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Theorem lcmf0val 16655
Description: The value, by convention, of the least common multiple for a set containing 0 is 0. (Contributed by AV, 21-Apr-2020.) (Proof shortened by AV, 16-Sep-2020.)
Assertion
Ref Expression
lcmf0val ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → (lcm𝑍) = 0)

Proof of Theorem lcmf0val
Dummy variables 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lcmf 16624 . 2 lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )))
2 eleq2 2827 . . . 4 (𝑧 = 𝑍 → (0 ∈ 𝑧 ↔ 0 ∈ 𝑍))
3 raleq 3320 . . . . . 6 (𝑧 = 𝑍 → (∀𝑚𝑧 𝑚𝑛 ↔ ∀𝑚𝑍 𝑚𝑛))
43rabbidv 3440 . . . . 5 (𝑧 = 𝑍 → {𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛} = {𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛})
54infeq1d 9514 . . . 4 (𝑧 = 𝑍 → inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛}, ℝ, < ))
62, 5ifbieq2d 4556 . . 3 (𝑧 = 𝑍 → if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )) = if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛}, ℝ, < )))
7 iftrue 4536 . . . 4 (0 ∈ 𝑍 → if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛}, ℝ, < )) = 0)
87adantl 481 . . 3 ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛}, ℝ, < )) = 0)
96, 8sylan9eqr 2796 . 2 (((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) ∧ 𝑧 = 𝑍) → if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )) = 0)
10 zex 12619 . . . . . 6 ℤ ∈ V
1110ssex 5326 . . . . 5 (𝑍 ⊆ ℤ → 𝑍 ∈ V)
12 elpwg 4607 . . . . 5 (𝑍 ∈ V → (𝑍 ∈ 𝒫 ℤ ↔ 𝑍 ⊆ ℤ))
1311, 12syl 17 . . . 4 (𝑍 ⊆ ℤ → (𝑍 ∈ 𝒫 ℤ ↔ 𝑍 ⊆ ℤ))
1413ibir 268 . . 3 (𝑍 ⊆ ℤ → 𝑍 ∈ 𝒫 ℤ)
1514adantr 480 . 2 ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → 𝑍 ∈ 𝒫 ℤ)
16 simpr 484 . 2 ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → 0 ∈ 𝑍)
171, 9, 15, 16fvmptd2 7023 1 ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → (lcm𝑍) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  {crab 3432  Vcvv 3477  wss 3962  ifcif 4530  𝒫 cpw 4604   class class class wbr 5147  cfv 6562  infcinf 9478  cr 11151  0cc0 11152   < clt 11292  cn 12263  cz 12610  cdvds 16286  lcmclcmf 16622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-cnex 11208  ax-resscn 11209
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-sup 9479  df-inf 9480  df-neg 11492  df-z 12611  df-lcmf 16624
This theorem is referenced by:  lcmfcl  16661  lcmfeq0b  16663  dvdslcmf  16664  lcmftp  16669  lcmfunsnlem2  16673
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