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Theorem lcmfunsnlem2lem2 15983

Description: Lemma 2 for lcmfunsnlem2 15984. (Contributed by AV, 26-Aug-2020.)

**Assertion**
Ref	Expression
lcmfunsnlem2lem2	⊢ (((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))))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))

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

Proof of Theorem lcmfunsnlem2lem2 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 4125	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑖 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑖 ∈ {𝑛}))
2		elun 4125	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑖 ∈ 𝑦 ∨ 𝑖 ∈ {𝑧}))
3		simp1 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑧 ∈ ℤ)
4	3	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑧 ∈ ℤ)
5	4	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ 𝑦 ∨ 𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑧 ∈ ℤ)
6		sneq 4577	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑧 → {𝑛} = {𝑧})
7	6	uneq2d 4139	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (𝑦 ∪ {𝑛}) = (𝑦 ∪ {𝑧}))
8	7	fveq2d 6674	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑧 → (lcm‘(𝑦 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑧})))
9		oveq2 7164	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑧 → ((lcm‘𝑦) lcm 𝑛) = ((lcm‘𝑦) lcm 𝑧))
10	8, 9	eqeq12d 2837	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑧 → ((lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
11	10	rspcv 3618	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ℤ → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
12	5, 11	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ 𝑦 ∨ 𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
13		breq1 5069	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 𝑖 → (𝑘 ∥ (lcm‘𝑦) ↔ 𝑖 ∥ (lcm‘𝑦)))
14	13	rspcv 3618	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ 𝑦 → (∀𝑘 ∈ 𝑦 𝑘 ∥ (lcm‘𝑦) → 𝑖 ∥ (lcm‘𝑦)))
15		dvdslcmf 15975	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ∀𝑘 ∈ 𝑦 𝑘 ∥ (lcm‘𝑦))
16	15	3adant1 1126	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ∀𝑘 ∈ 𝑦 𝑘 ∥ (lcm‘𝑦))
17	16	adantr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ∀𝑘 ∈ 𝑦 𝑘 ∥ (lcm‘𝑦))
18	14, 17	impel 508	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ (lcm‘𝑦))
19		lcmfcl 15972	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℕ₀)
20	19	nn0zd 12086	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
21	20	3adant1 1126	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
22	21	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (lcm‘𝑦) ∈ ℤ)
23		lcmcl 15945	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 lcm 𝑛) ∈ ℕ₀)
24	3, 23	sylan 582	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑧 lcm 𝑛) ∈ ℕ₀)
25	24	nn0zd 12086	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑧 lcm 𝑛) ∈ ℤ)
26	22, 25	jca 514	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm‘𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ))
27	26	adantl 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm‘𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ))
28		dvdslcm 15942	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((lcm‘𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ) → ((lcm‘𝑦) ∥ ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)) ∧ (𝑧 lcm 𝑛) ∥ ((lcm‘𝑦) lcm (𝑧 lcm 𝑛))))
29	28	simpld 497	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((lcm‘𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ) → (lcm‘𝑦) ∥ ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)))
30	27, 29	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (lcm‘𝑦) ∥ ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)))
31		ssel 3961	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ⊆ ℤ → (𝑖 ∈ 𝑦 → 𝑖 ∈ ℤ))
32	31	3ad2ant2 1130	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑖 ∈ 𝑦 → 𝑖 ∈ ℤ))
33	32	adantr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑖 ∈ 𝑦 → 𝑖 ∈ ℤ))
34	33	impcom 410	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∈ ℤ)
35	22	adantl 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (lcm‘𝑦) ∈ ℤ)
36	25	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (𝑧 lcm 𝑛) ∈ ℤ)
37		lcmcl 15945	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((lcm‘𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ) → ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)) ∈ ℕ₀)
38	35, 36, 37	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)) ∈ ℕ₀)
39	38	nn0zd 12086	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)) ∈ ℤ)
40		dvdstr 15646	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ ℤ ∧ (lcm‘𝑦) ∈ ℤ ∧ ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)) ∈ ℤ) → ((𝑖 ∥ (lcm‘𝑦) ∧ (lcm‘𝑦) ∥ ((lcm‘𝑦) lcm (𝑧 lcm 𝑛))) → 𝑖 ∥ ((lcm‘𝑦) lcm (𝑧 lcm 𝑛))))
41	34, 35, 39, 40	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((𝑖 ∥ (lcm‘𝑦) ∧ (lcm‘𝑦) ∥ ((lcm‘𝑦) lcm (𝑧 lcm 𝑛))) → 𝑖 ∥ ((lcm‘𝑦) lcm (𝑧 lcm 𝑛))))
42	18, 30, 41	mp2and 697	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)))
43	4	adantl 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑧 ∈ ℤ)
44		simprr 771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℤ)
45		lcmass 15958	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((lcm‘𝑦) lcm 𝑧) lcm 𝑛) = ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)))
46	35, 43, 44, 45	syl3anc 1367	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (((lcm‘𝑦) lcm 𝑧) lcm 𝑛) = ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)))
47	42, 46	breqtrrd 5094	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛))
48	47	ex 415	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ 𝑦 → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
49		elsni 4584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ {𝑧} → 𝑖 = 𝑧)
50	21, 3	jca 514	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
51	50	adantr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
52		dvdslcm 15942	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((lcm‘𝑦) ∥ ((lcm‘𝑦) lcm 𝑧) ∧ 𝑧 ∥ ((lcm‘𝑦) lcm 𝑧)))
53	52	simprd 498	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → 𝑧 ∥ ((lcm‘𝑦) lcm 𝑧))
54	51, 53	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑧 ∥ ((lcm‘𝑦) lcm 𝑧))
55	19	3adant1 1126	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℕ₀)
56	55	nn0zd 12086	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
57		lcmcl 15945	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((lcm‘𝑦) lcm 𝑧) ∈ ℕ₀)
58	56, 3, 57	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm‘𝑦) lcm 𝑧) ∈ ℕ₀)
59	58	nn0zd 12086	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm‘𝑦) lcm 𝑧) ∈ ℤ)
60		dvdslcm 15942	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((lcm‘𝑦) lcm 𝑧) ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((lcm‘𝑦) lcm 𝑧) ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛) ∧ 𝑛 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
61	60	simpld 497	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((lcm‘𝑦) lcm 𝑧) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((lcm‘𝑦) lcm 𝑧) ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛))
62	59, 61	sylan 582	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm‘𝑦) lcm 𝑧) ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛))
63	59	adantr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm‘𝑦) lcm 𝑧) ∈ ℤ)
64		lcmcl 15945	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((lcm‘𝑦) lcm 𝑧) ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((lcm‘𝑦) lcm 𝑧) lcm 𝑛) ∈ ℕ₀)
65	59, 64	sylan 582	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (((lcm‘𝑦) lcm 𝑧) lcm 𝑛) ∈ ℕ₀)
66	65	nn0zd 12086	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (((lcm‘𝑦) lcm 𝑧) lcm 𝑛) ∈ ℤ)
67		dvdstr 15646	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℤ ∧ ((lcm‘𝑦) lcm 𝑧) ∈ ℤ ∧ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛) ∈ ℤ) → ((𝑧 ∥ ((lcm‘𝑦) lcm 𝑧) ∧ ((lcm‘𝑦) lcm 𝑧) ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)) → 𝑧 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
68	4, 63, 66, 67	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((𝑧 ∥ ((lcm‘𝑦) lcm 𝑧) ∧ ((lcm‘𝑦) lcm 𝑧) ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)) → 𝑧 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
69	54, 62, 68	mp2and 697	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑧 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛))
70		breq1 5069	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑧 → (𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛) ↔ 𝑧 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
71	69, 70	syl5ibr 248	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑧 → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
72	49, 71	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ {𝑧} → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
73	48, 72	jaoi 853	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ 𝑦 ∨ 𝑖 ∈ {𝑧}) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
74	73	imp 409	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ 𝑦 ∨ 𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛))
75		oveq1 7163	. . . . . . . . . . . . . . . . 17 ⊢ ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (((lcm‘𝑦) lcm 𝑧) lcm 𝑛))
76	75	breq2d 5078	. . . . . . . . . . . . . . . 16 ⊢ ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → (𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ↔ 𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
77	74, 76	syl5ibrcom 249	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ 𝑦 ∨ 𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
78	12, 77	syld 47	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ 𝑦 ∨ 𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
79	78	ex 415	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ 𝑦 ∨ 𝑖 ∈ {𝑧}) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
80	2, 79	sylbi 219	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (𝑦 ∪ {𝑧}) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
81		elsni 4584	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ {𝑛} → 𝑖 = 𝑛)
82		simp2 1133	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
83		snssi 4741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ ℤ → {𝑧} ⊆ ℤ)
84	83	3ad2ant1 1129	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {𝑧} ⊆ ℤ)
85	82, 84	unssd 4162	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
86		simp3 1134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
87		snfi 8594	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {𝑧} ∈ Fin
88		unfi 8785	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
89	86, 87, 88	sylancl 588	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
90		lcmfcl 15972	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
91	85, 89, 90	syl2anc 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
92	91	nn0zd 12086	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
93	92	anim1i 616	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ))
94	93	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)) → ((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ))
95		dvdslcm 15942	. . . . . . . . . . . . . . . . 17 ⊢ (((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∧ 𝑛 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
96	94, 95	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∧ 𝑛 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
97	96	simprd 498	. . . . . . . . . . . . . . 15 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)) → 𝑛 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
98		breq1 5069	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → (𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ↔ 𝑛 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
99	97, 98	syl5ibr 248	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑛 → ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
100	99	expd 418	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑛 → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
101	81, 100	syl 17	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ {𝑛} → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
102	80, 101	jaoi 853	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑖 ∈ {𝑛}) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
103	1, 102	sylbi 219	. . . . . . . . . 10 ⊢ (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
104	103	com13 88	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
105	104	expd 418	. . . . . . . 8 ⊢ (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) → ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑛 ∈ ℤ → (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))
106	105	adantl 484	. . . . . . 7 ⊢ ((∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)) → ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑛 ∈ ℤ → (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))
107	106	impcom 410	. . . . . 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))) → (𝑛 ∈ ℤ → (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
108	107	impcom 410	. . . . 5 ⊢ ((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)))) → (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
109	108	adantl 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))))) → (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
110	109	ralrimiv 3181	. . 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))))) → ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
111		lcmfunsnlem2lem1 15982	. . 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))
112	93	adantr 483	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ))
113	85	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
114	89	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑦 ∪ {𝑧}) ∈ Fin)
115		df-nel 3124	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∉ 𝑦 ↔ ¬ 0 ∈ 𝑦)
116	115	biimpi 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∉ 𝑦 → ¬ 0 ∈ 𝑦)
117	116	3ad2ant1 1129	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ 𝑦)
118		elsni 4584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 ∈ {𝑧} → 0 = 𝑧)
119	118	eqcomd 2827	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ {𝑧} → 𝑧 = 0)
120	119	necon3ai 3041	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ≠ 0 → ¬ 0 ∈ {𝑧})
121	120	3ad2ant2 1130	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ {𝑧})
122		ioran 980	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
123	117, 121, 122	sylanbrc 585	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
124		elun 4125	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ (𝑦 ∪ {𝑧}) ↔ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
125	123, 124	sylnibr 331	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ (𝑦 ∪ {𝑧}))
126		df-nel 3124	. . . . . . . . . . . . . . . 16 ⊢ (0 ∉ (𝑦 ∪ {𝑧}) ↔ ¬ 0 ∈ (𝑦 ∪ {𝑧}))
127	125, 126	sylibr 236	. . . . . . . . . . . . . . 15 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → 0 ∉ (𝑦 ∪ {𝑧}))
128		lcmfn0cl 15970	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin ∧ 0 ∉ (𝑦 ∪ {𝑧})) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ)
129	113, 114, 127, 128	syl2an3an 1418	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ)
130	129	nnne0d 11688	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (lcm‘(𝑦 ∪ {𝑧})) ≠ 0)
131	130	neneqd 3021	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ¬ (lcm‘(𝑦 ∪ {𝑧})) = 0)
132		neneq 3022	. . . . . . . . . . . . . 14 ⊢ (𝑛 ≠ 0 → ¬ 𝑛 = 0)
133	132	3ad2ant3 1131	. . . . . . . . . . . . 13 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 𝑛 = 0)
134	133	adantl 484	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ¬ 𝑛 = 0)
135		ioran 980	. . . . . . . . . . . 12 ⊢ (¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0) ↔ (¬ (lcm‘(𝑦 ∪ {𝑧})) = 0 ∧ ¬ 𝑛 = 0))
136	131, 134, 135	sylanbrc 585	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0))
137		lcmn0cl 15941	. . . . . . . . . . 11 ⊢ ((((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0)) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ)
138	112, 136, 137	syl2anc 586	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ)
139		snssi 4741	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℤ → {𝑛} ⊆ ℤ)
140	139	adantl 484	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → {𝑛} ⊆ ℤ)
141	113, 140	unssd 4162	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
142	141	adantr 483	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
143	87, 88	mpan2 689	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
144		snfi 8594	. . . . . . . . . . . . . . 15 ⊢ {𝑛} ∈ Fin
145		unfi 8785	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∪ {𝑧}) ∈ Fin ∧ {𝑛} ∈ Fin) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
146	143, 144, 145	sylancl 588	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
147	146	3ad2ant3 1131	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
148	147	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
149	148	adantr 483	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
150		elun 4125	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}))
151		nnel 3132	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 0 ∉ 𝑦 ↔ 0 ∈ 𝑦)
152	151	biimpri 230	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ 𝑦 → ¬ 0 ∉ 𝑦)
153	152	3mix1d 1332	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ 𝑦 → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
154		nne 3020	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑧 ≠ 0 ↔ 𝑧 = 0)
155	119, 154	sylibr 236	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ {𝑧} → ¬ 𝑧 ≠ 0)
156	155	3mix2d 1333	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ {𝑧} → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
157	153, 156	jaoi 853	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ 𝑦 ∨ 0 ∈ {𝑧}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
158	124, 157	sylbi 219	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ (𝑦 ∪ {𝑧}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
159		elsni 4584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ {𝑛} → 0 = 𝑛)
160	159	eqcomd 2827	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ {𝑛} → 𝑛 = 0)
161		nne 3020	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑛 ≠ 0 ↔ 𝑛 = 0)
162	160, 161	sylibr 236	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ {𝑛} → ¬ 𝑛 ≠ 0)
163	162	3mix3d 1334	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ {𝑛} → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
164	158, 163	jaoi 853	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
165	150, 164	sylbi 219	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
166		3ianor 1103	. . . . . . . . . . . . . . 15 ⊢ (¬ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ↔ (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
167	165, 166	sylibr 236	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → ¬ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0))
168	167	con2i 141	. . . . . . . . . . . . 13 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
169		df-nel 3124	. . . . . . . . . . . . 13 ⊢ (0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ ¬ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
170	168, 169	sylibr 236	. . . . . . . . . . . 12 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
171	170	adantl 484	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
172	142, 149, 171	3jca 1124	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛})))
173	138, 172	jca 514	. . . . . . . . 9 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ ∧ (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))))
174	173	ex 415	. . . . . . . 8 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ ∧ (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛})))))
175	174	ex 415	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑛 ∈ ℤ → ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ ∧ (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))))))
176	175	adantr 483	. . . . . 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))) → (𝑛 ∈ ℤ → ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ ∧ (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))))))
177	176	impcom 410	. . . . 5 ⊢ ((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) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ ∧ (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛})))))
178	177	impcom 410	. . . 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))))) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ ∧ (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))))
179		lcmf 15977	. . . 4 ⊢ ((((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ ∧ (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) ↔ (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∧ ∀𝑘 ∈ ℕ (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘))))
180	178, 179	syl 17	. . 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))))) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) ↔ (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∧ ∀𝑘 ∈ ℕ (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ 𝑘 → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘))))
181	110, 111, 180	mpbir2and 711	. 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))))) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})))
182	181	eqcomd 2827	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))))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∨ w3o 1082 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∉ wnel 3123 ∀wral 3138 ∪ cun 3934 ⊆ wss 3936 {csn 4567 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 0cc0 10537 ≤ cle 10676 ℕcn 11638 ℕ₀cn0 11898 ℤcz 11982 ∥ cdvds 15607 lcm clcm 15932 lcmclcmf 15933

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-prod 15260 df-dvds 15608 df-gcd 15844 df-lcm 15934 df-lcmf 15935

This theorem is referenced by: lcmfunsnlem2 15984

W3C validator