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Theorem lcmfunsnlem2lem1 16080

Description: Lemma 1 for lcmfunsnlem2 16082. (Contributed by AV, 26-Aug-2020.)

**Assertion**
Ref	Expression
lcmfunsnlem2lem1	⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((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 ∈ ℕ (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘))

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

Proof of Theorem lcmfunsnlem2lem1 Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1920	. . 3 ⊢ Ⅎ𝑘(0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)
2		nfv 1920	. . . 4 ⊢ Ⅎ𝑘 𝑛 ∈ ℤ
3		nfv 1920	. . . . 5 ⊢ Ⅎ𝑘(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)
4		nfra1 3131	. . . . . 6 ⊢ Ⅎ𝑘∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)
5		nfv 1920	. . . . . 6 ⊢ Ⅎ𝑘∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)
6	4, 5	nfan 1905	. . . . 5 ⊢ Ⅎ𝑘(∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))
7	3, 6	nfan 1905	. . . 4 ⊢ 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)))
8	2, 7	nfan 1905	. . 3 ⊢ Ⅎ𝑘(𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))))
9	1, 8	nfan 1905	. 2 ⊢ Ⅎ𝑘((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)))))
10		simprr 773	. . . . . . . . . . . 12 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℕ)
11		simp2 1138	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
12		snssi 4697	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℤ → {𝑧} ⊆ ℤ)
13	12	3ad2ant1 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {𝑧} ⊆ ℤ)
14	11, 13	unssd 4077	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
15		simp3 1139	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
16		snfi 8643	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑧} ∈ Fin
17		unfi 8772	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
18	15, 16, 17	sylancl 589	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
19	14, 18	jca 515	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin))
20		lcmfcl 16070	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
21	19, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
22	21	nn0zd 12167	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
23	22	adantl 485	. . . . . . . . . . . . 13 ⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
24	23	adantr 484	. . . . . . . . . . . 12 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
25		simprl 771	. . . . . . . . . . . 12 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → 𝑛 ∈ ℤ)
26	10, 24, 25	3jca 1129	. . . . . . . . . . 11 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ))
27	14	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
28	18	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ∈ Fin)
29		df-nel 3039	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 ∉ 𝑦 ↔ ¬ 0 ∈ 𝑦)
30	29	biimpi 219	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 ∉ 𝑦 → ¬ 0 ∈ 𝑦)
31		elsni 4534	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 ∈ {𝑧} → 0 = 𝑧)
32	31	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 ∈ {𝑧} → 𝑧 = 0)
33	32	necon3ai 2959	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ≠ 0 → ¬ 0 ∈ {𝑧})
34	30, 33	anim12i 616	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0) → (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
35	34	3adant3 1133	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
36		df-nel 3039	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 ∉ (𝑦 ∪ {𝑧}) ↔ ¬ 0 ∈ (𝑦 ∪ {𝑧}))
37		ioran 983	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
38		elun 4040	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 ∈ (𝑦 ∪ {𝑧}) ↔ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
39	37, 38	xchnxbir 336	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 0 ∈ (𝑦 ∪ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
40	36, 39	bitri 278	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∉ (𝑦 ∪ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
41	35, 40	sylibr 237	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → 0 ∉ (𝑦 ∪ {𝑧}))
42	41	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∉ (𝑦 ∪ {𝑧}))
43	27, 28, 42	3jca 1129	. . . . . . . . . . . . . . . . 17 ⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin ∧ 0 ∉ (𝑦 ∪ {𝑧})))
44	43	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin ∧ 0 ∉ (𝑦 ∪ {𝑧})))
45		lcmfn0cl 16068	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin ∧ 0 ∉ (𝑦 ∪ {𝑧})) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ)
46	44, 45	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ)
47	46	nnne0d 11767	. . . . . . . . . . . . . 14 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (lcm‘(𝑦 ∪ {𝑧})) ≠ 0)
48	47	neneqd 2939	. . . . . . . . . . . . 13 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → ¬ (lcm‘(𝑦 ∪ {𝑧})) = 0)
49		neneq 2940	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ≠ 0 → ¬ 𝑛 = 0)
50	49	3ad2ant3 1136	. . . . . . . . . . . . . 14 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 𝑛 = 0)
51	50	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → ¬ 𝑛 = 0)
52	48, 51	jca 515	. . . . . . . . . . . 12 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (¬ (lcm‘(𝑦 ∪ {𝑧})) = 0 ∧ ¬ 𝑛 = 0))
53		ioran 983	. . . . . . . . . . . 12 ⊢ (¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0) ↔ (¬ (lcm‘(𝑦 ∪ {𝑧})) = 0 ∧ ¬ 𝑛 = 0))
54	52, 53	sylibr 237	. . . . . . . . . . 11 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0))
55	26, 54	jca 515	. . . . . . . . . 10 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → ((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0)))
56	55	exp43 440	. . . . . . . . 9 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑛 ∈ ℤ → (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0))))))
57	56	adantrd 495	. . . . . . . 8 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → (𝑛 ∈ ℤ → (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0))))))
58	57	com23 86	. . . . . . 7 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (𝑛 ∈ ℤ → (((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 ∈ ℕ → ((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0))))))
59	58	imp32 422	. . . . . 6 ⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((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 ∈ ℕ → ((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0))))
60	59	imp 410	. . . . 5 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((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 ∈ ℕ) → ((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0)))
61	60	adantr 484	. . . 4 ⊢ (((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((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 ∈ ℕ) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → ((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0)))
62		sneq 4527	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑧 → {𝑛} = {𝑧})
63	62	uneq2d 4054	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑧 → (𝑦 ∪ {𝑛}) = (𝑦 ∪ {𝑧}))
64	63	fveq2d 6679	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (lcm‘(𝑦 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑧})))
65		oveq2 7179	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → ((lcm‘𝑦) lcm 𝑛) = ((lcm‘𝑦) lcm 𝑧))
66	64, 65	eqeq12d 2754	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑧 → ((lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
67	66	rspcv 3522	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℤ → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
68	67	3ad2ant1 1134	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
69		nnz 12086	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
70	69	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
71	70	adantl 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℤ)
72		lcmfcl 16070	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℕ₀)
73	72	nn0zd 12167	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
74	73	3adant1 1131	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
75	74	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (lcm‘𝑦) ∈ ℤ)
76		simpll1 1213	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → 𝑧 ∈ ℤ)
77	71, 75, 76	3jca 1129	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (𝑘 ∈ ℤ ∧ (lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
78	77	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → (𝑘 ∈ ℤ ∧ (lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
79		elun1 4067	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑚 ∈ 𝑦 → 𝑚 ∈ (𝑦 ∪ {𝑧}))
80	79	orcd 872	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑚 ∈ 𝑦 → (𝑚 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑚 ∈ {𝑛}))
81		elun 4040	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑚 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑚 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑚 ∈ {𝑛}))
82	80, 81	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑚 ∈ 𝑦 → 𝑚 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
83		breq1 5034	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑖 = 𝑚 → (𝑖 ∥ 𝑘 ↔ 𝑚 ∥ 𝑘))
84	83	rspcv 3522	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑚 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → 𝑚 ∥ 𝑘))
85	82, 84	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑚 ∈ 𝑦 → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → 𝑚 ∥ 𝑘))
86	85	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (𝑚 ∈ 𝑦 → 𝑚 ∥ 𝑘))
87	86	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → (𝑚 ∈ 𝑦 → 𝑚 ∥ 𝑘))
88	87	ralrimiv 3095	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘)
89	88	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) ∧ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0))) → ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘)
90		breq2 5035	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 𝑙 → (𝑚 ∥ 𝑘 ↔ 𝑚 ∥ 𝑙))
91	90	ralbidv 3109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 = 𝑙 → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙))
92		breq2 5035	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 = 𝑙 → ((lcm‘𝑦) ∥ 𝑘 ↔ (lcm‘𝑦) ∥ 𝑙))
93	91, 92	imbi12d 348	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 = 𝑙 → ((∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ↔ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙)))
94	93	cbvralvw 3349	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ↔ ∀𝑙 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙))
95	70	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → 𝑘 ∈ ℤ)
96	95	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) ∧ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0))) → 𝑘 ∈ ℤ)
97		breq2 5035	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑙 = 𝑘 → (𝑚 ∥ 𝑙 ↔ 𝑚 ∥ 𝑘))
98	97	ralbidv 3109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑙 = 𝑘 → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 ↔ ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘))
99		breq2 5035	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑙 = 𝑘 → ((lcm‘𝑦) ∥ 𝑙 ↔ (lcm‘𝑦) ∥ 𝑘))
100	98, 99	imbi12d 348	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑙 = 𝑘 → ((∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙) ↔ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
101	100	rspcv 3522	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 ∈ ℤ → (∀𝑙 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙) → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
102	96, 101	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) ∧ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0))) → (∀𝑙 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙) → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
103	94, 102	syl5bi 245	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) ∧ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0))) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
104	89, 103	mpid 44	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) ∧ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0))) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) → (lcm‘𝑦) ∥ 𝑘))
105	104	exp31 423	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) → (lcm‘𝑦) ∥ 𝑘))))
106	105	com24 95	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) → (((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘))))
107	106	imp 410	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) → (((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
108	107	impl 459	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘))
109	108	imp 410	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → (lcm‘𝑦) ∥ 𝑘)
110		vsnid 4554	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑧 ∈ {𝑧}
111	110	olci 865	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧})
112		elun 4040	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧}))
113	111, 112	mpbir 234	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
114	113	orci 864	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑧 ∈ {𝑛})
115		elun 4040	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑧 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑧 ∈ {𝑛}))
116	114, 115	mpbir 234	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑧 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})
117		breq1 5034	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑧 → (𝑖 ∥ 𝑘 ↔ 𝑧 ∥ 𝑘))
118	117	rspcv 3522	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → 𝑧 ∥ 𝑘))
119	116, 118	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → 𝑧 ∥ 𝑘))
120	119	imp 410	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → 𝑧 ∥ 𝑘)
121	109, 120	jca 515	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → ((lcm‘𝑦) ∥ 𝑘 ∧ 𝑧 ∥ 𝑘))
122		lcmdvds 16050	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℤ ∧ (lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((lcm‘𝑦) ∥ 𝑘 ∧ 𝑧 ∥ 𝑘) → ((lcm‘𝑦) lcm 𝑧) ∥ 𝑘))
123	78, 121, 122	sylc 65	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → ((lcm‘𝑦) lcm 𝑧) ∥ 𝑘)
124		breq1 5034	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘 ↔ ((lcm‘𝑦) lcm 𝑧) ∥ 𝑘))
125	123, 124	syl5ibr 249	. . . . . . . . . . . . . . . . . . 19 ⊢ ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → ((((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
126	125	expd 419	. . . . . . . . . . . . . . . . . 18 ⊢ ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
127	126	exp5j 449	. . . . . . . . . . . . . . . . 17 ⊢ ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) → ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))))
128	127	com12 32	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) → ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))))
129	68, 128	syld 47	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) → ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))))
130	129	com23 86	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))))
131	130	imp32 422	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))
132	131	expd 419	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → (𝑛 ∈ ℤ → (𝑘 ∈ ℕ → ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))))
133	132	com34 91	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → (𝑛 ∈ ℤ → ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (𝑘 ∈ ℕ → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))))
134	133	com12 32	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (𝑘 ∈ ℕ → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))))
135	134	imp 410	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)))) → ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (𝑘 ∈ ℕ → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))
136	135	com12 32	. . . . . . . 8 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ((𝑛 ∈ ℤ ∧ ((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 ∈ ℕ → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))
137	136	imp 410	. . . . . . 7 ⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((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 ∈ ℕ → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
138	137	imp 410	. . . . . 6 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((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 ∈ ℕ) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
139	138	imp 410	. . . . 5 ⊢ (((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((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 ∈ ℕ) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)
140		vsnid 4554	. . . . . . . . 9 ⊢ 𝑛 ∈ {𝑛}
141	140	olci 865	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑛 ∈ {𝑛})
142		elun 4040	. . . . . . . 8 ⊢ (𝑛 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑛 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑛 ∈ {𝑛}))
143	141, 142	mpbir 234	. . . . . . 7 ⊢ 𝑛 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})
144		breq1 5034	. . . . . . . 8 ⊢ (𝑖 = 𝑛 → (𝑖 ∥ 𝑘 ↔ 𝑛 ∥ 𝑘))
145	144	rspcv 3522	. . . . . . 7 ⊢ (𝑛 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → 𝑛 ∥ 𝑘))
146	143, 145	mp1i 13	. . . . . 6 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((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 ∈ ℕ) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → 𝑛 ∥ 𝑘))
147	146	imp 410	. . . . 5 ⊢ (((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((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 ∈ ℕ) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → 𝑛 ∥ 𝑘)
148	139, 147	jca 515	. . . 4 ⊢ (((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((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 ∈ ℕ) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘 ∧ 𝑛 ∥ 𝑘))
149		lcmledvds 16041	. . . 4 ⊢ (((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0)) → (((lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘 ∧ 𝑛 ∥ 𝑘) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘))
150	61, 148, 149	sylc 65	. . 3 ⊢ (((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((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 ∈ ℕ) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘)
151	150	exp31 423	. 2 ⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((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 ∈ ℕ → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘)))
152	9, 151	ralrimi 3128	1 ⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((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 ∈ ℕ (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ∉ wnel 3038 ∀wral 3053 ∪ cun 3842 ⊆ wss 3844 {csn 4517 class class class wbr 5031 ‘cfv 6340 (class class class)co 7171 Fincfn 8556 0cc0 10616 ≤ cle 10755 ℕcn 11717 ℕ₀cn0 11977 ℤcz 12063 ∥ cdvds 15700 lcm clcm 16030 lcmclcmf 16031

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-inf2 9178 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693 ax-pre-sup 10694

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-om 7601 df-1st 7715 df-2nd 7716 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-1o 8132 df-er 8321 df-en 8557 df-dom 8558 df-sdom 8559 df-fin 8560 df-sup 8980 df-inf 8981 df-oi 9048 df-card 9442 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-div 11377 df-nn 11718 df-2 11780 df-3 11781 df-n0 11978 df-z 12064 df-uz 12326 df-rp 12474 df-fz 12983 df-fzo 13126 df-fl 13254 df-mod 13330 df-seq 13462 df-exp 13523 df-hash 13784 df-cj 14549 df-re 14550 df-im 14551 df-sqrt 14685 df-abs 14686 df-clim 14936 df-prod 15353 df-dvds 15701 df-gcd 15939 df-lcm 16032 df-lcmf 16033

This theorem is referenced by: lcmfunsnlem2lem2 16081

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