Theorem lcmval 12256

Description: Value of the lcm operator. (𝑀 lcm 𝑁) is the least common multiple of 𝑀 and 𝑁. If either 𝑀 or 𝑁 is 0, the result is defined conventionally as 0. Contrast with df-gcd 12146 and gcdval 12151. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)

Assertion

Ref	Expression
lcmval	⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )))

Distinct variable groups:   𝑛,𝑀   𝑛,𝑁

Proof of Theorem lcmval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lcm 12254	. . 3 ⊢ lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )))
2	1	a1i 9	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ))))
3		eqeq1 2203	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 = 0 ↔ 𝑀 = 0))
4	3	orbi1d 792	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑥 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑦 = 0)))
5		breq1 4037	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝑥 ∥ 𝑛 ↔ 𝑀 ∥ 𝑛))
6	5	anbi1d 465	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛) ↔ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)))
7	6	rabbidv 2752	. . . . . 6 ⊢ (𝑥 = 𝑀 → {𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)})
8	7	infeq1d 7087	. . . . 5 ⊢ (𝑥 = 𝑀 → inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ))
9	4, 8	ifbieq2d 3586	. . . 4 ⊢ (𝑥 = 𝑀 → if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )))
10		eqeq1 2203	. . . . . 6 ⊢ (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
11	10	orbi2d 791	. . . . 5 ⊢ (𝑦 = 𝑁 → ((𝑀 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑁 = 0)))
12		breq1 4037	. . . . . . . 8 ⊢ (𝑦 = 𝑁 → (𝑦 ∥ 𝑛 ↔ 𝑁 ∥ 𝑛))
13	12	anbi2d 464	. . . . . . 7 ⊢ (𝑦 = 𝑁 → ((𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛) ↔ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)))
14	13	rabbidv 2752	. . . . . 6 ⊢ (𝑦 = 𝑁 → {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)})
15	14	infeq1d 7087	. . . . 5 ⊢ (𝑦 = 𝑁 → inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ))
16	11, 15	ifbieq2d 3586	. . . 4 ⊢ (𝑦 = 𝑁 → if((𝑀 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )))
17	9, 16	sylan9eq 2249	. . 3 ⊢ ((𝑥 = 𝑀 ∧ 𝑦 = 𝑁) → if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )))
18	17	adantl 277	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑥 = 𝑀 ∧ 𝑦 = 𝑁)) → if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )))
19		simpl 109	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℤ)
20		simpr 110	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
21		c0ex 8037	. . . 4 ⊢ 0 ∈ V
22	21	a1i 9	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → 0 ∈ V)
23		1zzd 9370	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 1 ∈ ℤ)
24		nnuz 9654	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
25	24	rabeqi 2756	. . . . 5 ⊢ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)} = {𝑛 ∈ (ℤ≥‘1) ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}
26		dvdsmul1 11995	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∥ (𝑀 · 𝑁))
27	26	adantr 276	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∥ (𝑀 · 𝑁))
28		simpll 527	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∈ ℤ)
29		simplr 528	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∈ ℤ)
30	28, 29	zmulcld 9471	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 · 𝑁) ∈ ℤ)
31		dvdsabsb 11992	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (𝑀 ∥ (𝑀 · 𝑁) ↔ 𝑀 ∥ (abs‘(𝑀 · 𝑁))))
32	28, 30, 31	syl2anc 411	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∥ (𝑀 · 𝑁) ↔ 𝑀 ∥ (abs‘(𝑀 · 𝑁))))
33	27, 32	mpbid 147	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∥ (abs‘(𝑀 · 𝑁)))
34		dvdsmul2 11996	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁))
35	34	adantr 276	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∥ (𝑀 · 𝑁))
36		dvdsabsb 11992	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (𝑁 ∥ (𝑀 · 𝑁) ↔ 𝑁 ∥ (abs‘(𝑀 · 𝑁))))
37	29, 30, 36	syl2anc 411	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑁 ∥ (𝑀 · 𝑁) ↔ 𝑁 ∥ (abs‘(𝑀 · 𝑁))))
38	35, 37	mpbid 147	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∥ (abs‘(𝑀 · 𝑁)))
39	28	zcnd 9466	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∈ ℂ)
40	29	zcnd 9466	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∈ ℂ)
41	39, 40	absmuld 11376	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁)))
42		simpr 110	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 = 0 ∨ 𝑁 = 0))
43		ioran 753	. . . . . . . . . . . . 13 ⊢ (¬ (𝑀 = 0 ∨ 𝑁 = 0) ↔ (¬ 𝑀 = 0 ∧ ¬ 𝑁 = 0))
44	42, 43	sylib 122	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (¬ 𝑀 = 0 ∧ ¬ 𝑁 = 0))
45	44	simpld 112	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ 𝑀 = 0)
46	45	neneqad 2446	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ≠ 0)
47		nnabscl 11282	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
48	28, 46, 47	syl2anc 411	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘𝑀) ∈ ℕ)
49	44	simprd 114	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ 𝑁 = 0)
50	49	neneqad 2446	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ≠ 0)
51		nnabscl 11282	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
52	29, 50, 51	syl2anc 411	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘𝑁) ∈ ℕ)
53	48, 52	nnmulcld 9056	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((abs‘𝑀) · (abs‘𝑁)) ∈ ℕ)
54	41, 53	eqeltrd 2273	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘(𝑀 · 𝑁)) ∈ ℕ)
55		breq2 4038	. . . . . . . . 9 ⊢ (𝑛 = (abs‘(𝑀 · 𝑁)) → (𝑀 ∥ 𝑛 ↔ 𝑀 ∥ (abs‘(𝑀 · 𝑁))))
56		breq2 4038	. . . . . . . . 9 ⊢ (𝑛 = (abs‘(𝑀 · 𝑁)) → (𝑁 ∥ 𝑛 ↔ 𝑁 ∥ (abs‘(𝑀 · 𝑁))))
57	55, 56	anbi12d 473	. . . . . . . 8 ⊢ (𝑛 = (abs‘(𝑀 · 𝑁)) → ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) ↔ (𝑀 ∥ (abs‘(𝑀 · 𝑁)) ∧ 𝑁 ∥ (abs‘(𝑀 · 𝑁)))))
58	57	elrab3 2921	. . . . . . 7 ⊢ ((abs‘(𝑀 · 𝑁)) ∈ ℕ → ((abs‘(𝑀 · 𝑁)) ∈ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)} ↔ (𝑀 ∥ (abs‘(𝑀 · 𝑁)) ∧ 𝑁 ∥ (abs‘(𝑀 · 𝑁)))))
59	54, 58	syl 14	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((abs‘(𝑀 · 𝑁)) ∈ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)} ↔ (𝑀 ∥ (abs‘(𝑀 · 𝑁)) ∧ 𝑁 ∥ (abs‘(𝑀 · 𝑁)))))
60	33, 38, 59	mpbir2and 946	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘(𝑀 · 𝑁)) ∈ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)})
61		elfzelz 10117	. . . . . . 7 ⊢ (𝑛 ∈ (1...(abs‘(𝑀 · 𝑁))) → 𝑛 ∈ ℤ)
62		zdvdsdc 11994	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) → DECID 𝑀 ∥ 𝑛)
63	28, 61, 62	syl2an 289	. . . . . 6 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) ∧ 𝑛 ∈ (1...(abs‘(𝑀 · 𝑁)))) → DECID 𝑀 ∥ 𝑛)
64		zdvdsdc 11994	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → DECID 𝑁 ∥ 𝑛)
65	29, 61, 64	syl2an 289	. . . . . 6 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) ∧ 𝑛 ∈ (1...(abs‘(𝑀 · 𝑁)))) → DECID 𝑁 ∥ 𝑛)
66	63, 65	dcand 934	. . . . 5 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) ∧ 𝑛 ∈ (1...(abs‘(𝑀 · 𝑁)))) → DECID (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))
67	23, 25, 60, 66	infssuzcldc 10342	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)})
68	67	elexd 2776	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ) ∈ V)
69		lcmmndc 12255	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∨ 𝑁 = 0))
70	22, 68, 69	ifcldadc 3591	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )) ∈ V)
71	2, 18, 19, 20, 70	ovmpod 6054	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = if((𝑀 = 0 ∨ 𝑁 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < )))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  {crab 2479  Vcvv 2763  ifcif 3562   class class class wbr 4034  ‘cfv 5259  (class class class)co 5925   ∈ cmpo 5927  infcinf 7058  ℝcr 7895  0cc0 7896  1c1 7897   · cmul 7901   < clt 8078  ℕcn 9007  ℤcz 9343  ℤ_≥cuz 9618  ...cfz 10100  abscabs 11179   ∥ cdvds 11969   lcm clcm 12253

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 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  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 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-dvds 11970  df-lcm 12254

This theorem is referenced by:  lcmcom  12257  lcm0val  12258  lcmn0val  12259  lcmass  12278