Theorem lcmf0val 15173

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) → (lcm‘𝑍) = 0)

Proof of Theorem lcmf0val

Dummy variables 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lcmf 15142	. . 3 ⊢ lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < )))
2	1	a1i 11	. 2 ⊢ ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < ))))
3		eleq2 2677	. . . 4 ⊢ (𝑧 = 𝑍 → (0 ∈ 𝑧 ↔ 0 ∈ 𝑍))
4		raleq 3115	. . . . . 6 ⊢ (𝑧 = 𝑍 → (∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛 ↔ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛))
5	4	rabbidv 3164	. . . . 5 ⊢ (𝑧 = 𝑍 → {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛} = {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛})
6	5	infeq1d 8266	. . . 4 ⊢ (𝑧 = 𝑍 → inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ))
7	3, 6	ifbieq2d 4061	. . 3 ⊢ (𝑧 = 𝑍 → if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < )) = if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < )))
8		iftrue 4042	. . . 4 ⊢ (0 ∈ 𝑍 → if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < )) = 0)
9	8	adantl 481	. . 3 ⊢ ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < )) = 0)
10	7, 9	sylan9eqr 2666	. 2 ⊢ (((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) ∧ 𝑧 = 𝑍) → if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < )) = 0)
11		zex 11263	. . . . . 6 ⊢ ℤ ∈ V
12	11	ssex 4730	. . . . 5 ⊢ (𝑍 ⊆ ℤ → 𝑍 ∈ V)
13		elpwg 4116	. . . . 5 ⊢ (𝑍 ∈ V → (𝑍 ∈ 𝒫 ℤ ↔ 𝑍 ⊆ ℤ))
14	12, 13	syl 17	. . . 4 ⊢ (𝑍 ⊆ ℤ → (𝑍 ∈ 𝒫 ℤ ↔ 𝑍 ⊆ ℤ))
15	14	ibir 256	. . 3 ⊢ (𝑍 ⊆ ℤ → 𝑍 ∈ 𝒫 ℤ)
16	15	adantr 480	. 2 ⊢ ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → 𝑍 ∈ 𝒫 ℤ)
17		simpr 476	. 2 ⊢ ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → 0 ∈ 𝑍)
18	2, 10, 16, 17	fvmptd 6197	1 ⊢ ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → (lcm‘𝑍) = 0)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  ifcif 4036  𝒫 cpw 4108   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  infcinf 8230  ℝcr 9814  0cc0 9815   < clt 9953  ℕcn 10897  ℤcz 11254   ∥ cdvds 14821  lcmclcmf 15140

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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-cnex 9871  ax-resscn 9872

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-ral 2901  df-rex 2902  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-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-sup 8231  df-inf 8232  df-neg 10148  df-z 11255  df-lcmf 15142

This theorem is referenced by:  lcmfcl  15179  lcmfeq0b  15181  dvdslcmf  15182  lcmftp  15187  lcmfunsnlem2  15191