Theorem lcmass 12253

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 768	. . 3 ⊢ (((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0) ↔ (𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0)))
2		anass 401	. . . . . 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 2748	. . . 4 ⊢ {𝑥 ∈ ℕ ∣ ((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ∧ 𝑃 ∥ 𝑥)} = {𝑥 ∈ ℕ ∣ (𝑁 ∥ 𝑥 ∧ (𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥))}
5	4	infeq1i 7079	. . 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 3584	. 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 12240	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℕ0)
8	7	3adant3 1019	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℕ0)
9	8	nn0zd 9446	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℤ)
10		simp3 1001	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑃 ∈ ℤ)
11		lcmval 12231	. . . 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 411	. . 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 12239	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 lcm 𝑀) = 0 ↔ (𝑁 = 0 ∨ 𝑀 = 0)))
14	13	3adant3 1019	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) = 0 ↔ (𝑁 = 0 ∨ 𝑀 = 0)))
15	14	orbi1d 792	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (((𝑁 lcm 𝑀) = 0 ∨ 𝑃 = 0) ↔ ((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0)))
16	15	bicomd 141	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0) ↔ ((𝑁 lcm 𝑀) = 0 ∨ 𝑃 = 0)))
17		nnz 9345	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
18	17	adantl 277	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℤ)
19		simp1 999	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑁 ∈ ℤ)
20	19	adantr 276	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
21		simpl2 1003	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑀 ∈ ℤ)
22		lcmdvdsb 12252	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ↔ (𝑁 lcm 𝑀) ∥ 𝑥))
23	18, 20, 21, 22	syl3anc 1249	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → ((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ↔ (𝑁 lcm 𝑀) ∥ 𝑥))
24	23	anbi1d 465	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → (((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ∧ 𝑃 ∥ 𝑥) ↔ ((𝑁 lcm 𝑀) ∥ 𝑥 ∧ 𝑃 ∥ 𝑥)))
25	24	rabbidva 2751	. . . . 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 7078	. . . 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 3585	. . 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 2232	. 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 12240	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℕ0)
30	29	3adant1 1017	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℕ0)
31	30	nn0zd 9446	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℤ)
32		lcmval 12231	. . . 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 411	. . 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 12239	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀 lcm 𝑃) = 0 ↔ (𝑀 = 0 ∨ 𝑃 = 0)))
35	34	3adant1 1017	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀 lcm 𝑃) = 0 ↔ (𝑀 = 0 ∨ 𝑃 = 0)))
36	35	orbi2d 791	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 = 0 ∨ (𝑀 lcm 𝑃) = 0) ↔ (𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0))))
37	36	bicomd 141	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0)) ↔ (𝑁 = 0 ∨ (𝑀 lcm 𝑃) = 0)))
38	10	adantr 276	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑃 ∈ ℤ)
39		lcmdvdsb 12252	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥) ↔ (𝑀 lcm 𝑃) ∥ 𝑥))
40	18, 21, 38, 39	syl3anc 1249	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → ((𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥) ↔ (𝑀 lcm 𝑃) ∥ 𝑥))
41	40	anbi2d 464	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → ((𝑁 ∥ 𝑥 ∧ (𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥)) ↔ (𝑁 ∥ 𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)))
42	41	rabbidva 2751	. . . . 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 7078	. . . 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 3585	. . 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 2232	. 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 2255	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) lcm 𝑃) = (𝑁 lcm (𝑀 lcm 𝑃)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  {crab 2479  ifcif 3561   class class class wbr 4033  (class class class)co 5922  infcinf 7049  ℝcr 7878  0cc0 7879   < clt 8061  ℕcn 8990  ℕ₀cn0 9249  ℤcz 9326   ∥ cdvds 11952   lcm clcm 12228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-dvds 11953  df-gcd 12121  df-lcm 12229

This theorem is referenced by: (None)