Theorem lcmval 12628

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 12518 and gcdval 12523. (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 12626	. . 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 2236	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 = 0 ↔ 𝑀 = 0))
4	3	orbi1d 796	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑥 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑦 = 0)))
5		breq1 4089	. . . . . . . 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 2789	. . . . . 6 ⊢ (𝑥 = 𝑀 → {𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)})
8	7	infeq1d 7205	. . . . 5 ⊢ (𝑥 = 𝑀 → inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ))
9	4, 8	ifbieq2d 3628	. . . 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 2236	. . . . . 6 ⊢ (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
11	10	orbi2d 795	. . . . 5 ⊢ (𝑦 = 𝑁 → ((𝑀 = 0 ∨ 𝑦 = 0) ↔ (𝑀 = 0 ∨ 𝑁 = 0)))
12		breq1 4089	. . . . . . . 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 2789	. . . . . 6 ⊢ (𝑦 = 𝑁 → {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)} = {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)})
15	14	infeq1d 7205	. . . . 5 ⊢ (𝑦 = 𝑁 → inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ))
16	11, 15	ifbieq2d 3628	. . . 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 2282	. . 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 8166	. . . 4 ⊢ 0 ∈ V
22	21	a1i 9	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → 0 ∈ V)
23		1zzd 9499	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 1 ∈ ℤ)
24		nnuz 9785	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
25	24	rabeqi 2793	. . . . 5 ⊢ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)} = {𝑛 ∈ (ℤ≥‘1) ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}
26		dvdsmul1 12367	. . . . . . . 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 9601	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 · 𝑁) ∈ ℤ)
31		dvdsabsb 12364	. . . . . . . 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 12368	. . . . . . . 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 12364	. . . . . . . 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 9596	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ∈ ℂ)
40	29	zcnd 9596	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∈ ℂ)
41	39, 40	absmuld 11748	. . . . . . . 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 757	. . . . . . . . . . . . 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 2479	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑀 ≠ 0)
47		nnabscl 11654	. . . . . . . . . 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 2479	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ≠ 0)
51		nnabscl 11654	. . . . . . . . . 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 9185	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((abs‘𝑀) · (abs‘𝑁)) ∈ ℕ)
54	41, 53	eqeltrd 2306	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘(𝑀 · 𝑁)) ∈ ℕ)
55		breq2 4090	. . . . . . . . 9 ⊢ (𝑛 = (abs‘(𝑀 · 𝑁)) → (𝑀 ∥ 𝑛 ↔ 𝑀 ∥ (abs‘(𝑀 · 𝑁))))
56		breq2 4090	. . . . . . . . 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 2961	. . . . . . 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 950	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (abs‘(𝑀 · 𝑁)) ∈ {𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)})
61		elfzelz 10253	. . . . . . 7 ⊢ (𝑛 ∈ (1...(abs‘(𝑀 · 𝑁))) → 𝑛 ∈ ℤ)
62		zdvdsdc 12366	. . . . . . 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 12366	. . . . . . 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 938	. . . . 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 10488	. . . 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 2814	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → inf({𝑛 ∈ ℕ ∣ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛)}, ℝ, < ) ∈ V)
69		lcmmndc 12627	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∨ 𝑁 = 0))
70	22, 68, 69	ifcldadc 3633	. 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 6144	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 713  DECID wdc 839   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  {crab 2512  Vcvv 2800  ifcif 3603   class class class wbr 4086  ‘cfv 5324  (class class class)co 6013   ∈ cmpo 6015  infcinf 7176  ℝcr 8024  0cc0 8025  1c1 8026   · cmul 8030   < clt 8207  ℕcn 9136  ℤcz 9472  ℤ_≥cuz 9748  ...cfz 10236  abscabs 11551   ∥ cdvds 12341   lcm clcm 12625

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-sup 7177  df-inf 7178  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-n0 9396  df-z 9473  df-uz 9749  df-q 9847  df-rp 9882  df-fz 10237  df-fzo 10371  df-fl 10523  df-mod 10578  df-seqfrec 10703  df-exp 10794  df-cj 11396  df-re 11397  df-im 11398  df-rsqrt 11552  df-abs 11553  df-dvds 12342  df-lcm 12626

This theorem is referenced by:  lcmcom  12629  lcm0val  12630  lcmn0val  12631  lcmass  12650