Theorem lcmass 12087

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 767	. . 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 2724	. . . 4 ⊢ {𝑥 ∈ ℕ ∣ ((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ∧ 𝑃 ∥ 𝑥)} = {𝑥 ∈ ℕ ∣ (𝑁 ∥ 𝑥 ∧ (𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥))}
5	4	infeq1i 7014	. . 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 3559	. 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 12074	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℕ0)
8	7	3adant3 1017	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℕ0)
9	8	nn0zd 9375	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℤ)
10		simp3 999	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑃 ∈ ℤ)
11		lcmval 12065	. . . 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 12073	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 lcm 𝑀) = 0 ↔ (𝑁 = 0 ∨ 𝑀 = 0)))
14	13	3adant3 1017	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) = 0 ↔ (𝑁 = 0 ∨ 𝑀 = 0)))
15	14	orbi1d 791	. . . . 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 9274	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
18	17	adantl 277	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℤ)
19		simp1 997	. . . . . . . . 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 1001	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑀 ∈ ℤ)
22		lcmdvdsb 12086	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 ∥ 𝑥 ∧ 𝑀 ∥ 𝑥) ↔ (𝑁 lcm 𝑀) ∥ 𝑥))
23	18, 20, 21, 22	syl3anc 1238	. . . . . . 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 2727	. . . . 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 7013	. . . 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 3560	. . 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 2213	. 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 12074	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℕ0)
30	29	3adant1 1015	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℕ0)
31	30	nn0zd 9375	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℤ)
32		lcmval 12065	. . . 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 12073	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀 lcm 𝑃) = 0 ↔ (𝑀 = 0 ∨ 𝑃 = 0)))
35	34	3adant1 1015	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀 lcm 𝑃) = 0 ↔ (𝑀 = 0 ∨ 𝑃 = 0)))
36	35	orbi2d 790	. . . . 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 12086	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀 ∥ 𝑥 ∧ 𝑃 ∥ 𝑥) ↔ (𝑀 lcm 𝑃) ∥ 𝑥))
40	18, 21, 38, 39	syl3anc 1238	. . . . . . 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 2727	. . . . 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 7013	. . . 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 3560	. . 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 2213	. 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 2236	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 708   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  {crab 2459  ifcif 3536   class class class wbr 4005  (class class class)co 5877  infcinf 6984  ℝcr 7812  0cc0 7813   < clt 7994  ℕcn 8921  ℕ₀cn0 9178  ℤcz 9255   ∥ cdvds 11796   lcm clcm 12062

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-gcd 11946  df-lcm 12063

This theorem is referenced by: (None)