Theorem lcmfunsnlem1 16684

Description: Lemma for lcmfdvds 16689 and lcmfunsnlem 16688 (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 1913	. . 3 ⊢ Ⅎ𝑘(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)
2		nfra1 3290	. . . 4 ⊢ Ⅎ𝑘∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)
3		nfv 1913	. . . 4 ⊢ Ⅎ𝑘∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)
4	2, 3	nfan 1898	. . 3 ⊢ Ⅎ𝑘(∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))
5	1, 4	nfan 1898	. 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 5170	. . . . . . . 8 ⊢ (𝑘 = 𝑙 → (𝑚 ∥ 𝑘 ↔ 𝑚 ∥ 𝑙))
7	6	ralbidv 3184	. . . . . . 7 ⊢ (𝑘 = 𝑙 → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙))
8		breq2 5170	. . . . . . 7 ⊢ (𝑘 = 𝑙 → ((lcm‘𝑦) ∥ 𝑘 ↔ (lcm‘𝑦) ∥ 𝑙))
9	7, 8	imbi12d 344	. . . . . 6 ⊢ (𝑘 = 𝑙 → ((∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ↔ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙)))
10	9	cbvralvw 3243	. . . . 5 ⊢ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ↔ ∀𝑙 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙))
11		breq2 5170	. . . . . . . . . 10 ⊢ (𝑙 = 𝑘 → (𝑚 ∥ 𝑙 ↔ 𝑚 ∥ 𝑘))
12	11	ralbidv 3184	. . . . . . . . 9 ⊢ (𝑙 = 𝑘 → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 ↔ ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘))
13		breq2 5170	. . . . . . . . 9 ⊢ (𝑙 = 𝑘 → ((lcm‘𝑦) ∥ 𝑙 ↔ (lcm‘𝑦) ∥ 𝑘))
14	12, 13	imbi12d 344	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → ((∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙) ↔ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
15	14	rspcv 3631	. . . . . . 7 ⊢ (𝑘 ∈ ℤ → (∀𝑙 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙) → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
16	15	adantl 481	. . . . . 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 4658	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑧 → {𝑛} = {𝑧})
18	17	uneq2d 4191	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑧 → (𝑦 ∪ {𝑛}) = (𝑦 ∪ {𝑧}))
19	18	fveq2d 6924	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑧 → (lcm‘(𝑦 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑧})))
20		oveq2 7456	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑧 → ((lcm‘𝑦) lcm 𝑛) = ((lcm‘𝑦) lcm 𝑧))
21	19, 20	eqeq12d 2756	. . . . . . . . . 10 ⊢ (𝑛 = 𝑧 → ((lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
22	21	rspcv 3631	. . . . . . . . 9 ⊢ (𝑧 ∈ ℤ → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
23	22	3ad2ant1 1133	. . . . . . . 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 480	. . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℤ)
26		lcmfcl 16675	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℕ0)
27	26	nn0zd 12665	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
28	27	3adant1 1130	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
29	28	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (lcm‘𝑦) ∈ ℤ)
30		simpl1 1191	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → 𝑧 ∈ ℤ)
31	25, 29, 30	3jca 1128	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ℤ ∧ (lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
32	31	adantr 480	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 4201	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
35		ssralv 4077	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘))
36	34, 35	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘))
37	36	imim1d 82	. . . . . . . . . . . . . 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 417	. . . . . . . . . . . . 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 4682	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ {𝑧})
40	39	olcd 873	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℤ → (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧}))
41		elun 4176	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧}))
42	40, 41	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ (𝑦 ∪ {𝑧}))
43		breq1 5169	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑧 → (𝑚 ∥ 𝑘 ↔ 𝑧 ∥ 𝑘))
44	43	rspcv 3631	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → 𝑧 ∥ 𝑘))
45	42, 44	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℤ → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → 𝑧 ∥ 𝑘))
46	45	3ad2ant1 1133	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → 𝑧 ∥ 𝑘))
47	46	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → 𝑧 ∥ 𝑘))
48	47	adantr 480	. . . . . . . . . . . . . 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 406	. . . . . . . . . . . . 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 511	. . . . . . . . . . . 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 16655	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℤ ∧ (lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((lcm‘𝑦) ∥ 𝑘 ∧ 𝑧 ∥ 𝑘) → ((lcm‘𝑦) lcm 𝑧) ∥ 𝑘))
52	33, 50, 51	sylc 65	. . . . . . . . . . 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 5169	. . . . . . . . . . 11 ⊢ ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘 ↔ ((lcm‘𝑦) lcm 𝑧) ∥ 𝑘))
54	52, 53	syl5ibrcom 247	. . . . . . . . . 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 412	. . . . . . . . 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 86	. . . . . . . 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 412	. . . . . . 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 73	. . . . . 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 47	. . . . 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 242	. . . 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 410	. . 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 451	. 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 3263	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 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∪ cun 3974   ⊆ wss 3976  {csn 4648   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  ℤcz 12639   ∥ cdvds 16302   lcm clcm 16635  lcmclcmf 16636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-fz 13568  df-fzo 13712  df-fl 13843  df-mod 13921  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-prod 15952  df-dvds 16303  df-gcd 16541  df-lcm 16637  df-lcmf 16638

This theorem is referenced by:  lcmfunsnlem  16688