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Theorem lcmfunsnlem2 16699

Description: Lemma for lcmfunsn 16703 and lcmfunsnlem 16700 (Induction step part 2). (Contributed by AV, 26-Aug-2020.)

**Assertion**
Ref	Expression
lcmfunsnlem2	⊢ (((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 (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))

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

**Proof of Theorem lcmfunsnlem2**
Step	Hyp	Ref	Expression
1		nfv 1944	. . 3 ⊢ Ⅎ𝑛(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)
2		nfv 1944	. . . 4 ⊢ Ⅎ𝑛∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)
3		nfra1 3289	. . . 4 ⊢ Ⅎ𝑛∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)
4	2, 3	nfan 1929	. . 3 ⊢ Ⅎ𝑛(∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))
5	1, 4	nfan 1929	. 2 ⊢ Ⅎ𝑛((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)))
6		0z 12603	. . . . 5 ⊢ 0 ∈ ℤ
7		eqoreldif 4652	. . . . 5 ⊢ (0 ∈ ℤ → (𝑛 ∈ ℤ ↔ (𝑛 = 0 ∨ 𝑛 ∈ (ℤ ∖ {0}))))
8	6, 7	ax-mp 5	. . . 4 ⊢ (𝑛 ∈ ℤ ↔ (𝑛 = 0 ∨ 𝑛 ∈ (ℤ ∖ {0})))
9		simp2 1155	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
10		snssi 4752	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℤ → {𝑧} ⊆ ℤ)
11	10	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {𝑧} ⊆ ℤ)
12	9, 11	unssd 4146	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
13		snssi 4752	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℤ → {0} ⊆ ℤ)
14	6, 13	mp1i 14	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {0} ⊆ ℤ)
15	12, 14	unssd 4146	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ {𝑧}) ∪ {0}) ⊆ ℤ)
16		c0ex 11201	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
17	16	snid 4629	. . . . . . . . . . . . 13 ⊢ 0 ∈ {0}
18	17	olci 879	. . . . . . . . . . . 12 ⊢ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {0})
19		elun 4108	. . . . . . . . . . . 12 ⊢ (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {0}) ↔ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {0}))
20	18, 19	mpbir 234	. . . . . . . . . . 11 ⊢ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {0})
21		lcmf0val 16681	. . . . . . . . . . 11 ⊢ ((((𝑦 ∪ {𝑧}) ∪ {0}) ⊆ ℤ ∧ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {0})) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {0})) = 0)
22	15, 20, 21	sylancl 597	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {0})) = 0)
23	22	adantr 485	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {0})) = 0)
24		sneq 4600	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → {𝑛} = {0})
25	24	adantl 486	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → {𝑛} = {0})
26	25	uneq2d 4123	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) = ((𝑦 ∪ {𝑧}) ∪ {0}))
27	26	fveq2d 6887	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {0})))
28		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑛 = 0 → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 0))
29		snfi 9041	. . . . . . . . . . . . . . 15 ⊢ {𝑧} ∈ Fin
30		unfi 9156	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
31	29, 30	mpan2 703	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
32	31	3ad2ant3 1153	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
33		lcmfcl 16687	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
34	12, 32, 33	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
35	34	nn0zd 12617	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
36		lcm0val 16653	. . . . . . . . . . 11 ⊢ ((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ → ((lcm‘(𝑦 ∪ {𝑧})) lcm 0) = 0)
37	35, 36	syl 18	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 0) = 0)
38	28, 37	sylan9eqr 2820	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = 0)
39	23, 27, 38	3eqtr4d 2808	. . . . . . . 8 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
40	39	ex 417	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑛 = 0 → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
41	40	adantr 485	. . . . . 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 = 0 → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
42	41	com12 33	. . . . 5 ⊢ (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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
43	9	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 𝑦 ⊆ ℤ)
44	11	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → {𝑧} ⊆ ℤ)
45	43, 44	unssd 4146	. . . . . . . . . . . . 13 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
46		elun1 4136	. . . . . . . . . . . . . 14 ⊢ (0 ∈ 𝑦 → 0 ∈ (𝑦 ∪ {𝑧}))
47	46	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ (𝑦 ∪ {𝑧}))
48		lcmf0val 16681	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ 0 ∈ (𝑦 ∪ {𝑧})) → (lcm‘(𝑦 ∪ {𝑧})) = 0)
49	45, 47, 48	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘(𝑦 ∪ {𝑧})) = 0)
50	49	oveq2d 7428	. . . . . . . . . . 11 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))) = (𝑛 lcm 0))
51		eldifi 4086	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ ∖ {0}) → 𝑛 ∈ ℤ)
52		lcm0val 16653	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → (𝑛 lcm 0) = 0)
53	51, 52	syl 18	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ ∖ {0}) → (𝑛 lcm 0) = 0)
54	53	ad2antlr 739	. . . . . . . . . . 11 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm 0) = 0)
55	50, 54	eqtrd 2798	. . . . . . . . . 10 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))) = 0)
56		simp3 1156	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
57	56, 29, 30	sylancl 597	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
58	12, 57, 33	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
59	58	nn0zd 12617	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
60	51	adantl 486	. . . . . . . . . . 11 ⊢ ((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) → 𝑛 ∈ ℤ)
61		lcmcom 16652	. . . . . . . . . . 11 ⊢ (((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))))
62	59, 60, 61	syl2anr 608	. . . . . . . . . 10 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))))
63	12	adantl 486	. . . . . . . . . . . 12 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
64	51	snssd 4753	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ ∖ {0}) → {𝑛} ⊆ ℤ)
65	64	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → {𝑛} ⊆ ℤ)
66	63, 65	unssd 4146	. . . . . . . . . . 11 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
67	46	orcd 886	. . . . . . . . . . . . 13 ⊢ (0 ∈ 𝑦 → (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}))
68		elun 4108	. . . . . . . . . . . . 13 ⊢ (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}))
69	67, 68	sylibr 237	. . . . . . . . . . . 12 ⊢ (0 ∈ 𝑦 → 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
70	69	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
71		lcmf0val 16681	. . . . . . . . . . 11 ⊢ ((((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = 0)
72	66, 70, 71	syl2anc 595	. . . . . . . . . 10 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = 0)
73	55, 62, 72	3eqtr4rd 2809	. . . . . . . . 9 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
74	73	a1d 26	. . . . . . . 8 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {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)) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
75	74	expimpd 458	. . . . . . 7 ⊢ ((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
76	75	ex 417	. . . . . 6 ⊢ (0 ∈ 𝑦 → (𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
77		elsng 4604	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ ℤ → (0 ∈ {𝑧} ↔ 0 = 𝑧))
78		eqcom 2770	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 = 𝑧 ↔ 𝑧 = 0)
79	77, 78	bitrdi 290	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ ℤ → (0 ∈ {𝑧} ↔ 𝑧 = 0))
80	6, 79	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ {𝑧} ↔ 𝑧 = 0)
81	80	biimpri 231	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 0 ∈ {𝑧})
82	81	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ {𝑧})
83	82	olcd 887	. . . . . . . . . . . . . 14 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
84		elun 4108	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (𝑦 ∪ {𝑧}) ↔ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
85	83, 84	sylibr 237	. . . . . . . . . . . . 13 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ (𝑦 ∪ {𝑧}))
86	12, 85, 48	syl2an2 698	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘(𝑦 ∪ {𝑧})) = 0)
87	86	oveq2d 7428	. . . . . . . . . . 11 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))) = (𝑛 lcm 0))
88	51	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 𝑛 ∈ ℤ)
89	88, 52	syl 18	. . . . . . . . . . 11 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm 0) = 0)
90	87, 89	eqtrd 2798	. . . . . . . . . 10 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))) = 0)
91	59, 88, 61	syl2an2 698	. . . . . . . . . 10 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))))
92	12	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
93	64	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → {𝑛} ⊆ ℤ)
94	92, 93	unssd 4146	. . . . . . . . . . 11 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
95		sneq 4600	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 0 → {𝑧} = {0})
96	17, 95	eleqtrrid 2870	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 0 ∈ {𝑧})
97	96	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ {𝑧})
98	97	olcd 887	. . . . . . . . . . . . . 14 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
99	98, 84	sylibr 237	. . . . . . . . . . . . 13 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ (𝑦 ∪ {𝑧}))
100	99	orcd 886	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}))
101	100, 68	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
102	94, 101, 71	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = 0)
103	90, 91, 102	3eqtr4rd 2809	. . . . . . . . 9 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
104	103	a1d 26	. . . . . . . 8 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {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)) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
105	104	expimpd 458	. . . . . . 7 ⊢ ((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
106	105	ex 417	. . . . . 6 ⊢ (𝑧 = 0 → (𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
107		ioran 999	. . . . . . . 8 ⊢ (¬ (0 ∈ 𝑦 ∨ 𝑧 = 0) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 𝑧 = 0))
108		df-nel 3065	. . . . . . . . 9 ⊢ (0 ∉ 𝑦 ↔ ¬ 0 ∈ 𝑦)
109		df-ne 2959	. . . . . . . . 9 ⊢ (𝑧 ≠ 0 ↔ ¬ 𝑧 = 0)
110	108, 109	anbi12i 639	. . . . . . . 8 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 𝑧 = 0))
111	107, 110	bitr4i 281	. . . . . . 7 ⊢ (¬ (0 ∈ 𝑦 ∨ 𝑧 = 0) ↔ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0))
112		eldif 3916	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ ∖ {0}) ↔ (𝑛 ∈ ℤ ∧ ¬ 𝑛 ∈ {0}))
113		velsn 4606	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {0} ↔ 𝑛 = 0)
114	113	bicomi 227	. . . . . . . . . . 11 ⊢ (𝑛 = 0 ↔ 𝑛 ∈ {0})
115	114	necon3abii 3004	. . . . . . . . . 10 ⊢ (𝑛 ≠ 0 ↔ ¬ 𝑛 ∈ {0})
116		lcmfunsnlem2lem2 16698	. . . . . . . . . . . 12 ⊢ (((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))
117	116	exp520 1376	. . . . . . . . . . 11 ⊢ (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))))))
118	117	imp 411	. . . . . . . . . 10 ⊢ ((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)))))
119	115, 118	biimtrrid 246	. . . . . . . . 9 ⊢ ((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)))))
120	119	impcomd 416	. . . . . . . 8 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0) → ((𝑛 ∈ ℤ ∧ ¬ 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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
121	112, 120	biimtrid 245	. . . . . . 7 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0) → (𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
122	111, 121	sylbi 220	. . . . . 6 ⊢ (¬ (0 ∈ 𝑦 ∨ 𝑧 = 0) → (𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
123	76, 106, 122	ecase3 1048	. . . . 5 ⊢ (𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
124	42, 123	jaoi 870	. . . 4 ⊢ ((𝑛 = 0 ∨ 𝑛 ∈ (ℤ ∖ {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))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
125	8, 124	sylbi 220	. . 3 ⊢ (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)))
126	125	com12 33	. 2 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → (𝑛 ∈ ℤ → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
127	5, 126	ralrimi 3263	1 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∉ wnel 3064 ∀wral 3079 ∖ cdif 3903 ∪ cun 3904 ⊆ wss 3906 {csn 4590 class class class wbr 5110 ‘cfv 6538 (class class class)co 7412 Fincfn 8944 0cc0 11101 ℕ₀cn0 12505 ℤcz 12592 ∥ cdvds 16311 lcm clcm 16647 lcmclcmf 16648

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-sup 9403 df-inf 9404 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-z 12593 df-uz 12864 df-rp 13018 df-fz 13537 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-prod 15960 df-dvds 16312 df-gcd 16554 df-lcm 16649 df-lcmf 16650

This theorem is referenced by: lcmfunsnlem 16700

W3C validator