Theorem lcmfunsnlem1 16691

Description: Lemma for lcmfdvds 16696 and lcmfunsnlem 16695 (Induction step part 1). (Contributed by AV, 25-Aug-2020.)

Assertion

Ref	Expression
lcmfunsnlem1	⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → ∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))

Distinct variable groups:   𝑦,𝑚,𝑧   𝑘,𝑛,𝑦,𝑧   𝑘,𝑚

Proof of Theorem lcmfunsnlem1

Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1941	. . 3 ⊢ Ⅎ𝑘(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)
2		nfra1 3295	. . . 4 ⊢ Ⅎ𝑘∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)
3		nfv 1941	. . . 4 ⊢ Ⅎ𝑘∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)
4	2, 3	nfan 1926	. . 3 ⊢ Ⅎ𝑘(∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))
5	1, 4	nfan 1926	. 2 ⊢ Ⅎ𝑘((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)))
6		breq2 5114	. . . . . . . 8 ⊢ (𝑘 = 𝑙 → (𝑚 ∥ 𝑘 ↔ 𝑚 ∥ 𝑙))
7	6	ralbidv 3194	. . . . . . 7 ⊢ (𝑘 = 𝑙 → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙))
8		breq2 5114	. . . . . . 7 ⊢ (𝑘 = 𝑙 → ((lcm‘𝑦) ∥ 𝑘 ↔ (lcm‘𝑦) ∥ 𝑙))
9	7, 8	imbi12d 347	. . . . . 6 ⊢ (𝑘 = 𝑙 → ((∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ↔ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙)))
10	9	cbvralvw 3249	. . . . 5 ⊢ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ↔ ∀𝑙 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙))
11		breq2 5114	. . . . . . . . . 10 ⊢ (𝑙 = 𝑘 → (𝑚 ∥ 𝑙 ↔ 𝑚 ∥ 𝑘))
12	11	ralbidv 3194	. . . . . . . . 9 ⊢ (𝑙 = 𝑘 → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 ↔ ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘))
13		breq2 5114	. . . . . . . . 9 ⊢ (𝑙 = 𝑘 → ((lcm‘𝑦) ∥ 𝑙 ↔ (lcm‘𝑦) ∥ 𝑘))
14	12, 13	imbi12d 347	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → ((∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙) ↔ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
15	14	rspcv 3586	. . . . . . 7 ⊢ (𝑘 ∈ ℤ → (∀𝑙 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙) → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
16	15	adantl 486	. . . . . 6 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑙 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙) → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
17		sneq 4601	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑧 → {𝑛} = {𝑧})
18	17	uneq2d 4130	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑧 → (𝑦 ∪ {𝑛}) = (𝑦 ∪ {𝑧}))
19	18	fveq2d 6883	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑧 → (lcm‘(𝑦 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑧})))
20		oveq2 7416	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑧 → ((lcm‘𝑦) lcm 𝑛) = ((lcm‘𝑦) lcm 𝑧))
21	19, 20	eqeq12d 2785	. . . . . . . . . 10 ⊢ (𝑛 = 𝑧 → ((lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
22	21	rspcv 3586	. . . . . . . . 9 ⊢ (𝑧 ∈ ℤ → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
23	22	3ad2ant1 1149	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
24	23	adantr 485	. . . . . . 7 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
25		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℤ)
26		lcmfcl 16682	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℕ0)
27	26	nn0zd 12612	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
28	27	3adant1 1146	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
29	28	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (lcm‘𝑦) ∈ ℤ)
30		simpl1 1208	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → 𝑧 ∈ ℤ)
31	25, 29, 30	3jca 1144	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ℤ ∧ (lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
32	31	adantr 485	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) → (𝑘 ∈ ℤ ∧ (lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
33	32	adantr 485	. . . . . . . . . . . 12 ⊢ (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘) → (𝑘 ∈ ℤ ∧ (lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
34		ssun1 4139	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
35		ssralv 4014	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘))
36	34, 35	mp1i 14	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘))
37	36	imim1d 83	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → ((∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
38	37	imp31 422	. . . . . . . . . . . . 13 ⊢ (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘) → (lcm‘𝑦) ∥ 𝑘)
39		snidg 4628	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ {𝑧})
40	39	olcd 887	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℤ → (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧}))
41		elun 4115	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧}))
42	40, 41	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ (𝑦 ∪ {𝑧}))
43		breq1 5113	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑧 → (𝑚 ∥ 𝑘 ↔ 𝑧 ∥ 𝑘))
44	43	rspcv 3586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → 𝑧 ∥ 𝑘))
45	42, 44	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℤ → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → 𝑧 ∥ 𝑘))
46	45	3ad2ant1 1149	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → 𝑧 ∥ 𝑘))
47	46	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → 𝑧 ∥ 𝑘))
48	47	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → 𝑧 ∥ 𝑘))
49	48	imp 411	. . . . . . . . . . . . 13 ⊢ (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘) → 𝑧 ∥ 𝑘)
50	38, 49	jca 520	. . . . . . . . . . . 12 ⊢ (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘) → ((lcm‘𝑦) ∥ 𝑘 ∧ 𝑧 ∥ 𝑘))
51		lcmdvds 16662	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ (lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((lcm‘𝑦) ∥ 𝑘 ∧ 𝑧 ∥ 𝑘) → ((lcm‘𝑦) lcm 𝑧) ∥ 𝑘))
52	33, 50, 51	sylc 66	. . . . . . . . . . 11 ⊢ (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘) → ((lcm‘𝑦) lcm 𝑧) ∥ 𝑘)
53		breq1 5113	. . . . . . . . . . 11 ⊢ ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘 ↔ ((lcm‘𝑦) lcm 𝑧) ∥ 𝑘))
54	52, 53	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘) → ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
55	54	ex 417	. . . . . . . . 9 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
56	55	com23 87	. . . . . . . 8 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) → ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
57	56	ex 417	. . . . . . 7 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → ((∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) → ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))
58	24, 57	syl5d 74	. . . . . 6 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → ((∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))
59	16, 58	syld 48	. . . . 5 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑙 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))
60	10, 59	biimtrid 245	. . . 4 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))
61	60	impd 415	. . 3 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → ((∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
62	61	impancom 456	. 2 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → (𝑘 ∈ ℤ → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
63	5, 62	ralrimi 3269	1 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → ∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ∪ cun 3911   ⊆ wss 3913  {csn 4591   class class class wbr 5110  ‘cfv 6533  (class class class)co 7408  Fincfn 8939  ℤcz 12587   ∥ cdvds 16306   lcm clcm 16642  lcmclcmf 16643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-inf2 9606  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173  ax-pre-sup 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-oi 9468  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-div 11868  df-nn 12230  df-2 12299  df-3 12300  df-n0 12501  df-z 12588  df-uz 12859  df-rp 13013  df-fz 13532  df-fzo 13679  df-fl 13821  df-mod 13899  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-prod 15954  df-dvds 16307  df-gcd 16549  df-lcm 16644  df-lcmf 16645

This theorem is referenced by:  lcmfunsnlem  16695