Theorem lcmfunsnlem 15975

Description: Lemma for lcmfdvds 15976 and lcmfunsn 15978. These two theorems must be proven simultaneously by induction on the cardinality of a finite set 𝑌, because they depend on each other. This can be seen by the two parts lcmfunsnlem1 15971 and lcmfunsnlem2 15974 of the induction step, each of them using both induction hypotheses. (Contributed by AV, 26-Aug-2020.)

Assertion

Ref	Expression
lcmfunsnlem	⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘 → (lcm‘𝑌) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm‘𝑌) lcm 𝑛)))

Distinct variable group:   𝑘,𝑛,𝑚,𝑌

Proof of Theorem lcmfunsnlem

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3940	. . . 4 ⊢ (𝑥 = ∅ → (𝑥 ⊆ ℤ ↔ ∅ ⊆ ℤ))
2		raleq 3358	. . . . . . 7 ⊢ (𝑥 = ∅ → (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘))
3		fveq2 6645	. . . . . . . 8 ⊢ (𝑥 = ∅ → (lcm‘𝑥) = (lcm‘∅))
4	3	breq1d 5040	. . . . . . 7 ⊢ (𝑥 = ∅ → ((lcm‘𝑥) ∥ 𝑘 ↔ (lcm‘∅) ∥ 𝑘))
5	2, 4	imbi12d 348	. . . . . 6 ⊢ (𝑥 = ∅ → ((∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘)))
6	5	ralbidv 3162	. . . . 5 ⊢ (𝑥 = ∅ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘)))
7		uneq1 4083	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ∪ {𝑛}) = (∅ ∪ {𝑛}))
8	7	fveq2d 6649	. . . . . . 7 ⊢ (𝑥 = ∅ → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(∅ ∪ {𝑛})))
9	3	oveq1d 7150	. . . . . . 7 ⊢ (𝑥 = ∅ → ((lcm‘𝑥) lcm 𝑛) = ((lcm‘∅) lcm 𝑛))
10	8, 9	eqeq12d 2814	. . . . . 6 ⊢ (𝑥 = ∅ → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
11	10	ralbidv 3162	. . . . 5 ⊢ (𝑥 = ∅ → (∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
12	6, 11	anbi12d 633	. . . 4 ⊢ (𝑥 = ∅ → ((∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛)) ↔ (∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))))
13	1, 12	imbi12d 348	. . 3 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛))) ↔ (∅ ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))))
14		sseq1 3940	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ ℤ ↔ 𝑦 ⊆ ℤ))
15		raleq 3358	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘))
16		fveq2 6645	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (lcm‘𝑥) = (lcm‘𝑦))
17	16	breq1d 5040	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((lcm‘𝑥) ∥ 𝑘 ↔ (lcm‘𝑦) ∥ 𝑘))
18	15, 17	imbi12d 348	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
19	18	ralbidv 3162	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
20		uneq1 4083	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∪ {𝑛}) = (𝑦 ∪ {𝑛}))
21	20	fveq2d 6649	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑛})))
22	16	oveq1d 7150	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((lcm‘𝑥) lcm 𝑛) = ((lcm‘𝑦) lcm 𝑛))
23	21, 22	eqeq12d 2814	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)))
24	23	ralbidv 3162	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)))
25	19, 24	anbi12d 633	. . . 4 ⊢ (𝑥 = 𝑦 → ((∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛)) ↔ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))))
26	14, 25	imbi12d 348	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛))) ↔ (𝑦 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)))))
27		sseq1 3940	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ ℤ ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ))
28		raleq 3358	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘))
29		fveq2 6645	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘𝑥) = (lcm‘(𝑦 ∪ {𝑧})))
30	29	breq1d 5040	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘𝑥) ∥ 𝑘 ↔ (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
31	28, 30	imbi12d 348	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
32	31	ralbidv 3162	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
33		uneq1 4083	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∪ {𝑛}) = ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
34	33	fveq2d 6649	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})))
35	29	oveq1d 7150	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘𝑥) lcm 𝑛) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
36	34, 35	eqeq12d 2814	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
37	36	ralbidv 3162	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
38	32, 37	anbi12d 633	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛)) ↔ (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
39	27, 38	imbi12d 348	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛))) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))
40		sseq1 3940	. . . 4 ⊢ (𝑥 = 𝑌 → (𝑥 ⊆ ℤ ↔ 𝑌 ⊆ ℤ))
41		raleq 3358	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘))
42		fveq2 6645	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (lcm‘𝑥) = (lcm‘𝑌))
43	42	breq1d 5040	. . . . . . 7 ⊢ (𝑥 = 𝑌 → ((lcm‘𝑥) ∥ 𝑘 ↔ (lcm‘𝑌) ∥ 𝑘))
44	41, 43	imbi12d 348	. . . . . 6 ⊢ (𝑥 = 𝑌 → ((∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ (∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘 → (lcm‘𝑌) ∥ 𝑘)))
45	44	ralbidv 3162	. . . . 5 ⊢ (𝑥 = 𝑌 → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘 → (lcm‘𝑌) ∥ 𝑘)))
46		uneq1 4083	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (𝑥 ∪ {𝑛}) = (𝑌 ∪ {𝑛}))
47	46	fveq2d 6649	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(𝑌 ∪ {𝑛})))
48	42	oveq1d 7150	. . . . . . 7 ⊢ (𝑥 = 𝑌 → ((lcm‘𝑥) lcm 𝑛) = ((lcm‘𝑌) lcm 𝑛))
49	47, 48	eqeq12d 2814	. . . . . 6 ⊢ (𝑥 = 𝑌 → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm‘𝑌) lcm 𝑛)))
50	49	ralbidv 3162	. . . . 5 ⊢ (𝑥 = 𝑌 → (∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm‘𝑌) lcm 𝑛)))
51	45, 50	anbi12d 633	. . . 4 ⊢ (𝑥 = 𝑌 → ((∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛)) ↔ (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘 → (lcm‘𝑌) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm‘𝑌) lcm 𝑛))))
52	40, 51	imbi12d 348	. . 3 ⊢ (𝑥 = 𝑌 → ((𝑥 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛))) ↔ (𝑌 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘 → (lcm‘𝑌) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm‘𝑌) lcm 𝑛)))))
53		lcmf0 15968	. . . . . . . 8 ⊢ (lcm‘∅) = 1
54		1dvds 15616	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ → 1 ∥ 𝑘)
55	53, 54	eqbrtrid 5065	. . . . . . 7 ⊢ (𝑘 ∈ ℤ → (lcm‘∅) ∥ 𝑘)
56	55	a1d 25	. . . . . 6 ⊢ (𝑘 ∈ ℤ → (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘))
57	56	adantl 485	. . . . 5 ⊢ ((∅ ⊆ ℤ ∧ 𝑘 ∈ ℤ) → (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘))
58	57	ralrimiva 3149	. . . 4 ⊢ (∅ ⊆ ℤ → ∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘))
59		uncom 4080	. . . . . . . . . 10 ⊢ (∅ ∪ {𝑛}) = ({𝑛} ∪ ∅)
60		un0 4298	. . . . . . . . . 10 ⊢ ({𝑛} ∪ ∅) = {𝑛}
61	59, 60	eqtri 2821	. . . . . . . . 9 ⊢ (∅ ∪ {𝑛}) = {𝑛}
62	61	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (∅ ∪ {𝑛}) = {𝑛})
63	62	fveq2d 6649	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → (lcm‘(∅ ∪ {𝑛})) = (lcm‘{𝑛}))
64		lcmfsn 15969	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → (lcm‘{𝑛}) = (abs‘𝑛))
65	53	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → (lcm‘∅) = 1)
66	65	oveq1d 7150	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → ((lcm‘∅) lcm 𝑛) = (1 lcm 𝑛))
67		1z 12000	. . . . . . . . 9 ⊢ 1 ∈ ℤ
68		lcmcom 15927	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (1 lcm 𝑛) = (𝑛 lcm 1))
69	67, 68	mpan 689	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (1 lcm 𝑛) = (𝑛 lcm 1))
70		lcm1 15944	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (𝑛 lcm 1) = (abs‘𝑛))
71	66, 69, 70	3eqtrrd 2838	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → (abs‘𝑛) = ((lcm‘∅) lcm 𝑛))
72	63, 64, 71	3eqtrd 2837	. . . . . 6 ⊢ (𝑛 ∈ ℤ → (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
73	72	adantl 485	. . . . 5 ⊢ ((∅ ⊆ ℤ ∧ 𝑛 ∈ ℤ) → (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
74	73	ralrimiva 3149	. . . 4 ⊢ (∅ ⊆ ℤ → ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
75	58, 74	jca 515	. . 3 ⊢ (∅ ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
76		unss 4111	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ)
77		simpl 486	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑦 ⊆ ℤ)
78	76, 77	sylbir 238	. . . . . . 7 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑦 ⊆ ℤ)
79	78	adantl 485	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ ℤ) → 𝑦 ⊆ ℤ)
80		vex 3444	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
81	80	snss 4679	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℤ ↔ {𝑧} ⊆ ℤ)
82		lcmfunsnlem1 15971	. . . . . . . . . . . 12 ⊢ (((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 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
83		lcmfunsnlem2 15974	. . . . . . . . . . . 12 ⊢ (((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))
84	82, 83	jca 515	. . . . . . . . . . 11 ⊢ (((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 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
85	84	3exp1 1349	. . . . . . . . . 10 ⊢ (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 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))))
86	81, 85	sylbir 238	. . . . . . . . 9 ⊢ ({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 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))))
87	86	impcom 411	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → (𝑦 ∈ Fin → ((∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))
88	76, 87	sylbir 238	. . . . . . 7 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℤ → (𝑦 ∈ Fin → ((∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))
89	88	impcom 411	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ ℤ) → ((∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
90	79, 89	embantd 59	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ ℤ) → ((𝑦 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
91	90	ex 416	. . . 4 ⊢ (𝑦 ∈ Fin → ((𝑦 ∪ {𝑧}) ⊆ ℤ → ((𝑦 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))
92	91	com23 86	. . 3 ⊢ (𝑦 ∈ Fin → ((𝑦 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛))) → ((𝑦 ∪ {𝑧}) ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))
93	13, 26, 39, 52, 75, 92	findcard2 8742	. 2 ⊢ (𝑌 ∈ Fin → (𝑌 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘 → (lcm‘𝑌) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm‘𝑌) lcm 𝑛))))
94	93	impcom 411	1 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘 → (lcm‘𝑌) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm‘𝑌) lcm 𝑛)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∪ cun 3879   ⊆ wss 3881  ∅c0 4243  {csn 4525   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Fincfn 8492  1c1 10527  ℤcz 11969  abscabs 14585   ∥ cdvds 15599   lcm clcm 15922  lcmclcmf 15923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-inf 8891  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-fzo 13029  df-fl 13157  df-mod 13233  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-prod 15252  df-dvds 15600  df-gcd 15834  df-lcm 15924  df-lcmf 15925

This theorem is referenced by:  lcmfdvds  15976  lcmfunsn  15978