Theorem lcmass 12028

Description: Associative law for lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)

Assertion

Ref	Expression
lcmass	⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) lcm 𝑃) = (𝑁 lcm (𝑀 lcm 𝑃)))

Proof of Theorem lcmass

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orass 762	. . 3 ⊢ (((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0) ↔ (𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0)))
2		anass 399	. . . . . 6 ⊢ (((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ∧ 𝑃 ∥ 𝑥) ↔ (𝑁 ∥ 𝑥 ∧ (𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥)))
3	2	a1i 9	. . . . 5 ⊢ (𝑥 ∈ ℕ → (((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ∧ 𝑃 ∥ 𝑥) ↔ (𝑁 ∥ 𝑥 ∧ (𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥))))
4	3	rabbiia 2715	. . . 4 ⊢ {𝑥 ∈ ℕ ∣ ((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ∧ 𝑃 ∥ 𝑥)} = {𝑥 ∈ ℕ ∣ (𝑁 ∥ 𝑥 ∧ (𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥))}
5	4	infeq1i 6987	. . 3 ⊢ inf({𝑥 ∈ ℕ ∣ ((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ∧ 𝑃 ∥ 𝑥)}, ℝ, < ) = inf({𝑥 ∈ ℕ ∣ (𝑁 ∥ 𝑥 ∧ (𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥))}, ℝ, < )
6	1, 5	ifbieq2i 3548	. 2 ⊢ if(((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0), 0, inf({𝑥 ∈ ℕ ∣ ((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ∧ 𝑃 ∥ 𝑥)}, ℝ, < )) = if((𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0)), 0, inf({𝑥 ∈ ℕ ∣ (𝑁 ∥ 𝑥 ∧ (𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥))}, ℝ, < ))
7		lcmcl 12015	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℕ0)
8	7	3adant3 1012	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℕ0)
9	8	nn0zd 9321	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℤ)
10		simp3 994	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑃 ∈ ℤ)
11		lcmval 12006	. . . 4 ⊢ (((𝑁 lcm 𝑀) ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) lcm 𝑃) = if(((𝑁 lcm 𝑀) = 0 ∨ 𝑃 = 0), 0, inf({𝑥 ∈ ℕ ∣ ((𝑁 lcm 𝑀) ∥ 𝑥 ∧ 𝑃 ∥ 𝑥)}, ℝ, < )))
12	9, 10, 11	syl2anc 409	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) lcm 𝑃) = if(((𝑁 lcm 𝑀) = 0 ∨ 𝑃 = 0), 0, inf({𝑥 ∈ ℕ ∣ ((𝑁 lcm 𝑀) ∥ 𝑥 ∧ 𝑃 ∥ 𝑥)}, ℝ, < )))
13		lcmeq0 12014	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 lcm 𝑀) = 0 ↔ (𝑁 = 0 ∨ 𝑀 = 0)))
14	13	3adant3 1012	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) = 0 ↔ (𝑁 = 0 ∨ 𝑀 = 0)))
15	14	orbi1d 786	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (((𝑁 lcm 𝑀) = 0 ∨ 𝑃 = 0) ↔ ((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0)))
16	15	bicomd 140	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0) ↔ ((𝑁 lcm 𝑀) = 0 ∨ 𝑃 = 0)))
17		nnz 9220	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
18	17	adantl 275	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℤ)
19		simp1 992	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑁 ∈ ℤ)
20	19	adantr 274	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
21		simpl2 996	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑀 ∈ ℤ)
22		lcmdvdsb 12027	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ↔ (𝑁 lcm 𝑀) ∥ 𝑥))
23	18, 20, 21, 22	syl3anc 1233	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → ((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ↔ (𝑁 lcm 𝑀) ∥ 𝑥))
24	23	anbi1d 462	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → (((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ∧ 𝑃 ∥ 𝑥) ↔ ((𝑁 lcm 𝑀) ∥ 𝑥 ∧ 𝑃 ∥ 𝑥)))
25	24	rabbidva 2718	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → {𝑥 ∈ ℕ ∣ ((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ∧ 𝑃 ∥ 𝑥)} = {𝑥 ∈ ℕ ∣ ((𝑁 lcm 𝑀) ∥ 𝑥 ∧ 𝑃 ∥ 𝑥)})
26	25	infeq1d 6986	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → inf({𝑥 ∈ ℕ ∣ ((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ∧ 𝑃 ∥ 𝑥)}, ℝ, < ) = inf({𝑥 ∈ ℕ ∣ ((𝑁 lcm 𝑀) ∥ 𝑥 ∧ 𝑃 ∥ 𝑥)}, ℝ, < ))
27	16, 26	ifbieq2d 3549	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → if(((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0), 0, inf({𝑥 ∈ ℕ ∣ ((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ∧ 𝑃 ∥ 𝑥)}, ℝ, < )) = if(((𝑁 lcm 𝑀) = 0 ∨ 𝑃 = 0), 0, inf({𝑥 ∈ ℕ ∣ ((𝑁 lcm 𝑀) ∥ 𝑥 ∧ 𝑃 ∥ 𝑥)}, ℝ, < )))
28	12, 27	eqtr4d 2206	. 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) lcm 𝑃) = if(((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0), 0, inf({𝑥 ∈ ℕ ∣ ((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ∧ 𝑃 ∥ 𝑥)}, ℝ, < )))
29		lcmcl 12015	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℕ0)
30	29	3adant1 1010	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℕ0)
31	30	nn0zd 9321	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℤ)
32		lcmval 12006	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 lcm 𝑃) ∈ ℤ) → (𝑁 lcm (𝑀 lcm 𝑃)) = if((𝑁 = 0 ∨ (𝑀 lcm 𝑃) = 0), 0, inf({𝑥 ∈ ℕ ∣ (𝑁 ∥ 𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)}, ℝ, < )))
33	19, 31, 32	syl2anc 409	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 lcm (𝑀 lcm 𝑃)) = if((𝑁 = 0 ∨ (𝑀 lcm 𝑃) = 0), 0, inf({𝑥 ∈ ℕ ∣ (𝑁 ∥ 𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)}, ℝ, < )))
34		lcmeq0 12014	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀 lcm 𝑃) = 0 ↔ (𝑀 = 0 ∨ 𝑃 = 0)))
35	34	3adant1 1010	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀 lcm 𝑃) = 0 ↔ (𝑀 = 0 ∨ 𝑃 = 0)))
36	35	orbi2d 785	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 = 0 ∨ (𝑀 lcm 𝑃) = 0) ↔ (𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0))))
37	36	bicomd 140	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0)) ↔ (𝑁 = 0 ∨ (𝑀 lcm 𝑃) = 0)))
38	10	adantr 274	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑃 ∈ ℤ)
39		lcmdvdsb 12027	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥) ↔ (𝑀 lcm 𝑃) ∥ 𝑥))
40	18, 21, 38, 39	syl3anc 1233	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → ((𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥) ↔ (𝑀 lcm 𝑃) ∥ 𝑥))
41	40	anbi2d 461	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → ((𝑁 ∥ 𝑥 ∧ (𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥)) ↔ (𝑁 ∥ 𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)))
42	41	rabbidva 2718	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → {𝑥 ∈ ℕ ∣ (𝑁 ∥ 𝑥 ∧ (𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥))} = {𝑥 ∈ ℕ ∣ (𝑁 ∥ 𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)})
43	42	infeq1d 6986	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → inf({𝑥 ∈ ℕ ∣ (𝑁 ∥ 𝑥 ∧ (𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥))}, ℝ, < ) = inf({𝑥 ∈ ℕ ∣ (𝑁 ∥ 𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)}, ℝ, < ))
44	37, 43	ifbieq2d 3549	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → if((𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0)), 0, inf({𝑥 ∈ ℕ ∣ (𝑁 ∥ 𝑥 ∧ (𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥))}, ℝ, < )) = if((𝑁 = 0 ∨ (𝑀 lcm 𝑃) = 0), 0, inf({𝑥 ∈ ℕ ∣ (𝑁 ∥ 𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)}, ℝ, < )))
45	33, 44	eqtr4d 2206	. 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 lcm (𝑀 lcm 𝑃)) = if((𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0)), 0, inf({𝑥 ∈ ℕ ∣ (𝑁 ∥ 𝑥 ∧ (𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥))}, ℝ, < )))
46	6, 28, 45	3eqtr4a 2229	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) lcm 𝑃) = (𝑁 lcm (𝑀 lcm 𝑃)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  {crab 2452  ifcif 3525   class class class wbr 3987  (class class class)co 5851  infcinf 6957  ℝcr 7762  0cc0 7763   < clt 7943  ℕcn 8867  ℕ₀cn0 9124  ℤcz 9201   ∥ cdvds 11738   lcm clcm 12003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-mulrcl 7862  ax-addcom 7863  ax-mulcom 7864  ax-addass 7865  ax-mulass 7866  ax-distr 7867  ax-i2m1 7868  ax-0lt1 7869  ax-1rid 7870  ax-0id 7871  ax-rnegex 7872  ax-precex 7873  ax-cnre 7874  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-apti 7878  ax-pre-ltadd 7879  ax-pre-mulgt0 7880  ax-pre-mulext 7881  ax-arch 7882  ax-caucvg 7883

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-sup 6958  df-inf 6959  df-pnf 7945  df-mnf 7946  df-xr 7947  df-ltxr 7948  df-le 7949  df-sub 8081  df-neg 8082  df-reap 8483  df-ap 8490  df-div 8579  df-inn 8868  df-2 8926  df-3 8927  df-4 8928  df-n0 9125  df-z 9202  df-uz 9477  df-q 9568  df-rp 9600  df-fz 9955  df-fzo 10088  df-fl 10215  df-mod 10268  df-seqfrec 10391  df-exp 10465  df-cj 10795  df-re 10796  df-im 10797  df-rsqrt 10951  df-abs 10952  df-dvds 11739  df-gcd 11887  df-lcm 12004

This theorem is referenced by: (None)