Theorem lcmf0val 16641

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 16610	. 2 ⊢ lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < )))
2		eleq2 2823	. . . 4 ⊢ (𝑧 = 𝑍 → (0 ∈ 𝑧 ↔ 0 ∈ 𝑍))
3		raleq 3302	. . . . . 6 ⊢ (𝑧 = 𝑍 → (∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛 ↔ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛))
4	3	rabbidv 3423	. . . . 5 ⊢ (𝑧 = 𝑍 → {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛} = {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛})
5	4	infeq1d 9490	. . . 4 ⊢ (𝑧 = 𝑍 → inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ))
6	2, 5	ifbieq2d 4527	. . 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, < )))
7		iftrue 4506	. . . 4 ⊢ (0 ∈ 𝑍 → if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < )) = 0)
8	7	adantl 481	. . 3 ⊢ ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < )) = 0)
9	6, 8	sylan9eqr 2792	. 2 ⊢ (((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) ∧ 𝑧 = 𝑍) → if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < )) = 0)
10		zex 12597	. . . . . 6 ⊢ ℤ ∈ V
11	10	ssex 5291	. . . . 5 ⊢ (𝑍 ⊆ ℤ → 𝑍 ∈ V)
12		elpwg 4578	. . . . 5 ⊢ (𝑍 ∈ V → (𝑍 ∈ 𝒫 ℤ ↔ 𝑍 ⊆ ℤ))
13	11, 12	syl 17	. . . 4 ⊢ (𝑍 ⊆ ℤ → (𝑍 ∈ 𝒫 ℤ ↔ 𝑍 ⊆ ℤ))
14	13	ibir 268	. . 3 ⊢ (𝑍 ⊆ ℤ → 𝑍 ∈ 𝒫 ℤ)
15	14	adantr 480	. 2 ⊢ ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → 𝑍 ∈ 𝒫 ℤ)
16		simpr 484	. 2 ⊢ ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → 0 ∈ 𝑍)
17	1, 9, 15, 16	fvmptd2 6994	1 ⊢ ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → (lcm‘𝑍) = 0)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  {crab 3415  Vcvv 3459   ⊆ wss 3926  ifcif 4500  𝒫 cpw 4575   class class class wbr 5119  ‘cfv 6531  infcinf 9453  ℝcr 11128  0cc0 11129   < clt 11269  ℕcn 12240  ℤcz 12588   ∥ cdvds 16272  lcmclcmf 16608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-cnex 11185  ax-resscn 11186

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-sup 9454  df-inf 9455  df-neg 11469  df-z 12589  df-lcmf 16610

This theorem is referenced by:  lcmfcl  16647  lcmfeq0b  16649  dvdslcmf  16650  lcmftp  16655  lcmfunsnlem2  16659