lcmneg - Intuitionistic Logic Explorer

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Theorem lcmneg 12315

Description: Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.)

**Assertion**
Ref	Expression
lcmneg	⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (𝑀 lcm 𝑁))

**Proof of Theorem lcmneg**
Step	Hyp	Ref	Expression
1		lcm0val 12306	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = 0)
2		znegcl 9385	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
3		lcm0val 12306	. . . . . . . . 9 ⊢ (-𝑁 ∈ ℤ → (-𝑁 lcm 0) = 0)
4	2, 3	syl 14	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (-𝑁 lcm 0) = 0)
5	1, 4	eqtr4d 2240	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 lcm 0) = (-𝑁 lcm 0))
6	5	ad2antlr 489	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑁 lcm 0) = (-𝑁 lcm 0))
7		oveq2 5942	. . . . . . . 8 ⊢ (𝑀 = 0 → (𝑁 lcm 𝑀) = (𝑁 lcm 0))
8		oveq2 5942	. . . . . . . 8 ⊢ (𝑀 = 0 → (-𝑁 lcm 𝑀) = (-𝑁 lcm 0))
9	7, 8	eqeq12d 2219	. . . . . . 7 ⊢ (𝑀 = 0 → ((𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀) ↔ (𝑁 lcm 0) = (-𝑁 lcm 0)))
10	9	adantl 277	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀) ↔ (𝑁 lcm 0) = (-𝑁 lcm 0)))
11	6, 10	mpbird 167	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀))
12		lcmcom 12305	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = (𝑁 lcm 𝑀))
13		lcmcom 12305	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (-𝑁 lcm 𝑀))
14	2, 13	sylan2 286	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (-𝑁 lcm 𝑀))
15	12, 14	eqeq12d 2219	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = (𝑀 lcm -𝑁) ↔ (𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀)))
16	15	adantr 276	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑀 lcm 𝑁) = (𝑀 lcm -𝑁) ↔ (𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀)))
17	11, 16	mpbird 167	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
18		neg0 8300	. . . . . . . 8 ⊢ -0 = 0
19	18	oveq2i 5945	. . . . . . 7 ⊢ (𝑀 lcm -0) = (𝑀 lcm 0)
20	19	eqcomi 2208	. . . . . 6 ⊢ (𝑀 lcm 0) = (𝑀 lcm -0)
21		oveq2 5942	. . . . . 6 ⊢ (𝑁 = 0 → (𝑀 lcm 𝑁) = (𝑀 lcm 0))
22		negeq 8247	. . . . . . 7 ⊢ (𝑁 = 0 → -𝑁 = -0)
23	22	oveq2d 5950	. . . . . 6 ⊢ (𝑁 = 0 → (𝑀 lcm -𝑁) = (𝑀 lcm -0))
24	20, 21, 23	3eqtr4a 2263	. . . . 5 ⊢ (𝑁 = 0 → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
25	24	adantl 277	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
26	17, 25	jaodan 798	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
27		dvdslcm 12310	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ -𝑁 ∥ (𝑀 lcm -𝑁)))
28	2, 27	sylan2 286	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ -𝑁 ∥ (𝑀 lcm -𝑁)))
29		simpr 110	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
30		lcmcl 12313	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℕ₀)
31	2, 30	sylan2 286	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℕ₀)
32	31	nn0zd 9475	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℤ)
33		negdvdsb 12037	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 lcm -𝑁) ∈ ℤ) → (𝑁 ∥ (𝑀 lcm -𝑁) ↔ -𝑁 ∥ (𝑀 lcm -𝑁)))
34	29, 32, 33	syl2anc 411	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∥ (𝑀 lcm -𝑁) ↔ -𝑁 ∥ (𝑀 lcm -𝑁)))
35	34	anbi2d 464	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)) ↔ (𝑀 ∥ (𝑀 lcm -𝑁) ∧ -𝑁 ∥ (𝑀 lcm -𝑁))))
36	28, 35	mpbird 167	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)))
37	36	adantr 276	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)))
38		zcn 9359	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
39	38	negeq0d 8357	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 ↔ -𝑁 = 0))
40	39	orbi2d 791	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → ((𝑀 = 0 ∨ 𝑁 = 0) ↔ (𝑀 = 0 ∨ -𝑁 = 0)))
41	40	notbid 668	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (¬ (𝑀 = 0 ∨ 𝑁 = 0) ↔ ¬ (𝑀 = 0 ∨ -𝑁 = 0)))
42	41	biimpa 296	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 = 0 ∨ -𝑁 = 0))
43	42	adantll 476	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 = 0 ∨ -𝑁 = 0))
44		lcmn0cl 12309	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ -𝑁 = 0)) → (𝑀 lcm -𝑁) ∈ ℕ)
45	2, 44	sylanl2 403	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ -𝑁 = 0)) → (𝑀 lcm -𝑁) ∈ ℕ)
46	43, 45	syldan 282	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm -𝑁) ∈ ℕ)
47		simpl 109	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
48		3anass 984	. . . . . . 7 ⊢ (((𝑀 lcm -𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ↔ ((𝑀 lcm -𝑁) ∈ ℕ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)))
49	46, 47, 48	sylanbrc 417	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm -𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
50		simpr 110	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 = 0 ∨ 𝑁 = 0))
51		lcmledvds 12311	. . . . . 6 ⊢ ((((𝑀 lcm -𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)) → (𝑀 lcm 𝑁) ≤ (𝑀 lcm -𝑁)))
52	49, 50, 51	syl2anc 411	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)) → (𝑀 lcm 𝑁) ≤ (𝑀 lcm -𝑁)))
53	37, 52	mpd 13	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ≤ (𝑀 lcm -𝑁))
54		dvdslcm 12310	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
55	54	adantr 276	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
56		simplr 528	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∈ ℤ)
57		lcmn0cl 12309	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ ℕ)
58	57	nnzd 9476	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ ℤ)
59		negdvdsb 12037	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 lcm 𝑁) ∈ ℤ) → (𝑁 ∥ (𝑀 lcm 𝑁) ↔ -𝑁 ∥ (𝑀 lcm 𝑁)))
60	56, 58, 59	syl2anc 411	. . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑁 ∥ (𝑀 lcm 𝑁) ↔ -𝑁 ∥ (𝑀 lcm 𝑁)))
61	60	anbi2d 464	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)) ↔ (𝑀 ∥ (𝑀 lcm 𝑁) ∧ -𝑁 ∥ (𝑀 lcm 𝑁))))
62		lcmledvds 12311	. . . . . . . . . 10 ⊢ ((((𝑀 lcm 𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ -𝑁 = 0)) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ -𝑁 ∥ (𝑀 lcm 𝑁)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁)))
63	62	ex 115	. . . . . . . . 9 ⊢ (((𝑀 lcm 𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∨ -𝑁 = 0) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ -𝑁 ∥ (𝑀 lcm 𝑁)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁))))
64	2, 63	syl3an3 1284	. . . . . . . 8 ⊢ (((𝑀 lcm 𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∨ -𝑁 = 0) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ -𝑁 ∥ (𝑀 lcm 𝑁)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁))))
65	64	3expib 1208	. . . . . . 7 ⊢ ((𝑀 lcm 𝑁) ∈ ℕ → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∨ -𝑁 = 0) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ -𝑁 ∥ (𝑀 lcm 𝑁)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁)))))
66	57, 47, 43, 65	syl3c 63	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ -𝑁 ∥ (𝑀 lcm 𝑁)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁)))
67	61, 66	sylbid 150	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁)))
68	55, 67	mpd 13	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁))
69		lcmcl 12313	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ₀)
70	69	nn0red 9331	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℝ)
71	30	nn0red 9331	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℝ)
72	2, 71	sylan2 286	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℝ)
73	70, 72	letri3d 8170	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = (𝑀 lcm -𝑁) ↔ ((𝑀 lcm 𝑁) ≤ (𝑀 lcm -𝑁) ∧ (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁))))
74	73	adantr 276	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm 𝑁) = (𝑀 lcm -𝑁) ↔ ((𝑀 lcm 𝑁) ≤ (𝑀 lcm -𝑁) ∧ (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁))))
75	53, 68, 74	mpbir2and 946	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
76		lcmmndc 12303	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∨ 𝑁 = 0))
77		exmiddc 837	. . . 4 ⊢ (DECID (𝑀 = 0 ∨ 𝑁 = 0) → ((𝑀 = 0 ∨ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∨ 𝑁 = 0)))
78	76, 77	syl 14	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 = 0 ∨ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∨ 𝑁 = 0)))
79	26, 75, 78	mpjaodan 799	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
80	79	eqcomd 2210	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (𝑀 lcm 𝑁))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 DECID wdc 835 ∧ w3a 980 = wceq 1372 ∈ wcel 2175 class class class wbr 4043 (class class class)co 5934 ℝcr 7906 0cc0 7907 ≤ cle 8090 -cneg 8226 ℕcn 9018 ℕ₀cn0 9277 ℤcz 9354 ∥ cdvds 12017 lcm clcm 12301

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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-iinf 4634 ax-cnex 7998 ax-resscn 7999 ax-1cn 8000 ax-1re 8001 ax-icn 8002 ax-addcl 8003 ax-addrcl 8004 ax-mulcl 8005 ax-mulrcl 8006 ax-addcom 8007 ax-mulcom 8008 ax-addass 8009 ax-mulass 8010 ax-distr 8011 ax-i2m1 8012 ax-0lt1 8013 ax-1rid 8014 ax-0id 8015 ax-rnegex 8016 ax-precex 8017 ax-cnre 8018 ax-pre-ltirr 8019 ax-pre-ltwlin 8020 ax-pre-lttrn 8021 ax-pre-apti 8022 ax-pre-ltadd 8023 ax-pre-mulgt0 8024 ax-pre-mulext 8025 ax-arch 8026 ax-caucvg 8027

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4338 df-po 4341 df-iso 4342 df-iord 4411 df-on 4413 df-ilim 4414 df-suc 4416 df-iom 4637 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-rn 4684 df-res 4685 df-ima 4686 df-iota 5229 df-fun 5270 df-fn 5271 df-f 5272 df-f1 5273 df-fo 5274 df-f1o 5275 df-fv 5276 df-isom 5277 df-riota 5889 df-ov 5937 df-oprab 5938 df-mpo 5939 df-1st 6216 df-2nd 6217 df-recs 6381 df-frec 6467 df-sup 7068 df-inf 7069 df-pnf 8091 df-mnf 8092 df-xr 8093 df-ltxr 8094 df-le 8095 df-sub 8227 df-neg 8228 df-reap 8630 df-ap 8637 df-div 8728 df-inn 9019 df-2 9077 df-3 9078 df-4 9079 df-n0 9278 df-z 9355 df-uz 9631 df-q 9723 df-rp 9758 df-fz 10113 df-fzo 10247 df-fl 10394 df-mod 10449 df-seqfrec 10574 df-exp 10665 df-cj 11072 df-re 11073 df-im 11074 df-rsqrt 11228 df-abs 11229 df-dvds 12018 df-lcm 12302

This theorem is referenced by: neglcm 12316 lcmabs 12317

W3C validator