Theorem lcmfunsnlem2lem2 16552

Description: Lemma 2 for lcmfunsnlem2 16553. (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 4102	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑖 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑖 ∈ {𝑛}))
2		elun 4102	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑖 ∈ 𝑦 ∨ 𝑖 ∈ {𝑧}))
3		simp1 1136	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑧 ∈ ℤ)
4	3	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑧 ∈ ℤ)
5	4	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ 𝑦 ∨ 𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑧 ∈ ℤ)
6		sneq 4585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑧 → {𝑛} = {𝑧})
7	6	uneq2d 4117	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑧 → (𝑦 ∪ {𝑛}) = (𝑦 ∪ {𝑧}))
8	7	fveq2d 6832	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑧 → (lcm‘(𝑦 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑧})))
9		oveq2 7360	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑧 → ((lcm‘𝑦) lcm 𝑛) = ((lcm‘𝑦) lcm 𝑧))
10	8, 9	eqeq12d 2749	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑧 → ((lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧)))
11	10	rspcv 3569	. . . . . . . . . . . . . . . 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		ssel 3924	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ⊆ ℤ → (𝑖 ∈ 𝑦 → 𝑖 ∈ ℤ))
14	13	3ad2ant2 1134	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑖 ∈ 𝑦 → 𝑖 ∈ ℤ))
15	14	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑖 ∈ 𝑦 → 𝑖 ∈ ℤ))
16	15	impcom 407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∈ ℤ)
17		lcmfcl 16541	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℕ0)
18	17	nn0zd 12500	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
19	18	3adant1 1130	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
20	19	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (lcm‘𝑦) ∈ ℤ)
21	20	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (lcm‘𝑦) ∈ ℤ)
22		lcmcl 16514	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 lcm 𝑛) ∈ ℕ0)
23	3, 22	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑧 lcm 𝑛) ∈ ℕ0)
24	23	nn0zd 12500	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑧 lcm 𝑛) ∈ ℤ)
25	24	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (𝑧 lcm 𝑛) ∈ ℤ)
26		lcmcl 16514	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((lcm‘𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ) → ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)) ∈ ℕ0)
27	21, 25, 26	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)) ∈ ℕ0)
28	27	nn0zd 12500	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)) ∈ ℤ)
29		breq1 5096	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 𝑖 → (𝑘 ∥ (lcm‘𝑦) ↔ 𝑖 ∥ (lcm‘𝑦)))
30	29	rspcv 3569	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ 𝑦 → (∀𝑘 ∈ 𝑦 𝑘 ∥ (lcm‘𝑦) → 𝑖 ∥ (lcm‘𝑦)))
31		dvdslcmf 16544	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ∀𝑘 ∈ 𝑦 𝑘 ∥ (lcm‘𝑦))
32	31	3adant1 1130	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ∀𝑘 ∈ 𝑦 𝑘 ∥ (lcm‘𝑦))
33	32	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ∀𝑘 ∈ 𝑦 𝑘 ∥ (lcm‘𝑦))
34	30, 33	impel 505	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ (lcm‘𝑦))
35	20, 24	jca 511	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm‘𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ))
36	35	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm‘𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ))
37		dvdslcm 16511	. . . . . . . . . . . . . . . . . . . . . . 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))))
38	37	simpld 494	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((lcm‘𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ) → (lcm‘𝑦) ∥ ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)))
39	36, 38	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (lcm‘𝑦) ∥ ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)))
40	16, 21, 28, 34, 39	dvdstrd 16208	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)))
41	4	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑧 ∈ ℤ)
42		simprr 772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℤ)
43		lcmass 16527	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((lcm‘𝑦) lcm 𝑧) lcm 𝑛) = ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)))
44	21, 41, 42, 43	syl3anc 1373	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (((lcm‘𝑦) lcm 𝑧) lcm 𝑛) = ((lcm‘𝑦) lcm (𝑧 lcm 𝑛)))
45	40, 44	breqtrrd 5121	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ 𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛))
46	45	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ 𝑦 → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
47		elsni 4592	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ {𝑧} → 𝑖 = 𝑧)
48	17	3adant1 1130	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℕ0)
49	48	nn0zd 12500	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘𝑦) ∈ ℤ)
50		lcmcl 16514	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((lcm‘𝑦) lcm 𝑧) ∈ ℕ0)
51	49, 3, 50	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm‘𝑦) lcm 𝑧) ∈ ℕ0)
52	51	nn0zd 12500	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm‘𝑦) lcm 𝑧) ∈ ℤ)
53	52	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm‘𝑦) lcm 𝑧) ∈ ℤ)
54		lcmcl 16514	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((lcm‘𝑦) lcm 𝑧) ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((lcm‘𝑦) lcm 𝑧) lcm 𝑛) ∈ ℕ0)
55	52, 54	sylan 580	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (((lcm‘𝑦) lcm 𝑧) lcm 𝑛) ∈ ℕ0)
56	55	nn0zd 12500	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (((lcm‘𝑦) lcm 𝑧) lcm 𝑛) ∈ ℤ)
57	19, 3	jca 511	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
58	57	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
59		dvdslcm 16511	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((lcm‘𝑦) ∥ ((lcm‘𝑦) lcm 𝑧) ∧ 𝑧 ∥ ((lcm‘𝑦) lcm 𝑧)))
60	59	simprd 495	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((lcm‘𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → 𝑧 ∥ ((lcm‘𝑦) lcm 𝑧))
61	58, 60	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑧 ∥ ((lcm‘𝑦) lcm 𝑧))
62		dvdslcm 16511	. . . . . . . . . . . . . . . . . . . . . . 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)))
63	62	simpld 494	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((lcm‘𝑦) lcm 𝑧) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((lcm‘𝑦) lcm 𝑧) ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛))
64	52, 63	sylan 580	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm‘𝑦) lcm 𝑧) ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛))
65	4, 53, 56, 61, 64	dvdstrd 16208	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑧 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛))
66		breq1 5096	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑧 → (𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛) ↔ 𝑧 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
67	65, 66	imbitrrid 246	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑧 → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
68	47, 67	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ {𝑧} → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
69	46, 68	jaoi 857	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ 𝑦 ∨ 𝑖 ∈ {𝑧}) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
70	69	imp 406	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ 𝑦 ∨ 𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛))
71		oveq1 7359	. . . . . . . . . . . . . . . . 17 ⊢ ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (((lcm‘𝑦) lcm 𝑧) lcm 𝑛))
72	71	breq2d 5105	. . . . . . . . . . . . . . . 16 ⊢ ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm‘𝑦) lcm 𝑧) → (𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ↔ 𝑖 ∥ (((lcm‘𝑦) lcm 𝑧) lcm 𝑛)))
73	70, 72	syl5ibrcom 247	. . . . . . . . . . . . . . 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)))
74	12, 73	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)))
75	74	ex 412	. . . . . . . . . . . . 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))))
76	2, 75	sylbi 217	. . . . . . . . . . . 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))))
77		elsni 4592	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ {𝑛} → 𝑖 = 𝑛)
78		simp2 1137	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
79		snssi 4759	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ ℤ → {𝑧} ⊆ ℤ)
80	79	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {𝑧} ⊆ ℤ)
81	78, 80	unssd 4141	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
82		simp3 1138	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
83		snfi 8972	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {𝑧} ∈ Fin
84		unfi 9087	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
85	82, 83, 84	sylancl 586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
86		lcmfcl 16541	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ0)
87	81, 85, 86	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ0)
88	87	nn0zd 12500	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
89	88	anim1i 615	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ))
90	89	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)) → ((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ))
91		dvdslcm 16511	. . . . . . . . . . . . . . . . 17 ⊢ (((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∧ 𝑛 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
92	90, 91	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)))
93	92	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)) → 𝑛 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
94		breq1 5096	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → (𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ↔ 𝑛 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
95	93, 94	imbitrrid 246	. . . . . . . . . . . . . 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)))
96	95	expd 415	. . . . . . . . . . . . 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))))
97	77, 96	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))))
98	76, 97	jaoi 857	. . . . . . . . . . 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))))
99	1, 98	sylbi 217	. . . . . . . . . 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))))
100	99	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))))
101	100	expd 415	. . . . . . . 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)))))
102	101	adantl 481	. . . . . . 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)))))
103	102	impcom 407	. . . . . 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))))
104	103	impcom 407	. . . . 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)))
105	104	adantl 481	. . . 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)))
106	105	ralrimiv 3124	. . 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))
107		lcmfunsnlem2lem1 16551	. . 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))
108	89	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ))
109	81	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
110	85	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑦 ∪ {𝑧}) ∈ Fin)
111		df-nel 3034	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∉ 𝑦 ↔ ¬ 0 ∈ 𝑦)
112	111	biimpi 216	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∉ 𝑦 → ¬ 0 ∈ 𝑦)
113	112	3ad2ant1 1133	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ 𝑦)
114		elsni 4592	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 ∈ {𝑧} → 0 = 𝑧)
115	114	eqcomd 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ {𝑧} → 𝑧 = 0)
116	115	necon3ai 2954	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ≠ 0 → ¬ 0 ∈ {𝑧})
117	116	3ad2ant2 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ {𝑧})
118		ioran 985	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
119	113, 117, 118	sylanbrc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
120		elun 4102	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ (𝑦 ∪ {𝑧}) ↔ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
121	119, 120	sylnibr 329	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ (𝑦 ∪ {𝑧}))
122		df-nel 3034	. . . . . . . . . . . . . . . 16 ⊢ (0 ∉ (𝑦 ∪ {𝑧}) ↔ ¬ 0 ∈ (𝑦 ∪ {𝑧}))
123	121, 122	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → 0 ∉ (𝑦 ∪ {𝑧}))
124		lcmfn0cl 16539	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin ∧ 0 ∉ (𝑦 ∪ {𝑧})) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ)
125	109, 110, 123, 124	syl2an3an 1424	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ)
126	125	nnne0d 12182	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (lcm‘(𝑦 ∪ {𝑧})) ≠ 0)
127	126	neneqd 2934	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ¬ (lcm‘(𝑦 ∪ {𝑧})) = 0)
128		neneq 2935	. . . . . . . . . . . . . 14 ⊢ (𝑛 ≠ 0 → ¬ 𝑛 = 0)
129	128	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 𝑛 = 0)
130	129	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ¬ 𝑛 = 0)
131		ioran 985	. . . . . . . . . . . 12 ⊢ (¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0) ↔ (¬ (lcm‘(𝑦 ∪ {𝑧})) = 0 ∧ ¬ 𝑛 = 0))
132	127, 130, 131	sylanbrc 583	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0))
133		lcmn0cl 16510	. . . . . . . . . . 11 ⊢ ((((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0)) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ)
134	108, 132, 133	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ)
135		snssi 4759	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℤ → {𝑛} ⊆ ℤ)
136	135	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → {𝑛} ⊆ ℤ)
137	109, 136	unssd 4141	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
138	137	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
139	83, 84	mpan2 691	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
140		snfi 8972	. . . . . . . . . . . . . . 15 ⊢ {𝑛} ∈ Fin
141		unfi 9087	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∪ {𝑧}) ∈ Fin ∧ {𝑛} ∈ Fin) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
142	139, 140, 141	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
143	142	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
144	143	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
145	144	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
146		elun 4102	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}))
147		nnel 3043	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 0 ∉ 𝑦 ↔ 0 ∈ 𝑦)
148	147	biimpri 228	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ 𝑦 → ¬ 0 ∉ 𝑦)
149	148	3mix1d 1337	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ 𝑦 → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
150		nne 2933	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑧 ≠ 0 ↔ 𝑧 = 0)
151	115, 150	sylibr 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ {𝑧} → ¬ 𝑧 ≠ 0)
152	151	3mix2d 1338	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ {𝑧} → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
153	149, 152	jaoi 857	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ 𝑦 ∨ 0 ∈ {𝑧}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
154	120, 153	sylbi 217	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ (𝑦 ∪ {𝑧}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
155		elsni 4592	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ {𝑛} → 0 = 𝑛)
156	155	eqcomd 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ {𝑛} → 𝑛 = 0)
157		nne 2933	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑛 ≠ 0 ↔ 𝑛 = 0)
158	156, 157	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ {𝑛} → ¬ 𝑛 ≠ 0)
159	158	3mix3d 1339	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ {𝑛} → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
160	154, 159	jaoi 857	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
161	146, 160	sylbi 217	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
162		3ianor 1106	. . . . . . . . . . . . . . 15 ⊢ (¬ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ↔ (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
163	161, 162	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → ¬ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0))
164	163	con2i 139	. . . . . . . . . . . . 13 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
165		df-nel 3034	. . . . . . . . . . . . 13 ⊢ (0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ ¬ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
166	164, 165	sylibr 234	. . . . . . . . . . . 12 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
167	166	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
168	138, 145, 167	3jca 1128	. . . . . . . . . 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})))
169	134, 168	jca 511	. . . . . . . . 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}))))
170	169	ex 412	. . . . . . . 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})))))
171	170	ex 412	. . . . . . 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}))))))
172	171	adantr 480	. . . . . 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}))))))
173	172	impcom 407	. . . . 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})))))
174	173	impcom 407	. . . 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}))))
175		lcmf 16546	. . . 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))))
176	174, 175	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))))
177	106, 107, 176	mpbir2and 713	. 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})))
178	177	eqcomd 2739	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))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929   ∉ wnel 3033  ∀wral 3048   ∪ cun 3896   ⊆ wss 3898  {csn 4575   class class class wbr 5093  ‘cfv 6486  (class class class)co 7352  Fincfn 8875  0cc0 11013   ≤ cle 11154  ℕcn 12132  ℕ₀cn0 12388  ℤcz 12475   ∥ cdvds 16165   lcm clcm 16501  lcmclcmf 16502

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9538  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090  ax-pre-sup 11091

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-sup 9333  df-inf 9334  df-oi 9403  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-div 11782  df-nn 12133  df-2 12195  df-3 12196  df-n0 12389  df-z 12476  df-uz 12739  df-rp 12893  df-fz 13410  df-fzo 13557  df-fl 13698  df-mod 13776  df-seq 13911  df-exp 13971  df-hash 14240  df-cj 15008  df-re 15009  df-im 15010  df-sqrt 15144  df-abs 15145  df-clim 15397  df-prod 15813  df-dvds 16166  df-gcd 16408  df-lcm 16503  df-lcmf 16504

This theorem is referenced by:  lcmfunsnlem2  16553