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Theorem lcmf0val 16566
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 16535 . 2 lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )))
2 eleq2 2816 . . . 4 (𝑧 = 𝑍 → (0 ∈ 𝑧 ↔ 0 ∈ 𝑍))
3 raleq 3316 . . . . . 6 (𝑧 = 𝑍 → (∀𝑚𝑧 𝑚𝑛 ↔ ∀𝑚𝑍 𝑚𝑛))
43rabbidv 3434 . . . . 5 (𝑧 = 𝑍 → {𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛} = {𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛})
54infeq1d 9474 . . . 4 (𝑧 = 𝑍 → inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑍 𝑚𝑛}, ℝ, < ))
62, 5ifbieq2d 4549 . . 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 4529 . . . 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 2788 . 2 (((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) ∧ 𝑧 = 𝑍) → if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )) = 0)
10 zex 12571 . . . . . 6 ℤ ∈ V
1110ssex 5314 . . . . 5 (𝑍 ⊆ ℤ → 𝑍 ∈ V)
12 elpwg 4600 . . . . 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 7000 1 ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → (lcm𝑍) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3055  {crab 3426  Vcvv 3468  wss 3943  ifcif 4523  𝒫 cpw 4597   class class class wbr 5141  cfv 6537  infcinf 9438  cr 11111  0cc0 11112   < clt 11252  cn 12216  cz 12562  cdvds 16204  lcmclcmf 16533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-cnex 11168  ax-resscn 11169
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-sup 9439  df-inf 9440  df-neg 11451  df-z 12563  df-lcmf 16535
This theorem is referenced by:  lcmfcl  16572  lcmfeq0b  16574  dvdslcmf  16575  lcmftp  16580  lcmfunsnlem2  16584
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