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Theorem lcmf0val 15316
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 15285 . . 3 lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )))
21a1i 11 . 2 ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < ))))
3 eleq2 2688 . . . 4 (𝑧 = 𝑍 → (0 ∈ 𝑧 ↔ 0 ∈ 𝑍))
4 raleq 3133 . . . . . 6 (𝑧 = 𝑍 → (∀𝑚𝑧 𝑚𝑛 ↔ ∀𝑚𝑍 𝑚𝑛))
54rabbidv 3184 . . . . 5 (𝑧 = 𝑍 → {𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛} = {𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛})
65infeq1d 8368 . . . 4 (𝑧 = 𝑍 → inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛}, ℝ, < ))
73, 6ifbieq2d 4102 . . 3 (𝑧 = 𝑍 → if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )) = if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛}, ℝ, < )))
8 iftrue 4083 . . . 4 (0 ∈ 𝑍 → if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛}, ℝ, < )) = 0)
98adantl 482 . . 3 ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛}, ℝ, < )) = 0)
107, 9sylan9eqr 2676 . 2 (((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) ∧ 𝑧 = 𝑍) → if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )) = 0)
11 zex 11371 . . . . . 6 ℤ ∈ V
1211ssex 4793 . . . . 5 (𝑍 ⊆ ℤ → 𝑍 ∈ V)
13 elpwg 4157 . . . . 5 (𝑍 ∈ V → (𝑍 ∈ 𝒫 ℤ ↔ 𝑍 ⊆ ℤ))
1412, 13syl 17 . . . 4 (𝑍 ⊆ ℤ → (𝑍 ∈ 𝒫 ℤ ↔ 𝑍 ⊆ ℤ))
1514ibir 257 . . 3 (𝑍 ⊆ ℤ → 𝑍 ∈ 𝒫 ℤ)
1615adantr 481 . 2 ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → 𝑍 ∈ 𝒫 ℤ)
17 simpr 477 . 2 ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → 0 ∈ 𝑍)
182, 10, 16, 17fvmptd 6275 1 ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → (lcm𝑍) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  {crab 2913  Vcvv 3195  wss 3567  ifcif 4077  𝒫 cpw 4149   class class class wbr 4644  cmpt 4720  cfv 5876  infcinf 8332  cr 9920  0cc0 9921   < clt 10059  cn 11005  cz 11362  cdvds 14964  lcmclcmf 15283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-cnex 9977  ax-resscn 9978
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-sup 8333  df-inf 8334  df-neg 10254  df-z 11363  df-lcmf 15285
This theorem is referenced by:  lcmfcl  15322  lcmfeq0b  15324  dvdslcmf  15325  lcmftp  15330  lcmfunsnlem2  15334
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