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Theorem lcmfunsnlem2 16582

Description: Lemma for lcmfunsn 16586 and lcmfunsnlem 16583 (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 1916	. . 3 ⊢ Ⅎ𝑛(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)
2		nfv 1916	. . . 4 ⊢ Ⅎ𝑛∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)
3		nfra1 3280	. . . 4 ⊢ Ⅎ𝑛∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)
4	2, 3	nfan 1901	. . 3 ⊢ Ⅎ𝑛(∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))
5	1, 4	nfan 1901	. 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 12574	. . . . 5 ⊢ 0 ∈ ℤ
7		eqoreldif 4689	. . . . 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 1136	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
10		snssi 4812	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℤ → {𝑧} ⊆ ℤ)
11	10	3ad2ant1 1132	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {𝑧} ⊆ ℤ)
12	9, 11	unssd 4187	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
13		snssi 4812	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℤ → {0} ⊆ ℤ)
14	6, 13	mp1i 13	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {0} ⊆ ℤ)
15	12, 14	unssd 4187	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ {𝑧}) ∪ {0}) ⊆ ℤ)
16		c0ex 11213	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
17	16	snid 4665	. . . . . . . . . . . . 13 ⊢ 0 ∈ {0}
18	17	olci 863	. . . . . . . . . . . 12 ⊢ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {0})
19		elun 4149	. . . . . . . . . . . 12 ⊢ (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {0}) ↔ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {0}))
20	18, 19	mpbir 230	. . . . . . . . . . 11 ⊢ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {0})
21		lcmf0val 16564	. . . . . . . . . . 11 ⊢ ((((𝑦 ∪ {𝑧}) ∪ {0}) ⊆ ℤ ∧ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {0})) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {0})) = 0)
22	15, 20, 21	sylancl 585	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {0})) = 0)
23	22	adantr 480	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {0})) = 0)
24		sneq 4639	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → {𝑛} = {0})
25	24	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → {𝑛} = {0})
26	25	uneq2d 4164	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) = ((𝑦 ∪ {𝑧}) ∪ {0}))
27	26	fveq2d 6896	. . . . . . . . 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 9047	. . . . . . . . . . . . . . 15 ⊢ {𝑧} ∈ Fin
30		unfi 9175	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
31	29, 30	mpan2 688	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
32	31	3ad2ant3 1134	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
33		lcmfcl 16570	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
34	12, 32, 33	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
35	34	nn0zd 12589	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
36		lcm0val 16536	. . . . . . . . . . 11 ⊢ ((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ → ((lcm‘(𝑦 ∪ {𝑧})) lcm 0) = 0)
37	35, 36	syl 17	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 0) = 0)
38	28, 37	sylan9eqr 2793	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = 0)
39	23, 27, 38	3eqtr4d 2781	. . . . . . . 8 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
40	39	ex 412	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑛 = 0 → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
41	40	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 = 0 → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
42	41	com12 32	. . . . 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 481	. . . . . . . . . . . . . 14 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 𝑦 ⊆ ℤ)
44	11	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → {𝑧} ⊆ ℤ)
45	43, 44	unssd 4187	. . . . . . . . . . . . 13 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
46		elun1 4177	. . . . . . . . . . . . . 14 ⊢ (0 ∈ 𝑦 → 0 ∈ (𝑦 ∪ {𝑧}))
47	46	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ (𝑦 ∪ {𝑧}))
48		lcmf0val 16564	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ 0 ∈ (𝑦 ∪ {𝑧})) → (lcm‘(𝑦 ∪ {𝑧})) = 0)
49	45, 47, 48	syl2anc 583	. . . . . . . . . . . 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 4127	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ ∖ {0}) → 𝑛 ∈ ℤ)
52		lcm0val 16536	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → (𝑛 lcm 0) = 0)
53	51, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ ∖ {0}) → (𝑛 lcm 0) = 0)
54	53	ad2antlr 724	. . . . . . . . . . 11 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm 0) = 0)
55	50, 54	eqtrd 2771	. . . . . . . . . 10 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))) = 0)
56		simp3 1137	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
57	56, 29, 30	sylancl 585	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
58	12, 57, 33	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
59	58	nn0zd 12589	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
60	51	adantl 481	. . . . . . . . . . 11 ⊢ ((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) → 𝑛 ∈ ℤ)
61		lcmcom 16535	. . . . . . . . . . 11 ⊢ (((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))))
62	59, 60, 61	syl2anr 596	. . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
64	51	snssd 4813	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ ∖ {0}) → {𝑛} ⊆ ℤ)
65	64	ad2antlr 724	. . . . . . . . . . . 12 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → {𝑛} ⊆ ℤ)
66	63, 65	unssd 4187	. . . . . . . . . . 11 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
67	46	orcd 870	. . . . . . . . . . . . 13 ⊢ (0 ∈ 𝑦 → (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}))
68		elun 4149	. . . . . . . . . . . . 13 ⊢ (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}))
69	67, 68	sylibr 233	. . . . . . . . . . . 12 ⊢ (0 ∈ 𝑦 → 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
70	69	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
71		lcmf0val 16564	. . . . . . . . . . 11 ⊢ ((((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = 0)
72	66, 70, 71	syl2anc 583	. . . . . . . . . 10 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = 0)
73	55, 62, 72	3eqtr4rd 2782	. . . . . . . . 9 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
74	73	a1d 25	. . . . . . . 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 453	. . . . . . 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 412	. . . . . 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 4643	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ ℤ → (0 ∈ {𝑧} ↔ 0 = 𝑧))
78		eqcom 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 = 𝑧 ↔ 𝑧 = 0)
79	77, 78	bitrdi 286	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ ℤ → (0 ∈ {𝑧} ↔ 𝑧 = 0))
80	6, 79	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ {𝑧} ↔ 𝑧 = 0)
81	80	biimpri 227	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 0 ∈ {𝑧})
82	81	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ {𝑧})
83	82	olcd 871	. . . . . . . . . . . . . 14 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
84		elun 4149	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (𝑦 ∪ {𝑧}) ↔ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
85	83, 84	sylibr 233	. . . . . . . . . . . . 13 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ (𝑦 ∪ {𝑧}))
86	12, 85, 48	syl2an2 683	. . . . . . . . . . . 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 724	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 𝑛 ∈ ℤ)
89	88, 52	syl 17	. . . . . . . . . . 11 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm 0) = 0)
90	87, 89	eqtrd 2771	. . . . . . . . . 10 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))) = 0)
91	59, 88, 61	syl2an2 683	. . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
93	64	ad2antlr 724	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → {𝑛} ⊆ ℤ)
94	92, 93	unssd 4187	. . . . . . . . . . 11 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
95		sneq 4639	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 0 → {𝑧} = {0})
96	17, 95	eleqtrrid 2839	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 0 ∈ {𝑧})
97	96	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ {𝑧})
98	97	olcd 871	. . . . . . . . . . . . . 14 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
99	98, 84	sylibr 233	. . . . . . . . . . . . 13 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ (𝑦 ∪ {𝑧}))
100	99	orcd 870	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}))
101	100, 68	sylibr 233	. . . . . . . . . . 11 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
102	94, 101, 71	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = 0)
103	90, 91, 102	3eqtr4rd 2782	. . . . . . . . 9 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
104	103	a1d 25	. . . . . . . 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 453	. . . . . . 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 412	. . . . . 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 981	. . . . . . . 8 ⊢ (¬ (0 ∈ 𝑦 ∨ 𝑧 = 0) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 𝑧 = 0))
108		df-nel 3046	. . . . . . . . 9 ⊢ (0 ∉ 𝑦 ↔ ¬ 0 ∈ 𝑦)
109		df-ne 2940	. . . . . . . . 9 ⊢ (𝑧 ≠ 0 ↔ ¬ 𝑧 = 0)
110	108, 109	anbi12i 626	. . . . . . . 8 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 𝑧 = 0))
111	107, 110	bitr4i 277	. . . . . . 7 ⊢ (¬ (0 ∈ 𝑦 ∨ 𝑧 = 0) ↔ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0))
112		eldif 3959	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ ∖ {0}) ↔ (𝑛 ∈ ℤ ∧ ¬ 𝑛 ∈ {0}))
113		velsn 4645	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {0} ↔ 𝑛 = 0)
114	113	bicomi 223	. . . . . . . . . . 11 ⊢ (𝑛 = 0 ↔ 𝑛 ∈ {0})
115	114	necon3abii 2986	. . . . . . . . . 10 ⊢ (𝑛 ≠ 0 ↔ ¬ 𝑛 ∈ {0})
116		lcmfunsnlem2lem2 16581	. . . . . . . . . . . 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 1356	. . . . . . . . . . 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 406	. . . . . . . . . 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 242	. . . . . . . . 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 411	. . . . . . . 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 241	. . . . . . 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 216	. . . . . 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 1029	. . . . 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 854	. . . 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 216	. . 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 32	. 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 3253	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 205 ∧ wa 395 ∨ wo 844 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∉ wnel 3045 ∀wral 3060 ∖ cdif 3946 ∪ cun 3947 ⊆ wss 3949 {csn 4629 class class class wbr 5149 ‘cfv 6544 (class class class)co 7412 Fincfn 8942 0cc0 11113 ℕ₀cn0 12477 ℤcz 12563 ∥ cdvds 16202 lcm clcm 16530 lcmclcmf 16531

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9440 df-inf 9441 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-prod 15855 df-dvds 16203 df-gcd 16441 df-lcm 16532 df-lcmf 16533

This theorem is referenced by: lcmfunsnlem 16583

W3C validator