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Theorem lcmf0val 16505
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 16474 . 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 3312 . . . . . 6 (𝑧 = 𝑍 → (∀𝑚𝑧 𝑚𝑛 ↔ ∀𝑚𝑍 𝑚𝑛))
43rabbidv 3418 . . . . 5 (𝑧 = 𝑍 → {𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛} = {𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛})
54infeq1d 9420 . . . 4 (𝑧 = 𝑍 → inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛}, ℝ, < ))
62, 5ifbieq2d 4517 . . 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 4497 . . . 4 (0 ∈ 𝑍 → if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛}, ℝ, < )) = 0)
87adantl 483 . . 3 ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛}, ℝ, < )) = 0)
96, 8sylan9eqr 2799 . 2 (((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) ∧ 𝑧 = 𝑍) → if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )) = 0)
10 zex 12515 . . . . . 6 ℤ ∈ V
1110ssex 5283 . . . . 5 (𝑍 ⊆ ℤ → 𝑍 ∈ V)
12 elpwg 4568 . . . . 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 482 . 2 ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → 𝑍 ∈ 𝒫 ℤ)
16 simpr 486 . 2 ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → 0 ∈ 𝑍)
171, 9, 15, 16fvmptd2 6961 1 ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → (lcm𝑍) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  {crab 3410  Vcvv 3448  wss 3915  ifcif 4491  𝒫 cpw 4565   class class class wbr 5110  cfv 6501  infcinf 9384  cr 11057  0cc0 11058   < clt 11196  cn 12160  cz 12506  cdvds 16143  lcmclcmf 16472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-cnex 11114  ax-resscn 11115
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-sup 9385  df-inf 9386  df-neg 11395  df-z 12507  df-lcmf 16474
This theorem is referenced by:  lcmfcl  16511  lcmfeq0b  16513  dvdslcmf  16514  lcmftp  16519  lcmfunsnlem2  16523
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