = 𝑁 → (𝑦 ∥ 𝑛 ↔ 𝑁 ∥ 𝑛))
11	10	anbi2d 628	. . . . 5 ⊢ (𝑦 = 𝑁 → ((𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛) ↔ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)))
12	11	rabbidv 3412	. . . 4 ⊢ (𝑦 = 𝑁 → {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)})
13	12	infeq1d 9197	. . 3 ⊢ (𝑦 = 𝑁 → inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ))
14	9, 13	ifbieq2d 4490	. 2 ⊢ (𝑦 = 𝑁 → if((𝑀 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )))
15		df-lcm 16276	. 2 ⊢ lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )))
16		c0ex 10953	. . 3 ⊢ 0 ∈ V
17		ltso 11039	. . . 4 ⊢ < Or ℝ
18	17	infex 9213	. . 3 ⊢ inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ) ∈ V
19	16, 18	ifex 4514	. 2 ⊢ if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )) ∈ V
20	7, 14, 15, 19	ovmpo 7424	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1541   ∈ wcel 2109  {crab 3069  ifcif 4464   class class class wbr 5078  (class class class)co 7268  infcinf 9161  ℝcr 10854  0cc0 10855   < clt 10993  ℕcn 11956  ℤcz 12302   ∥ cdvds 15944   lcm clcm 16274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-mulcl 10917  ax-i2m1 10923  ax-pre-lttri 10929  ax-pre-lttrn 10930

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-po 5502  df-so 5503  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-er 8472  df-en 8708  df-dom 8709  df-sdom 8710  df-sup 9162  df-inf 9163  df-pnf 10995  df-mnf 10996  df-ltxr 10998  df-lcm 16276

This theorem is referenced by:  lcmcom  16279  lcm0val  16280  lcmn0val  16281  lcmass  16300  lcmfpr  16313