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Theorem lcmval 15507
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 15419 and gcdval 15420. (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 2764 . . . 4 (𝑥 = 𝑀 → (𝑥 = 0 ↔ 𝑀 = 0))
21orbi1d 741 . . 3 (𝑥 = 𝑀 → ((𝑥 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑦 = 0)))
3 breq1 4807 . . . . . 6 (𝑥 = 𝑀 → (𝑥𝑛𝑀𝑛))
43anbi1d 743 . . . . 5 (𝑥 = 𝑀 → ((𝑥𝑛𝑦𝑛) ↔ (𝑀𝑛𝑦𝑛)))
54rabbidv 3329 . . . 4 (𝑥 = 𝑀 → {𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)})
65infeq1d 8548 . . 3 (𝑥 = 𝑀 → inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)}, ℝ, < ))
72, 6ifbieq2d 4255 . 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 2764 . . . 4 (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
98orbi2d 740 . . 3 (𝑦 = 𝑁 → ((𝑀 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑁 = 0)))
10 breq1 4807 . . . . . 6 (𝑦 = 𝑁 → (𝑦𝑛𝑁𝑛))
1110anbi2d 742 . . . . 5 (𝑦 = 𝑁 → ((𝑀𝑛𝑦𝑛) ↔ (𝑀𝑛𝑁𝑛)))
1211rabbidv 3329 . . . 4 (𝑦 = 𝑁 → {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)})
1312infeq1d 8548 . . 3 (𝑦 = 𝑁 → inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < ))
149, 13ifbieq2d 4255 . 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 15505 . 2 lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )))
16 c0ex 10226 . . 3 0 ∈ V
17 ltso 10310 . . . 4 < Or ℝ
1817infex 8564 . . 3 inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < ) ∈ V
1916, 18ifex 4300 . 2 if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )) ∈ V
207, 14, 15, 19ovmpt2 6961 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  wo 382  wa 383   = wceq 1632  wcel 2139  {crab 3054  ifcif 4230   class class class wbr 4804  (class class class)co 6813  infcinf 8512  cr 10127  0cc0 10128   < clt 10266  cn 11212  cz 11569  cdvds 15182   lcm clcm 15503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-mulcl 10190  ax-i2m1 10196  ax-pre-lttri 10202  ax-pre-lttrn 10203
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-sup 8513  df-inf 8514  df-pnf 10268  df-mnf 10269  df-ltxr 10271  df-lcm 15505
This theorem is referenced by:  lcmcom  15508  lcm0val  15509  lcmn0val  15510  lcmass  15529  lcmfpr  15542
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