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Theorem lcmfunsnlem2lem1 16572

Description: Lemma 1 for lcmfunsnlem2 16574. (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 1909	. . 3 ⊢ Ⅎ𝑘(0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)
2		nfv 1909	. . . 4 ⊢ Ⅎ𝑘 𝑛 ∈ ℤ
3		nfv 1909	. . . . 5 ⊢ Ⅎ𝑘(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)
4		nfra1 3273	. . . . . 6 ⊢ Ⅎ𝑘∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)
5		nfv 1909	. . . . . 6 ⊢ Ⅎ𝑘∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)
6	4, 5	nfan 1894	. . . . 5 ⊢ Ⅎ𝑘(∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))
7	3, 6	nfan 1894	. . . 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 1894	. . 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 1894	. 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 770	. . . . . . . . . . . 12 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℕ)
11		simp2 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
12		snssi 4803	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℤ → {𝑧} ⊆ ℤ)
13	12	3ad2ant1 1130	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {𝑧} ⊆ ℤ)
14	11, 13	unssd 4178	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
15		simp3 1135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
16		snfi 9040	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑧} ∈ Fin
17		unfi 9168	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
18	15, 16, 17	sylancl 585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
19	14, 18	jca 511	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin))
20		lcmfcl 16562	. . . . . . . . . . . . . . . 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 12581	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
23	22	adantl 481	. . . . . . . . . . . . 13 ⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
24	23	adantr 480	. . . . . . . . . . . 12 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
25		simprl 768	. . . . . . . . . . . 12 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → 𝑛 ∈ ℤ)
26	10, 24, 25	3jca 1125	. . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
28	18	adantl 481	. . . . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 ∉ 𝑦 → ¬ 0 ∈ 𝑦)
31		elsni 4637	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 ∈ {𝑧} → 0 = 𝑧)
32	31	eqcomd 2730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 ∈ {𝑧} → 𝑧 = 0)
33	32	necon3ai 2957	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ≠ 0 → ¬ 0 ∈ {𝑧})
34	30, 33	anim12i 612	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0) → (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
35	34	3adant3 1129	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
36		df-nel 3039	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 ∉ (𝑦 ∪ {𝑧}) ↔ ¬ 0 ∈ (𝑦 ∪ {𝑧}))
37		ioran 980	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
38		elun 4140	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 ∈ (𝑦 ∪ {𝑧}) ↔ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
39	37, 38	xchnxbir 333	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 0 ∈ (𝑦 ∪ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
40	36, 39	bitri 275	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∉ (𝑦 ∪ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
41	35, 40	sylibr 233	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → 0 ∉ (𝑦 ∪ {𝑧}))
42	41	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∉ (𝑦 ∪ {𝑧}))
43	27, 28, 42	3jca 1125	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 16560	. . . . . . . . . . . . . . . 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 12259	. . . . . . . . . . . . . 14 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (lcm‘(𝑦 ∪ {𝑧})) ≠ 0)
48	47	neneqd 2937	. . . . . . . . . . . . 13 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → ¬ (lcm‘(𝑦 ∪ {𝑧})) = 0)
49		neneq 2938	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ≠ 0 → ¬ 𝑛 = 0)
50	49	3ad2ant3 1132	. . . . . . . . . . . . . 14 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 𝑛 = 0)
51	50	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ ((((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → ¬ 𝑛 = 0)
52	48, 51	jca 511	. . . . . . . . . . . 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 980	. . . . . . . . . . . 12 ⊢ (¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0) ↔ (¬ (lcm‘(𝑦 ∪ {𝑧})) = 0 ∧ ¬ 𝑛 = 0))
54	52, 53	sylibr 233	. . . . . . . . . . 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 511	. . . . . . . . . 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 436	. . . . . . . . 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 491	. . . . . . . 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 418	. . . . . 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 406	. . . . 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 480	. . . 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 4630	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑧 → {𝑛} = {𝑧})
63	62	uneq2d 4155	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑧 → (𝑦 ∪ {𝑛}) = (𝑦 ∪ {𝑧}))
64	63	fveq2d 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (lcm‘(𝑦 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑧})))
65		oveq2 7409	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → ((lcm‘𝑦) lcm 𝑛) = ((lcm‘𝑦) lcm 𝑧))
66	64, 65	eqeq12d 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑧 → ((lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
67	66	rspcv 3600	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℤ → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
68	67	3ad2ant1 1130	. . . . . . . . . . . . . . . 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 12576	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
70	69	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
71	70	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℤ)
72		lcmfcl 16562	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℕ₀)
73	72	nn0zd 12581	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
74	73	3adant1 1127	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
75	74	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . 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 1209	. . . . . . . . . . . . . . . . . . . . . . 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 1125	. . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . 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 4168	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑚 ∈ 𝑦 → 𝑚 ∈ (𝑦 ∪ {𝑧}))
80	79	orcd 870	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑚 ∈ 𝑦 → (𝑚 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑚 ∈ {𝑛}))
81		elun 4140	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑚 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑚 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑚 ∈ {𝑛}))
82	80, 81	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑚 ∈ 𝑦 → 𝑚 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
83		breq1 5141	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑖 = 𝑚 → (𝑖 ∥ 𝑘 ↔ 𝑚 ∥ 𝑘))
84	83	rspcv 3600	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → (𝑚 ∈ 𝑦 → 𝑚 ∥ 𝑘))
88	87	ralrimiv 3137	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘) → ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘)
89	88	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 𝑙 → (𝑚 ∥ 𝑘 ↔ 𝑚 ∥ 𝑙))
91	90	ralbidv 3169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 = 𝑙 → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙))
92		breq2 5142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 = 𝑙 → ((lcm‘𝑦) ∥ 𝑘 ↔ (lcm‘𝑦) ∥ 𝑙))
93	91, 92	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 = 𝑙 → ((∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ↔ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙)))
94	93	cbvralvw 3226	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ↔ ∀𝑙 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙))
95	70	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → 𝑘 ∈ ℤ)
96	95	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑙 = 𝑘 → (𝑚 ∥ 𝑙 ↔ 𝑚 ∥ 𝑘))
98	97	ralbidv 3169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑙 = 𝑘 → (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 ↔ ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘))
99		breq2 5142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑙 = 𝑘 → ((lcm‘𝑦) ∥ 𝑙 ↔ (lcm‘𝑦) ∥ 𝑘))
100	98, 99	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑙 = 𝑘 → ((∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑙 → (lcm‘𝑦) ∥ 𝑙) ↔ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
101	100	rspcv 3600	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	biimtrid 241	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . . . 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 455	. . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . 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 4657	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑧 ∈ {𝑧}
111	110	olci 863	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧})
112		elun 4140	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧}))
113	111, 112	mpbir 230	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
114	113	orci 862	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑧 ∈ {𝑛})
115		elun 4140	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑧 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑧 ∈ {𝑛}))
116	114, 115	mpbir 230	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑧 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})
117		breq1 5141	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑧 → (𝑖 ∥ 𝑘 ↔ 𝑧 ∥ 𝑘))
118	117	rspcv 3600	. . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . . . . . . . . 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 16542	. . . . . . . . . . . . . . . . . . . . 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 5141	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘 ↔ ((lcm‘𝑦) lcm 𝑧) ∥ 𝑘))
125	123, 124	imbitrrid 245	. . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 406	. . . . . . . . 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 406	. . . . . . 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 406	. . . . . 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 406	. . . . 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 4657	. . . . . . . . 9 ⊢ 𝑛 ∈ {𝑛}
141	140	olci 863	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑛 ∈ {𝑛})
142		elun 4140	. . . . . . . 8 ⊢ (𝑛 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑛 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑛 ∈ {𝑛}))
143	141, 142	mpbir 230	. . . . . . 7 ⊢ 𝑛 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})
144		breq1 5141	. . . . . . . 8 ⊢ (𝑖 = 𝑛 → (𝑖 ∥ 𝑘 ↔ 𝑛 ∥ 𝑘))
145	144	rspcv 3600	. . . . . . 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 406	. . . . 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 511	. . . 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 16533	. . . 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 419	. 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 3246	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 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∉ wnel 3038 ∀wral 3053 ∪ cun 3938 ⊆ wss 3940 {csn 4620 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 Fincfn 8935 0cc0 11106 ≤ cle 11246 ℕcn 12209 ℕ₀cn0 12469 ℤcz 12555 ∥ cdvds 16194 lcm clcm 16522 lcmclcmf 16523

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 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 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-prod 15847 df-dvds 16195 df-gcd 16433 df-lcm 16524 df-lcmf 16525

This theorem is referenced by: lcmfunsnlem2lem2 16573

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