lcmfunsnlem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lcmfunsnlem2		Structured version Visualization version GIF version

Theorem lcmfunsnlem2 16677

Description: Lemma for lcmfunsn 16681 and lcmfunsnlem 16678 (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 1914	. . 3 ⊢ Ⅎ𝑛(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)
2		nfv 1914	. . . 4 ⊢ Ⅎ𝑛∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)
3		nfra1 3284	. . . 4 ⊢ Ⅎ𝑛∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)
4	2, 3	nfan 1899	. . 3 ⊢ Ⅎ𝑛(∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))
5	1, 4	nfan 1899	. 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 12624	. . . . 5 ⊢ 0 ∈ ℤ
7		eqoreldif 4685	. . . . 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 1138	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
10		snssi 4808	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℤ → {𝑧} ⊆ ℤ)
11	10	3ad2ant1 1134	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {𝑧} ⊆ ℤ)
12	9, 11	unssd 4192	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
13		snssi 4808	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℤ → {0} ⊆ ℤ)
14	6, 13	mp1i 13	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {0} ⊆ ℤ)
15	12, 14	unssd 4192	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ {𝑧}) ∪ {0}) ⊆ ℤ)
16		c0ex 11255	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
17	16	snid 4662	. . . . . . . . . . . . 13 ⊢ 0 ∈ {0}
18	17	olci 867	. . . . . . . . . . . 12 ⊢ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {0})
19		elun 4153	. . . . . . . . . . . 12 ⊢ (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {0}) ↔ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {0}))
20	18, 19	mpbir 231	. . . . . . . . . . 11 ⊢ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {0})
21		lcmf0val 16659	. . . . . . . . . . 11 ⊢ ((((𝑦 ∪ {𝑧}) ∪ {0}) ⊆ ℤ ∧ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {0})) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {0})) = 0)
22	15, 20, 21	sylancl 586	. . . . . . . . . 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 4636	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → {𝑛} = {0})
25	24	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → {𝑛} = {0})
26	25	uneq2d 4168	. . . . . . . . . 10 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) = ((𝑦 ∪ {𝑧}) ∪ {0}))
27	26	fveq2d 6910	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {0})))
28		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑛 = 0 → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 0))
29		snfi 9083	. . . . . . . . . . . . . . 15 ⊢ {𝑧} ∈ Fin
30		unfi 9211	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
31	29, 30	mpan2 691	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
32	31	3ad2ant3 1136	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
33		lcmfcl 16665	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
34	12, 32, 33	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
35	34	nn0zd 12639	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
36		lcm0val 16631	. . . . . . . . . . 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 2799	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 = 0) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = 0)
39	23, 27, 38	3eqtr4d 2787	. . . . . . . 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 4192	. . . . . . . . . . . . 13 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
46		elun1 4182	. . . . . . . . . . . . . 14 ⊢ (0 ∈ 𝑦 → 0 ∈ (𝑦 ∪ {𝑧}))
47	46	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ (𝑦 ∪ {𝑧}))
48		lcmf0val 16659	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ 0 ∈ (𝑦 ∪ {𝑧})) → (lcm‘(𝑦 ∪ {𝑧})) = 0)
49	45, 47, 48	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘(𝑦 ∪ {𝑧})) = 0)
50	49	oveq2d 7447	. . . . . . . . . . 11 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))) = (𝑛 lcm 0))
51		eldifi 4131	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ ∖ {0}) → 𝑛 ∈ ℤ)
52		lcm0val 16631	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℤ → (𝑛 lcm 0) = 0)
53	51, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ ∖ {0}) → (𝑛 lcm 0) = 0)
54	53	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm 0) = 0)
55	50, 54	eqtrd 2777	. . . . . . . . . 10 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))) = 0)
56		simp3 1139	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
57	56, 29, 30	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
58	12, 57, 33	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ₀)
59	58	nn0zd 12639	. . . . . . . . . . 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 16630	. . . . . . . . . . 11 ⊢ (((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))))
62	59, 60, 61	syl2anr 597	. . . . . . . . . 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 4809	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ ∖ {0}) → {𝑛} ⊆ ℤ)
65	64	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → {𝑛} ⊆ ℤ)
66	63, 65	unssd 4192	. . . . . . . . . . 11 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
67	46	orcd 874	. . . . . . . . . . . . 13 ⊢ (0 ∈ 𝑦 → (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}))
68		elun 4153	. . . . . . . . . . . . 13 ⊢ (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}))
69	67, 68	sylibr 234	. . . . . . . . . . . 12 ⊢ (0 ∈ 𝑦 → 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
70	69	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
71		lcmf0val 16659	. . . . . . . . . . 11 ⊢ ((((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = 0)
72	66, 70, 71	syl2anc 584	. . . . . . . . . 10 ⊢ (((0 ∈ 𝑦 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = 0)
73	55, 62, 72	3eqtr4rd 2788	. . . . . . . . 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 4640	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 ∈ ℤ → (0 ∈ {𝑧} ↔ 0 = 𝑧))
78		eqcom 2744	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 = 𝑧 ↔ 𝑧 = 0)
79	77, 78	bitrdi 287	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ ℤ → (0 ∈ {𝑧} ↔ 𝑧 = 0))
80	6, 79	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ {𝑧} ↔ 𝑧 = 0)
81	80	biimpri 228	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 0 ∈ {𝑧})
82	81	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ {𝑧})
83	82	olcd 875	. . . . . . . . . . . . . 14 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
84		elun 4153	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (𝑦 ∪ {𝑧}) ↔ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
85	83, 84	sylibr 234	. . . . . . . . . . . . 13 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ (𝑦 ∪ {𝑧}))
86	12, 85, 48	syl2an2 686	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘(𝑦 ∪ {𝑧})) = 0)
87	86	oveq2d 7447	. . . . . . . . . . 11 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))) = (𝑛 lcm 0))
88	51	ad2antlr 727	. . . . . . . . . . . 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 2777	. . . . . . . . . 10 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑛 lcm (lcm‘(𝑦 ∪ {𝑧}))) = 0)
91	59, 88, 61	syl2an2 686	. . . . . . . . . 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 727	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → {𝑛} ⊆ ℤ)
94	92, 93	unssd 4192	. . . . . . . . . . 11 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
95		sneq 4636	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 0 → {𝑧} = {0})
96	17, 95	eleqtrrid 2848	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 0 ∈ {𝑧})
97	96	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ {𝑧})
98	97	olcd 875	. . . . . . . . . . . . . 14 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
99	98, 84	sylibr 234	. . . . . . . . . . . . 13 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ (𝑦 ∪ {𝑧}))
100	99	orcd 874	. . . . . . . . . . . 12 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}))
101	100, 68	sylibr 234	. . . . . . . . . . 11 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
102	94, 101, 71	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑧 = 0 ∧ 𝑛 ∈ (ℤ ∖ {0})) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = 0)
103	90, 91, 102	3eqtr4rd 2788	. . . . . . . . 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 986	. . . . . . . 8 ⊢ (¬ (0 ∈ 𝑦 ∨ 𝑧 = 0) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 𝑧 = 0))
108		df-nel 3047	. . . . . . . . 9 ⊢ (0 ∉ 𝑦 ↔ ¬ 0 ∈ 𝑦)
109		df-ne 2941	. . . . . . . . 9 ⊢ (𝑧 ≠ 0 ↔ ¬ 𝑧 = 0)
110	108, 109	anbi12i 628	. . . . . . . 8 ⊢ ((0 ∉ 𝑦 ∧ 𝑧 ≠ 0) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 𝑧 = 0))
111	107, 110	bitr4i 278	. . . . . . 7 ⊢ (¬ (0 ∈ 𝑦 ∨ 𝑧 = 0) ↔ (0 ∉ 𝑦 ∧ 𝑧 ≠ 0))
112		eldif 3961	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ ∖ {0}) ↔ (𝑛 ∈ ℤ ∧ ¬ 𝑛 ∈ {0}))
113		velsn 4642	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {0} ↔ 𝑛 = 0)
114	113	bicomi 224	. . . . . . . . . . 11 ⊢ (𝑛 = 0 ↔ 𝑛 ∈ {0})
115	114	necon3abii 2987	. . . . . . . . . 10 ⊢ (𝑛 ≠ 0 ↔ ¬ 𝑛 ∈ {0})
116		lcmfunsnlem2lem2 16676	. . . . . . . . . . . 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 1358	. . . . . . . . . . 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 243	. . . . . . . . 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 242	. . . . . . 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 217	. . . . . 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 1033	. . . . 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 858	. . . 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 217	. . 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 3257	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 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∉ wnel 3046 ∀wral 3061 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 {csn 4626 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 0cc0 11155 ℕ₀cn0 12526 ℤcz 12613 ∥ cdvds 16290 lcm clcm 16625 lcmclcmf 16626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 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 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-prod 15940 df-dvds 16291 df-gcd 16532 df-lcm 16627 df-lcmf 16628

This theorem is referenced by: lcmfunsnlem 16678

W3C validator