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Theorem lcmval 15143
 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 15055 and gcdval 15056. (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 2614 . . . 4 (𝑥 = 𝑀 → (𝑥 = 0 ↔ 𝑀 = 0))
21orbi1d 735 . . 3 (𝑥 = 𝑀 → ((𝑥 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑦 = 0)))
3 breq1 4586 . . . . . 6 (𝑥 = 𝑀 → (𝑥𝑛𝑀𝑛))
43anbi1d 737 . . . . 5 (𝑥 = 𝑀 → ((𝑥𝑛𝑦𝑛) ↔ (𝑀𝑛𝑦𝑛)))
54rabbidv 3164 . . . 4 (𝑥 = 𝑀 → {𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)})
65infeq1d 8266 . . 3 (𝑥 = 𝑀 → inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)}, ℝ, < ))
72, 6ifbieq2d 4061 . 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 2614 . . . 4 (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
98orbi2d 734 . . 3 (𝑦 = 𝑁 → ((𝑀 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑁 = 0)))
10 breq1 4586 . . . . . 6 (𝑦 = 𝑁 → (𝑦𝑛𝑁𝑛))
1110anbi2d 736 . . . . 5 (𝑦 = 𝑁 → ((𝑀𝑛𝑦𝑛) ↔ (𝑀𝑛𝑁𝑛)))
1211rabbidv 3164 . . . 4 (𝑦 = 𝑁 → {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)})
1312infeq1d 8266 . . 3 (𝑦 = 𝑁 → inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑦𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < ))
149, 13ifbieq2d 4061 . 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 15141 . 2 lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )))
16 c0ex 9913 . . 3 0 ∈ V
17 ltso 9997 . . . 4 < Or ℝ
1817infex 8282 . . 3 inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < ) ∈ V
1916, 18ifex 4106 . 2 if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀𝑛𝑁𝑛)}, ℝ, < )) ∈ V
207, 14, 15, 19ovmpt2 6694 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 1475   ∈ wcel 1977  {crab 2900  ifcif 4036   class class class wbr 4583  (class class class)co 6549  infcinf 8230  ℝcr 9814  0cc0 9815   < clt 9953  ℕcn 10897  ℤcz 11254   ∥ cdvds 14821   lcm clcm 15139 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-mulcl 9877  ax-i2m1 9883  ax-pre-lttri 9889  ax-pre-lttrn 9890 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-ltxr 9958  df-lcm 15141 This theorem is referenced by:  lcmcom  15144  lcm0val  15145  lcmn0val  15146  lcmass  15165  lcmfpr  15178
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