Theorem lcmf0val 15335

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 15304	. . 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 2690	. . . 4 ⊢ (𝑧 = 𝑍 → (0 ∈ 𝑧 ↔ 0 ∈ 𝑍))
4		raleq 3138	. . . . . 6 ⊢ (𝑧 = 𝑍 → (∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛 ↔ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛))
5	4	rabbidv 3189	. . . . 5 ⊢ (𝑧 = 𝑍 → {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛} = {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛})
6	5	infeq1d 8383	. . . 4 ⊢ (𝑧 = 𝑍 → inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < ))
7	3, 6	ifbieq2d 4111	. . 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 4092	. . . 4 ⊢ (0 ∈ 𝑍 → if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < )) = 0)
9	8	adantl 482	. . 3 ⊢ ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → if(0 ∈ 𝑍, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑍 𝑚 ∥ 𝑛}, ℝ, < )) = 0)
10	7, 9	sylan9eqr 2678	. 2 ⊢ (((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) ∧ 𝑧 = 𝑍) → if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < )) = 0)
11		zex 11386	. . . . . 6 ⊢ ℤ ∈ V
12	11	ssex 4802	. . . . 5 ⊢ (𝑍 ⊆ ℤ → 𝑍 ∈ V)
13		elpwg 4166	. . . . 5 ⊢ (𝑍 ∈ V → (𝑍 ∈ 𝒫 ℤ ↔ 𝑍 ⊆ ℤ))
14	12, 13	syl 17	. . . 4 ⊢ (𝑍 ⊆ ℤ → (𝑍 ∈ 𝒫 ℤ ↔ 𝑍 ⊆ ℤ))
15	14	ibir 257	. . 3 ⊢ (𝑍 ⊆ ℤ → 𝑍 ∈ 𝒫 ℤ)
16	15	adantr 481	. 2 ⊢ ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → 𝑍 ∈ 𝒫 ℤ)
17		simpr 477	. 2 ⊢ ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → 0 ∈ 𝑍)
18	2, 10, 16, 17	fvmptd 6288	1 ⊢ ((𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍) → (lcm‘𝑍) = 0)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ⊆ wss 3574  ifcif 4086  𝒫 cpw 4158   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  infcinf 8347  ℝcr 9935  0cc0 9936   < clt 10074  ℕcn 11020  ℤcz 11377   ∥ cdvds 14983  lcmclcmf 15302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-sup 8348  df-inf 8349  df-neg 10269  df-z 11378  df-lcmf 15304

This theorem is referenced by:  lcmfcl  15341  lcmfeq0b  15343  dvdslcmf  15344  lcmftp  15349  lcmfunsnlem2  15353