Theorem lcmfunsnlem 16562

Description: Lemma for lcmfdvds 16563 and lcmfunsn 16565. 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 16558 and lcmfunsnlem2 16561 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 3957	. . . 4 ⊢ (𝑥 = ∅ → (𝑥 ⊆ ℤ ↔ ∅ ⊆ ℤ))
2		raleq 3291	. . . . . . 7 ⊢ (𝑥 = ∅ → (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘))
3		fveq2 6831	. . . . . . . 8 ⊢ (𝑥 = ∅ → (lcm‘𝑥) = (lcm‘∅))
4	3	breq1d 5105	. . . . . . 7 ⊢ (𝑥 = ∅ → ((lcm‘𝑥) ∥ 𝑘 ↔ (lcm‘∅) ∥ 𝑘))
5	2, 4	imbi12d 344	. . . . . 6 ⊢ (𝑥 = ∅ → ((∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘)))
6	5	ralbidv 3157	. . . . 5 ⊢ (𝑥 = ∅ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘)))
7		uneq1 4112	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ∪ {𝑛}) = (∅ ∪ {𝑛}))
8	7	fveq2d 6835	. . . . . . 7 ⊢ (𝑥 = ∅ → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(∅ ∪ {𝑛})))
9	3	oveq1d 7370	. . . . . . 7 ⊢ (𝑥 = ∅ → ((lcm‘𝑥) lcm 𝑛) = ((lcm‘∅) lcm 𝑛))
10	8, 9	eqeq12d 2749	. . . . . 6 ⊢ (𝑥 = ∅ → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
11	10	ralbidv 3157	. . . . 5 ⊢ (𝑥 = ∅ → (∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
12	6, 11	anbi12d 632	. . . 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 344	. . 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 3957	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ ℤ ↔ 𝑦 ⊆ ℤ))
15		raleq 3291	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘))
16		fveq2 6831	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (lcm‘𝑥) = (lcm‘𝑦))
17	16	breq1d 5105	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((lcm‘𝑥) ∥ 𝑘 ↔ (lcm‘𝑦) ∥ 𝑘))
18	15, 17	imbi12d 344	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
19	18	ralbidv 3157	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
20		uneq1 4112	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∪ {𝑛}) = (𝑦 ∪ {𝑛}))
21	20	fveq2d 6835	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑛})))
22	16	oveq1d 7370	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((lcm‘𝑥) lcm 𝑛) = ((lcm‘𝑦) lcm 𝑛))
23	21, 22	eqeq12d 2749	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)))
24	23	ralbidv 3157	. . . . 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 632	. . . 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 344	. . 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 3957	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ ℤ ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ))
28		raleq 3291	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘))
29		fveq2 6831	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘𝑥) = (lcm‘(𝑦 ∪ {𝑧})))
30	29	breq1d 5105	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘𝑥) ∥ 𝑘 ↔ (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
31	28, 30	imbi12d 344	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
32	31	ralbidv 3157	. . . . 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 4112	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∪ {𝑛}) = ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
34	33	fveq2d 6835	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})))
35	29	oveq1d 7370	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘𝑥) lcm 𝑛) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
36	34, 35	eqeq12d 2749	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
37	36	ralbidv 3157	. . . . 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 632	. . . 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 344	. . 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 3957	. . . 4 ⊢ (𝑥 = 𝑌 → (𝑥 ⊆ ℤ ↔ 𝑌 ⊆ ℤ))
41		raleq 3291	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘))
42		fveq2 6831	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (lcm‘𝑥) = (lcm‘𝑌))
43	42	breq1d 5105	. . . . . . 7 ⊢ (𝑥 = 𝑌 → ((lcm‘𝑥) ∥ 𝑘 ↔ (lcm‘𝑌) ∥ 𝑘))
44	41, 43	imbi12d 344	. . . . . 6 ⊢ (𝑥 = 𝑌 → ((∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ (∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘 → (lcm‘𝑌) ∥ 𝑘)))
45	44	ralbidv 3157	. . . . 5 ⊢ (𝑥 = 𝑌 → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘 → (lcm‘𝑌) ∥ 𝑘)))
46		uneq1 4112	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (𝑥 ∪ {𝑛}) = (𝑌 ∪ {𝑛}))
47	46	fveq2d 6835	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(𝑌 ∪ {𝑛})))
48	42	oveq1d 7370	. . . . . . 7 ⊢ (𝑥 = 𝑌 → ((lcm‘𝑥) lcm 𝑛) = ((lcm‘𝑌) lcm 𝑛))
49	47, 48	eqeq12d 2749	. . . . . 6 ⊢ (𝑥 = 𝑌 → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm‘𝑌) lcm 𝑛)))
50	49	ralbidv 3157	. . . . 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 632	. . . 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 344	. . 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 16555	. . . . . . . 8 ⊢ (lcm‘∅) = 1
54		1dvds 16191	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ → 1 ∥ 𝑘)
55	53, 54	eqbrtrid 5130	. . . . . . 7 ⊢ (𝑘 ∈ ℤ → (lcm‘∅) ∥ 𝑘)
56	55	a1d 25	. . . . . 6 ⊢ (𝑘 ∈ ℤ → (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘))
57	56	adantl 481	. . . . 5 ⊢ ((∅ ⊆ ℤ ∧ 𝑘 ∈ ℤ) → (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘))
58	57	ralrimiva 3126	. . . 4 ⊢ (∅ ⊆ ℤ → ∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘))
59		uncom 4109	. . . . . . . . . 10 ⊢ (∅ ∪ {𝑛}) = ({𝑛} ∪ ∅)
60		un0 4345	. . . . . . . . . 10 ⊢ ({𝑛} ∪ ∅) = {𝑛}
61	59, 60	eqtri 2756	. . . . . . . . 9 ⊢ (∅ ∪ {𝑛}) = {𝑛}
62	61	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (∅ ∪ {𝑛}) = {𝑛})
63	62	fveq2d 6835	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → (lcm‘(∅ ∪ {𝑛})) = (lcm‘{𝑛}))
64		lcmfsn 16556	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → (lcm‘{𝑛}) = (abs‘𝑛))
65	53	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → (lcm‘∅) = 1)
66	65	oveq1d 7370	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → ((lcm‘∅) lcm 𝑛) = (1 lcm 𝑛))
67		1z 12512	. . . . . . . . 9 ⊢ 1 ∈ ℤ
68		lcmcom 16514	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (1 lcm 𝑛) = (𝑛 lcm 1))
69	67, 68	mpan 690	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (1 lcm 𝑛) = (𝑛 lcm 1))
70		lcm1 16531	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (𝑛 lcm 1) = (abs‘𝑛))
71	66, 69, 70	3eqtrrd 2773	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → (abs‘𝑛) = ((lcm‘∅) lcm 𝑛))
72	63, 64, 71	3eqtrd 2772	. . . . . 6 ⊢ (𝑛 ∈ ℤ → (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
73	72	adantl 481	. . . . 5 ⊢ ((∅ ⊆ ℤ ∧ 𝑛 ∈ ℤ) → (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
74	73	ralrimiva 3126	. . . 4 ⊢ (∅ ⊆ ℤ → ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
75	58, 74	jca 511	. . 3 ⊢ (∅ ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
76		unss 4141	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ)
77		simpl 482	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑦 ⊆ ℤ)
78	76, 77	sylbir 235	. . . . . . 7 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑦 ⊆ ℤ)
79	78	adantl 481	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ ℤ) → 𝑦 ⊆ ℤ)
80		vex 3442	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
81	80	snss 4738	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℤ ↔ {𝑧} ⊆ ℤ)
82		lcmfunsnlem1 16558	. . . . . . . . . . . 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 16561	. . . . . . . . . . . 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 511	. . . . . . . . . . 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 1353	. . . . . . . . . 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 235	. . . . . . . . 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 407	. . . . . . . 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 235	. . . . . . 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 407	. . . . . 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 412	. . . 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 9084	. 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 407	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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049   ∪ cun 3897   ⊆ wss 3899  ∅c0 4284  {csn 4577   class class class wbr 5095  ‘cfv 6489  (class class class)co 7355  Fincfn 8878  1c1 11017  ℤcz 12478  abscabs 15151   ∥ cdvds 16173   lcm clcm 16509  lcmclcmf 16510

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-inf2 9541  ax-cnex 11072  ax-resscn 11073  ax-1cn 11074  ax-icn 11075  ax-addcl 11076  ax-addrcl 11077  ax-mulcl 11078  ax-mulrcl 11079  ax-mulcom 11080  ax-addass 11081  ax-mulass 11082  ax-distr 11083  ax-i2m1 11084  ax-1ne0 11085  ax-1rid 11086  ax-rnegex 11087  ax-rrecex 11088  ax-cnre 11089  ax-pre-lttri 11090  ax-pre-lttrn 11091  ax-pre-ltadd 11092  ax-pre-mulgt0 11093  ax-pre-sup 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-er 8631  df-en 8879  df-dom 8880  df-sdom 8881  df-fin 8882  df-sup 9336  df-inf 9337  df-oi 9406  df-card 9842  df-pnf 11158  df-mnf 11159  df-xr 11160  df-ltxr 11161  df-le 11162  df-sub 11356  df-neg 11357  df-div 11785  df-nn 12136  df-2 12198  df-3 12199  df-n0 12392  df-z 12479  df-uz 12743  df-rp 12901  df-fz 13418  df-fzo 13565  df-fl 13706  df-mod 13784  df-seq 13919  df-exp 13979  df-hash 14248  df-cj 15016  df-re 15017  df-im 15018  df-sqrt 15152  df-abs 15153  df-clim 15405  df-prod 15821  df-dvds 16174  df-gcd 16416  df-lcm 16511  df-lcmf 16512

This theorem is referenced by:  lcmfdvds  16563  lcmfunsn  16565