Theorem lcmftp 15349

Description: The least common multiple of a triple of integers is the least common multiple of the third integer and the the least common multiple of the first two integers. Although there would be a shorter proof using lcmfunsn 15357, this explicit proof (not based on induction) should be kept. (Proof modification is discouraged.) (Contributed by AV, 23-Aug-2020.)

Assertion

Ref	Expression
lcmftp	⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (lcm‘{𝐴, 𝐵, 𝐶}) = ((𝐴 lcm 𝐵) lcm 𝐶))

Proof of Theorem lcmftp

Dummy variables 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0z 11388	. . . . . . 7 ⊢ 0 ∈ ℤ
2		eltpg 4227	. . . . . . 7 ⊢ (0 ∈ ℤ → (0 ∈ {𝐴, 𝐵, 𝐶} ↔ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶)))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ (0 ∈ {𝐴, 𝐵, 𝐶} ↔ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶))
4	3	biimpri 218	. . . . 5 ⊢ ((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → 0 ∈ {𝐴, 𝐵, 𝐶})
5		tpssi 4369	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → {𝐴, 𝐵, 𝐶} ⊆ ℤ)
6	4, 5	anim12ci 591	. . . 4 ⊢ (((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ({𝐴, 𝐵, 𝐶} ⊆ ℤ ∧ 0 ∈ {𝐴, 𝐵, 𝐶}))
7		lcmf0val 15335	. . . 4 ⊢ (({𝐴, 𝐵, 𝐶} ⊆ ℤ ∧ 0 ∈ {𝐴, 𝐵, 𝐶}) → (lcm‘{𝐴, 𝐵, 𝐶}) = 0)
8	6, 7	syl 17	. . 3 ⊢ (((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (lcm‘{𝐴, 𝐵, 𝐶}) = 0)
9		0zd 11389	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℤ → 0 ∈ ℤ)
10		lcmcom 15306	. . . . . . . . . 10 ⊢ ((0 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (0 lcm 𝐶) = (𝐶 lcm 0))
11	9, 10	mpancom 703	. . . . . . . . 9 ⊢ (𝐶 ∈ ℤ → (0 lcm 𝐶) = (𝐶 lcm 0))
12		lcm0val 15307	. . . . . . . . 9 ⊢ (𝐶 ∈ ℤ → (𝐶 lcm 0) = 0)
13	11, 12	eqtrd 2656	. . . . . . . 8 ⊢ (𝐶 ∈ ℤ → (0 lcm 𝐶) = 0)
14	13	eqcomd 2628	. . . . . . 7 ⊢ (𝐶 ∈ ℤ → 0 = (0 lcm 𝐶))
15	14	3ad2ant3 1084	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = (0 lcm 𝐶))
16	15	adantl 482	. . . . 5 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = (0 lcm 𝐶))
17		0zd 11389	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℤ → 0 ∈ ℤ)
18		lcmcom 15306	. . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 lcm 𝐵) = (𝐵 lcm 0))
19	17, 18	mpancom 703	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℤ → (0 lcm 𝐵) = (𝐵 lcm 0))
20		lcm0val 15307	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℤ → (𝐵 lcm 0) = 0)
21	19, 20	eqtrd 2656	. . . . . . . . 9 ⊢ (𝐵 ∈ ℤ → (0 lcm 𝐵) = 0)
22	21	eqcomd 2628	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → 0 = (0 lcm 𝐵))
23	22	3ad2ant2 1083	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = (0 lcm 𝐵))
24	23	adantl 482	. . . . . 6 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = (0 lcm 𝐵))
25	24	oveq1d 6665	. . . . 5 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (0 lcm 𝐶) = ((0 lcm 𝐵) lcm 𝐶))
26		oveq1 6657	. . . . . . 7 ⊢ (0 = 𝐴 → (0 lcm 𝐵) = (𝐴 lcm 𝐵))
27	26	oveq1d 6665	. . . . . 6 ⊢ (0 = 𝐴 → ((0 lcm 𝐵) lcm 𝐶) = ((𝐴 lcm 𝐵) lcm 𝐶))
28	27	adantr 481	. . . . 5 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((0 lcm 𝐵) lcm 𝐶) = ((𝐴 lcm 𝐵) lcm 𝐶))
29	16, 25, 28	3eqtrd 2660	. . . 4 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
30		lcm0val 15307	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → (𝐴 lcm 0) = 0)
31	30	eqcomd 2628	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 0 = (𝐴 lcm 0))
32	31	3ad2ant1 1082	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = (𝐴 lcm 0))
33	32	adantl 482	. . . . . 6 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = (𝐴 lcm 0))
34	33	oveq1d 6665	. . . . 5 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (0 lcm 𝐶) = ((𝐴 lcm 0) lcm 𝐶))
35	13	3ad2ant3 1084	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (0 lcm 𝐶) = 0)
36	35	adantl 482	. . . . 5 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (0 lcm 𝐶) = 0)
37		oveq2 6658	. . . . . . 7 ⊢ (0 = 𝐵 → (𝐴 lcm 0) = (𝐴 lcm 𝐵))
38	37	adantr 481	. . . . . 6 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 lcm 0) = (𝐴 lcm 𝐵))
39	38	oveq1d 6665	. . . . 5 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 0) lcm 𝐶) = ((𝐴 lcm 𝐵) lcm 𝐶))
40	34, 36, 39	3eqtr3d 2664	. . . 4 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
41		lcmcl 15314	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 lcm 𝐵) ∈ ℕ0)
42	41	nn0zd 11480	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 lcm 𝐵) ∈ ℤ)
43		lcm0val 15307	. . . . . . . 8 ⊢ ((𝐴 lcm 𝐵) ∈ ℤ → ((𝐴 lcm 𝐵) lcm 0) = 0)
44	43	eqcomd 2628	. . . . . . 7 ⊢ ((𝐴 lcm 𝐵) ∈ ℤ → 0 = ((𝐴 lcm 𝐵) lcm 0))
45	42, 44	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 0 = ((𝐴 lcm 𝐵) lcm 0))
46	45	3adant3 1081	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = ((𝐴 lcm 𝐵) lcm 0))
47		oveq2 6658	. . . . 5 ⊢ (0 = 𝐶 → ((𝐴 lcm 𝐵) lcm 0) = ((𝐴 lcm 𝐵) lcm 𝐶))
48	46, 47	sylan9eqr 2678	. . . 4 ⊢ ((0 = 𝐶 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
49	29, 40, 48	3jaoian 1393	. . 3 ⊢ (((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
50	8, 49	eqtrd 2656	. 2 ⊢ (((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (lcm‘{𝐴, 𝐵, 𝐶}) = ((𝐴 lcm 𝐵) lcm 𝐶))
51	42	3adant3 1081	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 lcm 𝐵) ∈ ℤ)
52		simp3 1063	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐶 ∈ ℤ)
53	51, 52	jca 554	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ))
54	53	adantl 482	. . . . . . . 8 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ))
55		dvdslcm 15311	. . . . . . . 8 ⊢ (((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
56	54, 55	syl 17	. . . . . . 7 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
57		dvdslcm 15311	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∥ (𝐴 lcm 𝐵) ∧ 𝐵 ∥ (𝐴 lcm 𝐵)))
58	57	3adant3 1081	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∥ (𝐴 lcm 𝐵) ∧ 𝐵 ∥ (𝐴 lcm 𝐵)))
59		simp1 1061	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐴 ∈ ℤ)
60		lcmcl 15314	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℕ0)
61	53, 60	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℕ0)
62	61	nn0zd 11480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ)
63	59, 51, 62	3jca 1242	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ ℤ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ))
64		dvdstr 15018	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ) → ((𝐴 ∥ (𝐴 lcm 𝐵) ∧ (𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
65	63, 64	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 ∥ (𝐴 lcm 𝐵) ∧ (𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
66	65	expd 452	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∥ (𝐴 lcm 𝐵) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))))
67	66	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∥ (𝐴 lcm 𝐵) → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))))
68	67	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∥ (𝐴 lcm 𝐵) ∧ 𝐵 ∥ (𝐴 lcm 𝐵)) → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))))
69	58, 68	mpcom 38	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
70	69	adantl 482	. . . . . . . . . . 11 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
71	70	com12 32	. . . . . . . . . 10 ⊢ ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
72	71	adantr 481	. . . . . . . . 9 ⊢ (((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
73	72	impcom 446	. . . . . . . 8 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))
74		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∥ (𝐴 lcm 𝐵) ∧ 𝐵 ∥ (𝐴 lcm 𝐵)) → 𝐵 ∥ (𝐴 lcm 𝐵))
75	57, 74	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∥ (𝐴 lcm 𝐵))
76	75	3adant3 1081	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐵 ∥ (𝐴 lcm 𝐵))
77	76	adantl 482	. . . . . . . . . . . 12 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐵 ∥ (𝐴 lcm 𝐵))
78		simp2 1062	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐵 ∈ ℤ)
79	78, 51, 62	3jca 1242	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵 ∈ ℤ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ))
80	79	adantl 482	. . . . . . . . . . . . 13 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐵 ∈ ℤ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ))
81		dvdstr 15018	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℤ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ) → ((𝐵 ∥ (𝐴 lcm 𝐵) ∧ (𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
82	80, 81	syl 17	. . . . . . . . . . . 12 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐵 ∥ (𝐴 lcm 𝐵) ∧ (𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
83	77, 82	mpand 711	. . . . . . . . . . 11 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
84	83	com12 32	. . . . . . . . . 10 ⊢ ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
85	84	adantr 481	. . . . . . . . 9 ⊢ (((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
86	85	impcom 446	. . . . . . . 8 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))) → 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))
87		simpr 477	. . . . . . . . 9 ⊢ (((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))
88	87	adantl 482	. . . . . . . 8 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))) → 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))
89	73, 86, 88	3jca 1242	. . . . . . 7 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))) → (𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
90	56, 89	mpdan 702	. . . . . 6 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
91		breq1 4656	. . . . . . . 8 ⊢ (𝑚 = 𝐴 → (𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
92		breq1 4656	. . . . . . . 8 ⊢ (𝑚 = 𝐵 → (𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
93		breq1 4656	. . . . . . . 8 ⊢ (𝑚 = 𝐶 → (𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
94	91, 92, 93	raltpg 4236	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ (𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))))
95	94	adantl 482	. . . . . 6 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ (𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))))
96	90, 95	mpbird 247	. . . . 5 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))
97		breq1 4656	. . . . . . . . 9 ⊢ (𝑚 = 𝐴 → (𝑚 ∥ 𝑘 ↔ 𝐴 ∥ 𝑘))
98		breq1 4656	. . . . . . . . 9 ⊢ (𝑚 = 𝐵 → (𝑚 ∥ 𝑘 ↔ 𝐵 ∥ 𝑘))
99		breq1 4656	. . . . . . . . 9 ⊢ (𝑚 = 𝐶 → (𝑚 ∥ 𝑘 ↔ 𝐶 ∥ 𝑘))
100	97, 98, 99	raltpg 4236	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ 𝑘 ↔ (𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘)))
101	100	ad2antlr 763	. . . . . . 7 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ 𝑘 ↔ (𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘)))
102		simpr 477	. . . . . . . . . . 11 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
103	51	ad2antlr 763	. . . . . . . . . . 11 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝐴 lcm 𝐵) ∈ ℤ)
104	52	ad2antlr 763	. . . . . . . . . . 11 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → 𝐶 ∈ ℤ)
105	102, 103, 104	3jca 1242	. . . . . . . . . 10 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℕ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ))
106	105	adantr 481	. . . . . . . . 9 ⊢ ((((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) ∧ (𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘)) → (𝑘 ∈ ℕ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ))
107		3ioran 1056	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ↔ (¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶))
108		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 = 𝐴 ↔ 𝐴 = 0)
109	108	notbii 310	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 0 = 𝐴 ↔ ¬ 𝐴 = 0)
110		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 = 𝐵 ↔ 𝐵 = 0)
111	110	notbii 310	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 0 = 𝐵 ↔ ¬ 𝐵 = 0)
112	109, 111	anbi12i 733	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) ↔ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
113	112	biimpi 206	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
114		ioran 511	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ (𝐴 = 0 ∨ 𝐵 = 0) ↔ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
115	113, 114	sylibr 224	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
116	115	3adant3 1081	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
117	107, 116	sylbi 207	. . . . . . . . . . . . . . . 16 ⊢ (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
118		id 22	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
119	118	3adant3 1081	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
120	117, 119	anim12ci 591	. . . . . . . . . . . . . . 15 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)))
121		lcmn0cl 15310	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (𝐴 lcm 𝐵) ∈ ℕ)
122	120, 121	syl 17	. . . . . . . . . . . . . 14 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 lcm 𝐵) ∈ ℕ)
123		nnne0 11053	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 lcm 𝐵) ∈ ℕ → (𝐴 lcm 𝐵) ≠ 0)
124	123	neneqd 2799	. . . . . . . . . . . . . 14 ⊢ ((𝐴 lcm 𝐵) ∈ ℕ → ¬ (𝐴 lcm 𝐵) = 0)
125	122, 124	syl 17	. . . . . . . . . . . . 13 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ (𝐴 lcm 𝐵) = 0)
126		eqcom 2629	. . . . . . . . . . . . . . . . . 18 ⊢ (0 = 𝐶 ↔ 𝐶 = 0)
127	126	notbii 310	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 0 = 𝐶 ↔ ¬ 𝐶 = 0)
128	127	biimpi 206	. . . . . . . . . . . . . . . 16 ⊢ (¬ 0 = 𝐶 → ¬ 𝐶 = 0)
129	128	3ad2ant3 1084	. . . . . . . . . . . . . . 15 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶) → ¬ 𝐶 = 0)
130	107, 129	sylbi 207	. . . . . . . . . . . . . 14 ⊢ (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ 𝐶 = 0)
131	130	adantr 481	. . . . . . . . . . . . 13 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ 𝐶 = 0)
132	125, 131	jca 554	. . . . . . . . . . . 12 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
133	132	adantr 481	. . . . . . . . . . 11 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
134	133	adantr 481	. . . . . . . . . 10 ⊢ ((((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) ∧ (𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘)) → (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
135		ioran 511	. . . . . . . . . 10 ⊢ (¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0) ↔ (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
136	134, 135	sylibr 224	. . . . . . . . 9 ⊢ ((((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) ∧ (𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘)) → ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0))
137	119	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
138		nnz 11399	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
139	137, 138	anim12ci 591	. . . . . . . . . . . . . 14 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ)))
140		3anass 1042	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (𝑘 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ)))
141	139, 140	sylibr 224	. . . . . . . . . . . . 13 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
142		lcmdvds 15321	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘) → (𝐴 lcm 𝐵) ∥ 𝑘))
143	141, 142	syl 17	. . . . . . . . . . . 12 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → ((𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘) → (𝐴 lcm 𝐵) ∥ 𝑘))
144	143	com12 32	. . . . . . . . . . 11 ⊢ ((𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘) → (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝐴 lcm 𝐵) ∥ 𝑘))
145	144	3adant3 1081	. . . . . . . . . 10 ⊢ ((𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘) → (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝐴 lcm 𝐵) ∥ 𝑘))
146	145	impcom 446	. . . . . . . . 9 ⊢ ((((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) ∧ (𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘)) → (𝐴 lcm 𝐵) ∥ 𝑘)
147		simp3 1063	. . . . . . . . . 10 ⊢ ((𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘) → 𝐶 ∥ 𝑘)
148	147	adantl 482	. . . . . . . . 9 ⊢ ((((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) ∧ (𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘)) → 𝐶 ∥ 𝑘)
149		lcmledvds 15312	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ℕ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0)) → (((𝐴 lcm 𝐵) ∥ 𝑘 ∧ 𝐶 ∥ 𝑘) → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘))
150	149	imp 445	. . . . . . . . 9 ⊢ ((((𝑘 ∈ ℕ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0)) ∧ ((𝐴 lcm 𝐵) ∥ 𝑘 ∧ 𝐶 ∥ 𝑘)) → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘)
151	106, 136, 146, 148, 150	syl22anc 1327	. . . . . . . 8 ⊢ ((((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) ∧ (𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘)) → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘)
152	151	ex 450	. . . . . . 7 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → ((𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘) → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘))
153	101, 152	sylbid 230	. . . . . 6 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ 𝑘 → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘))
154	153	ralrimiva 2966	. . . . 5 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ∀𝑘 ∈ ℕ (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ 𝑘 → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘))
155	96, 154	jca 554	. . . 4 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ ∀𝑘 ∈ ℕ (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ 𝑘 → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘)))
156	109	biimpi 206	. . . . . . . . . . . . . . . 16 ⊢ (¬ 0 = 𝐴 → ¬ 𝐴 = 0)
157	111	biimpi 206	. . . . . . . . . . . . . . . 16 ⊢ (¬ 0 = 𝐵 → ¬ 𝐵 = 0)
158	156, 157	anim12i 590	. . . . . . . . . . . . . . 15 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
159	158, 114	sylibr 224	. . . . . . . . . . . . . 14 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
160	159	3adant3 1081	. . . . . . . . . . . . 13 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
161	107, 160	sylbi 207	. . . . . . . . . . . 12 ⊢ (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
162	161, 119	anim12ci 591	. . . . . . . . . . 11 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)))
163	162, 121	syl 17	. . . . . . . . . 10 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 lcm 𝐵) ∈ ℕ)
164	163, 124	syl 17	. . . . . . . . 9 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ (𝐴 lcm 𝐵) = 0)
165	164, 131	jca 554	. . . . . . . 8 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
166	165, 135	sylibr 224	. . . . . . 7 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0))
167	54, 166	jca 554	. . . . . 6 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0)))
168		lcmn0cl 15310	. . . . . 6 ⊢ ((((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0)) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℕ)
169	167, 168	syl 17	. . . . 5 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℕ)
170	5	adantl 482	. . . . 5 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → {𝐴, 𝐵, 𝐶} ⊆ ℤ)
171		tpfi 8236	. . . . . 6 ⊢ {𝐴, 𝐵, 𝐶} ∈ Fin
172	171	a1i 11	. . . . 5 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → {𝐴, 𝐵, 𝐶} ∈ Fin)
173	3	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (0 ∈ {𝐴, 𝐵, 𝐶} ↔ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶)))
174	173	biimpd 219	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (0 ∈ {𝐴, 𝐵, 𝐶} → (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶)))
175	174	con3d 148	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ 0 ∈ {𝐴, 𝐵, 𝐶}))
176	175	impcom 446	. . . . . 6 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ 0 ∈ {𝐴, 𝐵, 𝐶})
177		df-nel 2898	. . . . . 6 ⊢ (0 ∉ {𝐴, 𝐵, 𝐶} ↔ ¬ 0 ∈ {𝐴, 𝐵, 𝐶})
178	176, 177	sylibr 224	. . . . 5 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 ∉ {𝐴, 𝐵, 𝐶})
179		lcmf 15346	. . . . 5 ⊢ ((((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℕ ∧ ({𝐴, 𝐵, 𝐶} ⊆ ℤ ∧ {𝐴, 𝐵, 𝐶} ∈ Fin ∧ 0 ∉ {𝐴, 𝐵, 𝐶})) → (((𝐴 lcm 𝐵) lcm 𝐶) = (lcm‘{𝐴, 𝐵, 𝐶}) ↔ (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ ∀𝑘 ∈ ℕ (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ 𝑘 → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘))))
180	169, 170, 172, 178, 179	syl13anc 1328	. . . 4 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (((𝐴 lcm 𝐵) lcm 𝐶) = (lcm‘{𝐴, 𝐵, 𝐶}) ↔ (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ ∀𝑘 ∈ ℕ (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ 𝑘 → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘))))
181	155, 180	mpbird 247	. . 3 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) lcm 𝐶) = (lcm‘{𝐴, 𝐵, 𝐶}))
182	181	eqcomd 2628	. 2 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (lcm‘{𝐴, 𝐵, 𝐶}) = ((𝐴 lcm 𝐵) lcm 𝐶))
183	50, 182	pm2.61ian 831	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (lcm‘{𝐴, 𝐵, 𝐶}) = ((𝐴 lcm 𝐵) lcm 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ∉ wnel 2897  ∀wral 2912   ⊆ wss 3574  {ctp 4181   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  0cc0 9936   ≤ cle 10075  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377   ∥ cdvds 14983   lcm clcm 15301  lcmclcmf 15302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636  df-dvds 14984  df-gcd 15217  df-lcm 15303  df-lcmf 15304

This theorem is referenced by:  lcmf2a3a4e12  15360