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Theorem lcmfunsnlem2 16847

Description: Lemma for lcmfunsn 16851 and lcmfunsnlem 16848 (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 1949	. . 3 ⊢ Ⅎ𝑛(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)
2		nfv 1949	. . . 4 ⊢ Ⅎ𝑛∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)
3		nfra1 3290	. . . 4 ⊢ Ⅎ𝑛∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)
4	2, 3	nfan 1934	. . 3 ⊢ Ⅎ𝑛(∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))
5	1, 4	nfan 1934	. 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 12734	. . . . 5 ⊢ 0 ∈ ℤ
7		eqoreldif 4646	. . . . 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 4746	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℤ → {𝑧} ⊆ ℤ)
11	10	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {𝑧} ⊆ ℤ)
12	9, 11	unssd 4193	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
13		snssi 4746	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℤ → {0} ⊆ ℤ)
14	6, 13	mp1i 14	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {0} ⊆ ℤ)
15	12, 14	unssd 4193	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ {𝑧}) ∪ {0}) ⊆ ℤ)
16		c0ex 11330	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
17	16	snid 4623	. . . . . . . . . . . . 13 ⊢ 0 ∈ {0}
18	17	olci 880	. . . . . . . . . . . 12 ⊢ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {0})
19		elun 4155	. . . . . . . . . . . 12 ⊢ (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {0}) ↔ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {0}))
20	18, 19	mpbir 234	. . . . . . . . . . 11 ⊢ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {0})
21		lcmf0val 16829	. . . . . . . . . . 11 ⊢ ((((𝑦 ∪ {𝑧}) ∪ {0}) ⊆ ℤ ∧ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {0})) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {0})) = 0)
22	15, 20, 21	sylancl 598	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {0})) = 0)
23	22	adantr 486	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {0})) = 0)
24		sneq 4594	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → {𝑛} = {0})
25	24	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → {𝑛} = {0})
26	25	uneq2d 4170	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) = ((𝑦 ∪ {𝑧}) ∪ {0}))
27	26	fveq2d 6915	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {0})))
28		oveq2 7454	. . . . . . . . . 10 ⊢ (𝑛 = 0 → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 0))
29		snfi 9102	. . . . . . . . . . . . . . 15 ⊢ {𝑧} ∈ Fin
30		unfi 9217	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
31	29, 30	mpan2 704	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
32	31	3ad2ant3 1153	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
33		lcmfcl 16835	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
34	12, 32, 33	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
35	34	nn0zd 12748	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
36		lcm0val 16801	. . . . . . . . . . 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 418	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑛 = 0 → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
41	40	adantr 486	. . . . . 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 487	. . . . . . . . . . . . . 14 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 𝑦 ⊆ ℤ)
44	11	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → {𝑧} ⊆ ℤ)
45	43, 44	unssd 4193	. . . . . . . . . . . . 13 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
46		elun1 4183	. . . . . . . . . . . . . 14 ⊢ (0 ∈ 𝑦 → 0 ∈ (𝑦 ∪ {𝑧}))
47	46	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ (𝑦 ∪ {𝑧}))
48		lcmf0val 16829	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ 0 ∈ (𝑦 ∪ {𝑧})) → (lcm‘(𝑦 ∪ {𝑧})) = 0)
49	45, 47, 48	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘(𝑦 ∪ {𝑧})) = 0)
50	49	oveq2d 7462	. . . . . . . . . . 11 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))) = (𝑛 lcm 0))
51		eldifi 4070	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ ∖ {0}) → 𝑛 ∈ ℤ)
52		lcm0val 16801	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → (𝑛 lcm 0) = 0)
53	51, 52	syl 18	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ ∖ {0}) → (𝑛 lcm 0) = 0)
54	53	ad2antlr 740	. . . . . . . . . . 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 598	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
58	12, 57, 33	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
59	58	nn0zd 12748	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
60	51	adantl 487	. . . . . . . . . . 11 ⊢ ((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) → 𝑛 ∈ ℤ)
61		lcmcom 16800	. . . . . . . . . . 11 ⊢ (((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))))
62	59, 60, 61	syl2anr 609	. . . . . . . . . 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 487	. . . . . . . . . . . 12 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
64	51	snssd 4747	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ ∖ {0}) → {𝑛} ⊆ ℤ)
65	64	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → {𝑛} ⊆ ℤ)
66	63, 65	unssd 4193	. . . . . . . . . . 11 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
67	46	orcd 887	. . . . . . . . . . . . 13 ⊢ (0 ∈ 𝑦 → (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}))
68		elun 4155	. . . . . . . . . . . . 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 739	. . . . . . . . . . 11 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
71		lcmf0val 16829	. . . . . . . . . . 11 ⊢ ((((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = 0)
72	66, 70, 71	syl2anc 596	. . . . . . . . . 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 459	. . . . . . 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 418	. . . . . 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 4598	. . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ {𝑧})
83	82	olcd 888	. . . . . . . . . . . . . 14 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
84		elun 4155	. . . . . . . . . . . . . 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 699	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘(𝑦 ∪ {𝑧})) = 0)
87	86	oveq2d 7462	. . . . . . . . . . 11 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))) = (𝑛 lcm 0))
88	51	ad2antlr 740	. . . . . . . . . . . 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 699	. . . . . . . . . 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 487	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
93	64	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → {𝑛} ⊆ ℤ)
94	92, 93	unssd 4193	. . . . . . . . . . 11 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
95		sneq 4594	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 0 → {𝑧} = {0})
96	17, 95	eleqtrrid 2871	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 0 ∈ {𝑧})
97	96	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ {𝑧})
98	97	olcd 888	. . . . . . . . . . . . . 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 887	. . . . . . . . . . . 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 596	. . . . . . . . . 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 459	. . . . . . 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 418	. . . . . 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 3066	. . . . . . . . 9 ⊢ (0 ∉ 𝑦 ↔ ¬ 0 ∈ 𝑦)
109		df-ne 2960	. . . . . . . . 9 ⊢ (𝑧 ≠ 0 ↔ ¬ 𝑧 = 0)
110	108, 109	anbi12i 640	. . . . . . . 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 4052	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ ∖ {0}) ↔ (𝑛 ∈ ℤ ∧ ¬ 𝑛 ∈ {0}))
113		velsn 4600	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {0} ↔ 𝑛 = 0)
114	113	bicomi 227	. . . . . . . . . . 11 ⊢ (𝑛 = 0 ↔ 𝑛 ∈ {0})
115	114	necon3abii 3005	. . . . . . . . . 10 ⊢ (𝑛 ≠ 0 ↔ ¬ 𝑛 ∈ {0})
116		lcmfunsnlem2lem2 16846	. . . . . . . . . . . 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 412	. . . . . . . . . 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 417	. . . . . . . 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 871	. . . 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 3264	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∉ wnel 3065 ∀wral 3080 ⊆ wss 3900 ∖ cdif 4048 ∪ cun 4152 {csn 4588 class class class wbr 5106 ‘cfv 6561 (class class class)co 7446 Fincfn 9004 0cc0 11230 ℕ₀cn0 12636 ℤcz 12723 ∥ cdvds 16454 lcm clcm 16795 lcmclcmf 16796

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-inf2 9672 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307 ax-pre-sup 11308

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-om 7905 df-1st 8028 df-2nd 8029 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-er 8748 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-sup 9464 df-inf 9465 df-oi 9534 df-card 10050 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-div 12004 df-nn 12366 df-2 12435 df-3 12436 df-n0 12637 df-z 12724 df-uz 12996 df-rp 13151 df-fz 13670 df-fzo 13819 df-fl 13962 df-mod 14040 df-seq 14175 df-exp 14235 df-hash 14505 df-cj 15298 df-re 15299 df-im 15300 df-sqrt 15434 df-abs 15435 df-clim 15687 df-prod 16105 df-dvds 16455 df-gcd 16697 df-lcm 16797 df-lcmf 16798

This theorem is used by: lcmfunsnlem 16848

W3C validator