Theorem lcmftp 16719

Description: The least common multiple of a triple of integers is the least common multiple of the third integer and the least common multiple of the first two integers. Although there would be a shorter proof using lcmfunsn 16727, 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 12620	. . . . . . 7 ⊢ 0 ∈ ℤ
2		eltpg 4657	. . . . . . 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 231	. . . . 5 ⊢ ((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → 0 ∈ {𝐴, 𝐵, 𝐶})
5		tpssi 4808	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → {𝐴, 𝐵, 𝐶} ⊆ ℤ)
6	4, 5	anim12ci 626	. . . 4 ⊢ (((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ({𝐴, 𝐵, 𝐶} ⊆ ℤ ∧ 0 ∈ {𝐴, 𝐵, 𝐶}))
7		lcmf0val 16705	. . . 4 ⊢ (({𝐴, 𝐵, 𝐶} ⊆ ℤ ∧ 0 ∈ {𝐴, 𝐵, 𝐶}) → (lcm‘{𝐴, 𝐵, 𝐶}) = 0)
8	6, 7	syl 18	. . 3 ⊢ (((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (lcm‘{𝐴, 𝐵, 𝐶}) = 0)
9		0zd 12621	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℤ → 0 ∈ ℤ)
10		lcmcom 16676	. . . . . . . . . 10 ⊢ ((0 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (0 lcm 𝐶) = (𝐶 lcm 0))
11	9, 10	mpancom 701	. . . . . . . . 9 ⊢ (𝐶 ∈ ℤ → (0 lcm 𝐶) = (𝐶 lcm 0))
12		lcm0val 16677	. . . . . . . . 9 ⊢ (𝐶 ∈ ℤ → (𝐶 lcm 0) = 0)
13	11, 12	eqtrd 2801	. . . . . . . 8 ⊢ (𝐶 ∈ ℤ → (0 lcm 𝐶) = 0)
14	13	eqcomd 2772	. . . . . . 7 ⊢ (𝐶 ∈ ℤ → 0 = (0 lcm 𝐶))
15	14	3ad2ant3 1153	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = (0 lcm 𝐶))
16	15	adantl 487	. . . . 5 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = (0 lcm 𝐶))
17		0zd 12621	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℤ → 0 ∈ ℤ)
18		lcmcom 16676	. . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 lcm 𝐵) = (𝐵 lcm 0))
19	17, 18	mpancom 701	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℤ → (0 lcm 𝐵) = (𝐵 lcm 0))
20		lcm0val 16677	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℤ → (𝐵 lcm 0) = 0)
21	19, 20	eqtrd 2801	. . . . . . . . 9 ⊢ (𝐵 ∈ ℤ → (0 lcm 𝐵) = 0)
22	21	eqcomd 2772	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → 0 = (0 lcm 𝐵))
23	22	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = (0 lcm 𝐵))
24	23	adantl 487	. . . . . 6 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = (0 lcm 𝐵))
25	24	oveq1d 7438	. . . . 5 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (0 lcm 𝐶) = ((0 lcm 𝐵) lcm 𝐶))
26		oveq1 7430	. . . . . . 7 ⊢ (0 = 𝐴 → (0 lcm 𝐵) = (𝐴 lcm 𝐵))
27	26	oveq1d 7438	. . . . . 6 ⊢ (0 = 𝐴 → ((0 lcm 𝐵) lcm 𝐶) = ((𝐴 lcm 𝐵) lcm 𝐶))
28	27	adantr 486	. . . . 5 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((0 lcm 𝐵) lcm 𝐶) = ((𝐴 lcm 𝐵) lcm 𝐶))
29	16, 25, 28	3eqtrd 2805	. . . 4 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
30		lcm0val 16677	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → (𝐴 lcm 0) = 0)
31	30	eqcomd 2772	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 0 = (𝐴 lcm 0))
32	31	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = (𝐴 lcm 0))
33	32	adantl 487	. . . . . 6 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = (𝐴 lcm 0))
34	33	oveq1d 7438	. . . . 5 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (0 lcm 𝐶) = ((𝐴 lcm 0) lcm 𝐶))
35	13	3ad2ant3 1153	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (0 lcm 𝐶) = 0)
36	35	adantl 487	. . . . 5 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (0 lcm 𝐶) = 0)
37		oveq2 7431	. . . . . . 7 ⊢ (0 = 𝐵 → (𝐴 lcm 0) = (𝐴 lcm 𝐵))
38	37	adantr 486	. . . . . 6 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 lcm 0) = (𝐴 lcm 𝐵))
39	38	oveq1d 7438	. . . . 5 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 0) lcm 𝐶) = ((𝐴 lcm 𝐵) lcm 𝐶))
40	34, 36, 39	3eqtr3d 2809	. . . 4 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
41		lcmcl 16684	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 lcm 𝐵) ∈ ℕ0)
42	41	nn0zd 12634	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 lcm 𝐵) ∈ ℤ)
43		lcm0val 16677	. . . . . . . 8 ⊢ ((𝐴 lcm 𝐵) ∈ ℤ → ((𝐴 lcm 𝐵) lcm 0) = 0)
44	43	eqcomd 2772	. . . . . . 7 ⊢ ((𝐴 lcm 𝐵) ∈ ℤ → 0 = ((𝐴 lcm 𝐵) lcm 0))
45	42, 44	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 0 = ((𝐴 lcm 𝐵) lcm 0))
46	45	3adant3 1150	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = ((𝐴 lcm 𝐵) lcm 0))
47		oveq2 7431	. . . . 5 ⊢ (0 = 𝐶 → ((𝐴 lcm 𝐵) lcm 0) = ((𝐴 lcm 𝐵) lcm 𝐶))
48	46, 47	sylan9eqr 2823	. . . 4 ⊢ ((0 = 𝐶 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
49	29, 40, 48	3jaoian 1457	. . 3 ⊢ (((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
50	8, 49	eqtrd 2801	. 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 1150	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 lcm 𝐵) ∈ ℤ)
52		simp3 1156	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐶 ∈ ℤ)
53	51, 52	jca 521	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ))
54	53	adantl 487	. . . . . . . 8 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ))
55		dvdslcm 16681	. . . . . . . 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 18	. . . . . . 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 16681	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∥ (𝐴 lcm 𝐵) ∧ 𝐵 ∥ (𝐴 lcm 𝐵)))
58	57	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∥ (𝐴 lcm 𝐵) ∧ 𝐵 ∥ (𝐴 lcm 𝐵)))
59		simp1 1154	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐴 ∈ ℤ)
60		lcmcl 16684	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℕ0)
61	53, 60	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℕ0)
62	61	nn0zd 12634	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ)
63	59, 51, 62	3jca 1146	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ ℤ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ))
64		dvdstr 16377	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 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 421	. . . . . . . . . . . . . . 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 33	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 39	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
70	69	adantl 487	. . . . . . . . . . 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 33	. . . . . . . . . 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 486	. . . . . . . . 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 413	. . . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∥ (𝐴 lcm 𝐵) ∧ 𝐵 ∥ (𝐴 lcm 𝐵)) → 𝐵 ∥ (𝐴 lcm 𝐵))
75	57, 74	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∥ (𝐴 lcm 𝐵))
76	75	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐵 ∥ (𝐴 lcm 𝐵))
77	76	adantl 487	. . . . . . . . . . . 12 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐵 ∥ (𝐴 lcm 𝐵))
78		simp2 1155	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐵 ∈ ℤ)
79	78, 51, 62	3jca 1146	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵 ∈ ℤ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ))
80	79	adantl 487	. . . . . . . . . . . . 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 16377	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 708	. . . . . . . . . . 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 33	. . . . . . . . . 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 486	. . . . . . . . 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 413	. . . . . . . 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 490	. . . . . . . . 9 ⊢ (((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))
88	87	adantl 487	. . . . . . . 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 1146	. . . . . . 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 700	. . . . . 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 5117	. . . . . . . 8 ⊢ (𝑚 = 𝐴 → (𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
92		breq1 5117	. . . . . . . 8 ⊢ (𝑚 = 𝐵 → (𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
93		breq1 5117	. . . . . . . 8 ⊢ (𝑚 = 𝐶 → (𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
94	91, 92, 93	raltpg 4669	. . . . . . 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 487	. . . . . 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 260	. . . . 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 5117	. . . . . . . . 9 ⊢ (𝑚 = 𝐴 → (𝑚 ∥ 𝑘 ↔ 𝐴 ∥ 𝑘))
98		breq1 5117	. . . . . . . . 9 ⊢ (𝑚 = 𝐵 → (𝑚 ∥ 𝑘 ↔ 𝐵 ∥ 𝑘))
99		breq1 5117	. . . . . . . . 9 ⊢ (𝑚 = 𝐶 → (𝑚 ∥ 𝑘 ↔ 𝐶 ∥ 𝑘))
100	97, 98, 99	raltpg 4669	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ 𝑘 ↔ (𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘)))
101	100	ad2antlr 740	. . . . . . 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 490	. . . . . . . . . . 11 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
103	51	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝐴 lcm 𝐵) ∈ ℤ)
104	52	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → 𝐶 ∈ ℤ)
105	102, 103, 104	3jca 1146	. . . . . . . . . 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 486	. . . . . . . . 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 1123	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ↔ (¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶))
108		eqcom 2773	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 = 𝐴 ↔ 𝐴 = 0)
109	108	notbii 323	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 0 = 𝐴 ↔ ¬ 𝐴 = 0)
110		eqcom 2773	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 = 𝐵 ↔ 𝐵 = 0)
111	110	notbii 323	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 0 = 𝐵 ↔ ¬ 𝐵 = 0)
112	109, 111	anbi12i 640	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) ↔ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
113	112	biimpi 219	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
114		ioran 999	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ (𝐴 = 0 ∨ 𝐵 = 0) ↔ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
115	113, 114	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
116	115	3adant3 1150	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
117	107, 116	sylbi 220	. . . . . . . . . . . . . . . 16 ⊢ (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
118		id 23	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
119	118	3adant3 1150	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
120	117, 119	anim12ci 626	. . . . . . . . . . . . . . 15 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)))
121		lcmn0cl 16680	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (𝐴 lcm 𝐵) ∈ ℕ)
122	120, 121	syl 18	. . . . . . . . . . . . . 14 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 lcm 𝐵) ∈ ℕ)
123		nnne0 12288	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 lcm 𝐵) ∈ ℕ → (𝐴 lcm 𝐵) ≠ 0)
124	123	neneqd 2966	. . . . . . . . . . . . . 14 ⊢ ((𝐴 lcm 𝐵) ∈ ℕ → ¬ (𝐴 lcm 𝐵) = 0)
125	122, 124	syl 18	. . . . . . . . . . . . 13 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ (𝐴 lcm 𝐵) = 0)
126		eqcom 2773	. . . . . . . . . . . . . . . . . 18 ⊢ (0 = 𝐶 ↔ 𝐶 = 0)
127	126	notbii 323	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 0 = 𝐶 ↔ ¬ 𝐶 = 0)
128	127	biimpi 219	. . . . . . . . . . . . . . . 16 ⊢ (¬ 0 = 𝐶 → ¬ 𝐶 = 0)
129	128	3ad2ant3 1153	. . . . . . . . . . . . . . 15 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶) → ¬ 𝐶 = 0)
130	107, 129	sylbi 220	. . . . . . . . . . . . . 14 ⊢ (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ 𝐶 = 0)
131	130	adantr 486	. . . . . . . . . . . . 13 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ 𝐶 = 0)
132	125, 131	jca 521	. . . . . . . . . . . 12 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
133	132	adantr 486	. . . . . . . . . . 11 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
134	133	adantr 486	. . . . . . . . . 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 999	. . . . . . . . . 10 ⊢ (¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0) ↔ (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
136	134, 135	sylibr 237	. . . . . . . . 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 487	. . . . . . . . . . . . . . 15 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
138		nnz 12630	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
139	137, 138	anim12ci 626	. . . . . . . . . . . . . 14 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ)))
140		3anass 1111	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (𝑘 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ)))
141	139, 140	sylibr 237	. . . . . . . . . . . . 13 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
142		lcmdvds 16691	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘) → (𝐴 lcm 𝐵) ∥ 𝑘))
143	141, 142	syl 18	. . . . . . . . . . . 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 33	. . . . . . . . . . 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 1150	. . . . . . . . . 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 413	. . . . . . . . 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 1156	. . . . . . . . . 10 ⊢ ((𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘) → 𝐶 ∥ 𝑘)
148	147	adantl 487	. . . . . . . . 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 16682	. . . . . . . . . 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 412	. . . . . . . . 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 852	. . . . . . . 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 418	. . . . . . 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 243	. . . . . 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 3160	. . . . 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 521	. . . 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 219	. . . . . . . . . . . . . . . 16 ⊢ (¬ 0 = 𝐴 → ¬ 𝐴 = 0)
157	111	biimpi 219	. . . . . . . . . . . . . . . 16 ⊢ (¬ 0 = 𝐵 → ¬ 𝐵 = 0)
158	156, 157	anim12i 625	. . . . . . . . . . . . . . 15 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
159	158, 114	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
160	159	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
161	107, 160	sylbi 220	. . . . . . . . . . . 12 ⊢ (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
162	161, 119	anim12ci 626	. . . . . . . . . . 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 18	. . . . . . . . . 10 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 lcm 𝐵) ∈ ℕ)
164	163, 124	syl 18	. . . . . . . . 9 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ (𝐴 lcm 𝐵) = 0)
165	164, 131	jca 521	. . . . . . . 8 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
166	165, 135	sylibr 237	. . . . . . 7 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0))
167	54, 166	jca 521	. . . . . 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 16680	. . . . . 6 ⊢ ((((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0)) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℕ)
169	167, 168	syl 18	. . . . 5 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℕ)
170	5	adantl 487	. . . . 5 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → {𝐴, 𝐵, 𝐶} ⊆ ℤ)
171		tpfi 9295	. . . . . 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 232	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (0 ∈ {𝐴, 𝐵, 𝐶} → (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶)))
175	174	con3d 153	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ 0 ∈ {𝐴, 𝐵, 𝐶}))
176	175	impcom 413	. . . . . 6 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ 0 ∈ {𝐴, 𝐵, 𝐶})
177		df-nel 3068	. . . . . 6 ⊢ (0 ∉ {𝐴, 𝐵, 𝐶} ↔ ¬ 0 ∈ {𝐴, 𝐵, 𝐶})
178	176, 177	sylibr 237	. . . . 5 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 ∉ {𝐴, 𝐵, 𝐶})
179		lcmf 16716	. . . . 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 1399	. . . 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 260	. . 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 2772	. 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 824	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (lcm‘{𝐴, 𝐵, 𝐶}) = ((𝐴 lcm 𝐵) lcm 𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ∉ wnel 3067  ∀wral 3082   ⊆ wss 3908  {ctp 4598   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  Fincfn 8952  0cc0 11118   ≤ cle 11262  ℕcn 12251  ℕ₀cn0 12522  ℤcz 12609   ∥ cdvds 16335   lcm clcm 16671  lcmclcmf 16672

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-inf2 9620  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195  ax-pre-sup 11196

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-inf 9413  df-oi 9482  df-card 9944  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-div 11890  df-nn 12252  df-2 12321  df-3 12322  df-n0 12523  df-z 12610  df-uz 12881  df-rp 13035  df-fz 13554  df-fzo 13702  df-fl 13845  df-mod 13923  df-seq 14058  df-exp 14118  df-hash 14387  df-cj 15176  df-re 15177  df-im 15178  df-sqrt 15312  df-abs 15313  df-clim 15565  df-prod 15984  df-dvds 16336  df-gcd 16578  df-lcm 16673  df-lcmf 16674

This theorem is used by:  lcmf2a3a4e12  16730