Theorem lcmftp 16563

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 16571, 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 12499	. . . . . . 7 ⊢ 0 ∈ ℤ
2		eltpg 4643	. . . . . . 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 228	. . . . 5 ⊢ ((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → 0 ∈ {𝐴, 𝐵, 𝐶})
5		tpssi 4794	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → {𝐴, 𝐵, 𝐶} ⊆ ℤ)
6	4, 5	anim12ci 614	. . . 4 ⊢ (((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ({𝐴, 𝐵, 𝐶} ⊆ ℤ ∧ 0 ∈ {𝐴, 𝐵, 𝐶}))
7		lcmf0val 16549	. . . 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 12500	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℤ → 0 ∈ ℤ)
10		lcmcom 16520	. . . . . . . . . 10 ⊢ ((0 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (0 lcm 𝐶) = (𝐶 lcm 0))
11	9, 10	mpancom 688	. . . . . . . . 9 ⊢ (𝐶 ∈ ℤ → (0 lcm 𝐶) = (𝐶 lcm 0))
12		lcm0val 16521	. . . . . . . . 9 ⊢ (𝐶 ∈ ℤ → (𝐶 lcm 0) = 0)
13	11, 12	eqtrd 2771	. . . . . . . 8 ⊢ (𝐶 ∈ ℤ → (0 lcm 𝐶) = 0)
14	13	eqcomd 2742	. . . . . . 7 ⊢ (𝐶 ∈ ℤ → 0 = (0 lcm 𝐶))
15	14	3ad2ant3 1135	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = (0 lcm 𝐶))
16	15	adantl 481	. . . . 5 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = (0 lcm 𝐶))
17		0zd 12500	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℤ → 0 ∈ ℤ)
18		lcmcom 16520	. . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 lcm 𝐵) = (𝐵 lcm 0))
19	17, 18	mpancom 688	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℤ → (0 lcm 𝐵) = (𝐵 lcm 0))
20		lcm0val 16521	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℤ → (𝐵 lcm 0) = 0)
21	19, 20	eqtrd 2771	. . . . . . . . 9 ⊢ (𝐵 ∈ ℤ → (0 lcm 𝐵) = 0)
22	21	eqcomd 2742	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → 0 = (0 lcm 𝐵))
23	22	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = (0 lcm 𝐵))
24	23	adantl 481	. . . . . 6 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = (0 lcm 𝐵))
25	24	oveq1d 7373	. . . . 5 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (0 lcm 𝐶) = ((0 lcm 𝐵) lcm 𝐶))
26		oveq1 7365	. . . . . . 7 ⊢ (0 = 𝐴 → (0 lcm 𝐵) = (𝐴 lcm 𝐵))
27	26	oveq1d 7373	. . . . . 6 ⊢ (0 = 𝐴 → ((0 lcm 𝐵) lcm 𝐶) = ((𝐴 lcm 𝐵) lcm 𝐶))
28	27	adantr 480	. . . . 5 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((0 lcm 𝐵) lcm 𝐶) = ((𝐴 lcm 𝐵) lcm 𝐶))
29	16, 25, 28	3eqtrd 2775	. . . 4 ⊢ ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
30		lcm0val 16521	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → (𝐴 lcm 0) = 0)
31	30	eqcomd 2742	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 0 = (𝐴 lcm 0))
32	31	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = (𝐴 lcm 0))
33	32	adantl 481	. . . . . 6 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = (𝐴 lcm 0))
34	33	oveq1d 7373	. . . . 5 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (0 lcm 𝐶) = ((𝐴 lcm 0) lcm 𝐶))
35	13	3ad2ant3 1135	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (0 lcm 𝐶) = 0)
36	35	adantl 481	. . . . 5 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (0 lcm 𝐶) = 0)
37		oveq2 7366	. . . . . . 7 ⊢ (0 = 𝐵 → (𝐴 lcm 0) = (𝐴 lcm 𝐵))
38	37	adantr 480	. . . . . 6 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 lcm 0) = (𝐴 lcm 𝐵))
39	38	oveq1d 7373	. . . . 5 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 0) lcm 𝐶) = ((𝐴 lcm 𝐵) lcm 𝐶))
40	34, 36, 39	3eqtr3d 2779	. . . 4 ⊢ ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
41		lcmcl 16528	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 lcm 𝐵) ∈ ℕ0)
42	41	nn0zd 12513	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 lcm 𝐵) ∈ ℤ)
43		lcm0val 16521	. . . . . . . 8 ⊢ ((𝐴 lcm 𝐵) ∈ ℤ → ((𝐴 lcm 𝐵) lcm 0) = 0)
44	43	eqcomd 2742	. . . . . . 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 1132	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = ((𝐴 lcm 𝐵) lcm 0))
47		oveq2 7366	. . . . 5 ⊢ (0 = 𝐶 → ((𝐴 lcm 𝐵) lcm 0) = ((𝐴 lcm 𝐵) lcm 𝐶))
48	46, 47	sylan9eqr 2793	. . . 4 ⊢ ((0 = 𝐶 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
49	29, 40, 48	3jaoian 1432	. . 3 ⊢ (((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
50	8, 49	eqtrd 2771	. 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 1132	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 lcm 𝐵) ∈ ℤ)
52		simp3 1138	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐶 ∈ ℤ)
53	51, 52	jca 511	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ))
54	53	adantl 481	. . . . . . . 8 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ))
55		dvdslcm 16525	. . . . . . . 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 16525	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∥ (𝐴 lcm 𝐵) ∧ 𝐵 ∥ (𝐴 lcm 𝐵)))
58	57	3adant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∥ (𝐴 lcm 𝐵) ∧ 𝐵 ∥ (𝐴 lcm 𝐵)))
59		simp1 1136	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐴 ∈ ℤ)
60		lcmcl 16528	. . . . . . . . . . . . . . . . . . . 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 12513	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ)
63	59, 51, 62	3jca 1128	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ ℤ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ))
64		dvdstr 16221	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 481	. . . . . . . . . . 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 480	. . . . . . . . 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 407	. . . . . . . 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 484	. . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐵 ∥ (𝐴 lcm 𝐵))
77	76	adantl 481	. . . . . . . . . . . 12 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐵 ∥ (𝐴 lcm 𝐵))
78		simp2 1137	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐵 ∈ ℤ)
79	78, 51, 62	3jca 1128	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵 ∈ ℤ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ))
80	79	adantl 481	. . . . . . . . . . . . 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 16221	. . . . . . . . . . . . 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 695	. . . . . . . . . . 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 480	. . . . . . . . 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 407	. . . . . . . 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 484	. . . . . . . . 9 ⊢ (((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))
88	87	adantl 481	. . . . . . . 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 1128	. . . . . . 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 687	. . . . . 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 5101	. . . . . . . 8 ⊢ (𝑚 = 𝐴 → (𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
92		breq1 5101	. . . . . . . 8 ⊢ (𝑚 = 𝐵 → (𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
93		breq1 5101	. . . . . . . 8 ⊢ (𝑚 = 𝐶 → (𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
94	91, 92, 93	raltpg 4655	. . . . . . 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 481	. . . . . 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 257	. . . . 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 5101	. . . . . . . . 9 ⊢ (𝑚 = 𝐴 → (𝑚 ∥ 𝑘 ↔ 𝐴 ∥ 𝑘))
98		breq1 5101	. . . . . . . . 9 ⊢ (𝑚 = 𝐵 → (𝑚 ∥ 𝑘 ↔ 𝐵 ∥ 𝑘))
99		breq1 5101	. . . . . . . . 9 ⊢ (𝑚 = 𝐶 → (𝑚 ∥ 𝑘 ↔ 𝐶 ∥ 𝑘))
100	97, 98, 99	raltpg 4655	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ 𝑘 ↔ (𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘)))
101	100	ad2antlr 727	. . . . . . 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 484	. . . . . . . . . . 11 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
103	51	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝐴 lcm 𝐵) ∈ ℤ)
104	52	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → 𝐶 ∈ ℤ)
105	102, 103, 104	3jca 1128	. . . . . . . . . 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 480	. . . . . . . . 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 1105	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ↔ (¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶))
108		eqcom 2743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 = 𝐴 ↔ 𝐴 = 0)
109	108	notbii 320	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 0 = 𝐴 ↔ ¬ 𝐴 = 0)
110		eqcom 2743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 = 𝐵 ↔ 𝐵 = 0)
111	110	notbii 320	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 0 = 𝐵 ↔ ¬ 𝐵 = 0)
112	109, 111	anbi12i 628	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) ↔ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
113	112	biimpi 216	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
114		ioran 985	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ (𝐴 = 0 ∨ 𝐵 = 0) ↔ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
115	113, 114	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
116	115	3adant3 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
117	107, 116	sylbi 217	. . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
120	117, 119	anim12ci 614	. . . . . . . . . . . . . . 15 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)))
121		lcmn0cl 16524	. . . . . . . . . . . . . . 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 12179	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 lcm 𝐵) ∈ ℕ → (𝐴 lcm 𝐵) ≠ 0)
124	123	neneqd 2937	. . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . 18 ⊢ (0 = 𝐶 ↔ 𝐶 = 0)
127	126	notbii 320	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 0 = 𝐶 ↔ ¬ 𝐶 = 0)
128	127	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ (¬ 0 = 𝐶 → ¬ 𝐶 = 0)
129	128	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶) → ¬ 𝐶 = 0)
130	107, 129	sylbi 217	. . . . . . . . . . . . . 14 ⊢ (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ 𝐶 = 0)
131	130	adantr 480	. . . . . . . . . . . . 13 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ 𝐶 = 0)
132	125, 131	jca 511	. . . . . . . . . . . 12 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
133	132	adantr 480	. . . . . . . . . . 11 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
134	133	adantr 480	. . . . . . . . . 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 985	. . . . . . . . . 10 ⊢ (¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0) ↔ (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
136	134, 135	sylibr 234	. . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
138		nnz 12509	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
139	137, 138	anim12ci 614	. . . . . . . . . . . . . 14 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ)))
140		3anass 1094	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (𝑘 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ)))
141	139, 140	sylibr 234	. . . . . . . . . . . . 13 ⊢ (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
142		lcmdvds 16535	. . . . . . . . . . . . 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 1132	. . . . . . . . . 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 407	. . . . . . . . 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 1138	. . . . . . . . . 10 ⊢ ((𝐴 ∥ 𝑘 ∧ 𝐵 ∥ 𝑘 ∧ 𝐶 ∥ 𝑘) → 𝐶 ∥ 𝑘)
148	147	adantl 481	. . . . . . . . 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 16526	. . . . . . . . . 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 406	. . . . . . . . 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 838	. . . . . . . 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 412	. . . . . . 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 240	. . . . . 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 3128	. . . . 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 511	. . . 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 216	. . . . . . . . . . . . . . . 16 ⊢ (¬ 0 = 𝐴 → ¬ 𝐴 = 0)
157	111	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ (¬ 0 = 𝐵 → ¬ 𝐵 = 0)
158	156, 157	anim12i 613	. . . . . . . . . . . . . . 15 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
159	158, 114	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
160	159	3adant3 1132	. . . . . . . . . . . . 13 ⊢ ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
161	107, 160	sylbi 217	. . . . . . . . . . . 12 ⊢ (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
162	161, 119	anim12ci 614	. . . . . . . . . . 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 511	. . . . . . . 8 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
166	165, 135	sylibr 234	. . . . . . 7 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0))
167	54, 166	jca 511	. . . . . 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 16524	. . . . . 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 481	. . . . 5 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → {𝐴, 𝐵, 𝐶} ⊆ ℤ)
171		tpfi 9226	. . . . . 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 229	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (0 ∈ {𝐴, 𝐵, 𝐶} → (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶)))
175	174	con3d 152	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ 0 ∈ {𝐴, 𝐵, 𝐶}))
176	175	impcom 407	. . . . . 6 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ 0 ∈ {𝐴, 𝐵, 𝐶})
177		df-nel 3037	. . . . . 6 ⊢ (0 ∉ {𝐴, 𝐵, 𝐶} ↔ ¬ 0 ∈ {𝐴, 𝐵, 𝐶})
178	176, 177	sylibr 234	. . . . 5 ⊢ ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 ∉ {𝐴, 𝐵, 𝐶})
179		lcmf 16560	. . . . 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 1374	. . . 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 257	. . 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 2742	. 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 811	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (lcm‘{𝐴, 𝐵, 𝐶}) = ((𝐴 lcm 𝐵) lcm 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ∉ wnel 3036  ∀wral 3051   ⊆ wss 3901  {ctp 4584   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Fincfn 8883  0cc0 11026   ≤ cle 11167  ℕcn 12145  ℕ₀cn0 12401  ℤcz 12488   ∥ cdvds 16179   lcm clcm 16515  lcmclcmf 16516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-n0 12402  df-z 12489  df-uz 12752  df-rp 12906  df-fz 13424  df-fzo 13571  df-fl 13712  df-mod 13790  df-seq 13925  df-exp 13985  df-hash 14254  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-clim 15411  df-prod 15827  df-dvds 16180  df-gcd 16422  df-lcm 16517  df-lcmf 16518

This theorem is referenced by:  lcmf2a3a4e12  16574