Theorem lcmfunsnlem1 16712

Description: Lemma for lcmfdvds 16717 and lcmfunsnlem 16716 (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 1947	. . 3 ⊢ Ⅎ𝑘(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)
2		nfra1 3291	. . . 4 ⊢ Ⅎ𝑘∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)
3		nfv 1947	. . . 4 ⊢ Ⅎ𝑘∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)
4	2, 3	nfan 1932	. . 3 ⊢ Ⅎ𝑘(∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))
5	1, 4	nfan 1932	. 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 5115	. . . . . . . 8 ⊢ (𝑘 = 𝑙 → (𝑚 ∥ 𝑘 ↔ 𝑚 ∥ 𝑙))
7	6	ralbidv 3190	. . . . . . 7 ⊢ (𝑘 = 𝑙 → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙))
8		breq2 5115	. . . . . . 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 3245	. . . . 5 ⊢ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ↔ ∀𝑙 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙))
11		breq2 5115	. . . . . . . . . 10 ⊢ (𝑙 = 𝑘 → (𝑚 ∥ 𝑙 ↔ 𝑚 ∥ 𝑘))
12	11	ralbidv 3190	. . . . . . . . 9 ⊢ (𝑙 = 𝑘 → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 ↔ ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘))
13		breq2 5115	. . . . . . . . 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 3579	. . . . . . 7 ⊢ (𝑘 ∈ ℤ → (∀𝑙 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙) → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
16	15	adantl 487	. . . . . 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 4122	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑧 → (𝑦 ∪ {𝑛}) = (𝑦 ∪ {𝑧}))
19	18	fveq2d 6889	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑧 → (lcm‘(𝑦 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑧})))
20		oveq2 7424	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑧 → ((lcm‘𝑦) lcm 𝑛) = ((lcm‘𝑦) lcm 𝑧))
21	19, 20	eqeq12d 2781	. . . . . . . . . 10 ⊢ (𝑛 = 𝑧 → ((lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
22	21	rspcv 3579	. . . . . . . . 9 ⊢ (𝑧 ∈ ℤ → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
23	22	3ad2ant1 1151	. . . . . . . 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 486	. . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℤ)
26		lcmfcl 16703	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℕ0)
27	26	nn0zd 12627	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
28	27	3adant1 1148	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
29	28	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (lcm‘𝑦) ∈ ℤ)
30		simpl1 1210	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → 𝑧 ∈ ℤ)
31	25, 29, 30	3jca 1146	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ℤ ∧ (lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
32	31	adantr 486	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 4131	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
35		ssralv 4007	. . . . . . . . . . . . . . . 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 423	. . . . . . . . . . . . 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 888	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℤ → (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧}))
41		elun 4107	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧}))
42	40, 41	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ (𝑦 ∪ {𝑧}))
43		breq1 5114	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑧 → (𝑚 ∥ 𝑘 ↔ 𝑧 ∥ 𝑘))
44	43	rspcv 3579	. . . . . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → 𝑧 ∥ 𝑘))
47	46	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → 𝑧 ∥ 𝑘))
48	47	adantr 486	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 521	. . . . . . . . . . . 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 16683	. . . . . . . . . . . 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 5114	. . . . . . . . . . 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 418	. . . . . . . . 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 418	. . . . . . 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 416	. . 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 457	. 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 3265	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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ∪ cun 3904   ⊆ wss 3906  {csn 4591   class class class wbr 5111  ‘cfv 6540  (class class class)co 7416  Fincfn 8945  ℤcz 12602   ∥ cdvds 16327   lcm clcm 16663  lcmclcmf 16664

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 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-inf2 9613  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188  ax-pre-sup 11189

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9405  df-inf 9406  df-oi 9475  df-card 9937  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-div 11883  df-nn 12245  df-2 12314  df-3 12315  df-n0 12516  df-z 12603  df-uz 12874  df-rp 13028  df-fz 13547  df-fzo 13695  df-fl 13838  df-mod 13916  df-seq 14051  df-exp 14111  df-hash 14380  df-cj 15169  df-re 15170  df-im 15171  df-sqrt 15305  df-abs 15306  df-clim 15558  df-prod 15976  df-dvds 16328  df-gcd 16570  df-lcm 16665  df-lcmf 16666

This theorem is used by:  lcmfunsnlem  16716