Theorem lcmfunsnlem 16658

Description: Lemma for lcmfdvds 16659 and lcmfunsn 16661. 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 16654 and lcmfunsnlem2 16657 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 3961	. . . 4 ⊢ (𝑥 = ∅ → (𝑥 ⊆ ℤ ↔ ∅ ⊆ ℤ))
2		raleq 3316	. . . . . . 7 ⊢ (𝑥 = ∅ → (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘))
3		fveq2 6863	. . . . . . . 8 ⊢ (𝑥 = ∅ → (lcm‘𝑥) = (lcm‘∅))
4	3	breq1d 5109	. . . . . . 7 ⊢ (𝑥 = ∅ → ((lcm‘𝑥) ∥ 𝑘 ↔ (lcm‘∅) ∥ 𝑘))
5	2, 4	imbi12d 346	. . . . . 6 ⊢ (𝑥 = ∅ → ((∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘)))
6	5	ralbidv 3184	. . . . 5 ⊢ (𝑥 = ∅ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘)))
7		uneq1 4114	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ∪ {𝑛}) = (∅ ∪ {𝑛}))
8	7	fveq2d 6867	. . . . . . 7 ⊢ (𝑥 = ∅ → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(∅ ∪ {𝑛})))
9	3	oveq1d 7407	. . . . . . 7 ⊢ (𝑥 = ∅ → ((lcm‘𝑥) lcm 𝑛) = ((lcm‘∅) lcm 𝑛))
10	8, 9	eqeq12d 2777	. . . . . 6 ⊢ (𝑥 = ∅ → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
11	10	ralbidv 3184	. . . . 5 ⊢ (𝑥 = ∅ → (∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
12	6, 11	anbi12d 641	. . . 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 346	. . 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 3961	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ ℤ ↔ 𝑦 ⊆ ℤ))
15		raleq 3316	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘))
16		fveq2 6863	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (lcm‘𝑥) = (lcm‘𝑦))
17	16	breq1d 5109	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((lcm‘𝑥) ∥ 𝑘 ↔ (lcm‘𝑦) ∥ 𝑘))
18	15, 17	imbi12d 346	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
19	18	ralbidv 3184	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑦 𝑚 ∥ 𝑘 → (lcm‘𝑦) ∥ 𝑘)))
20		uneq1 4114	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∪ {𝑛}) = (𝑦 ∪ {𝑛}))
21	20	fveq2d 6867	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑛})))
22	16	oveq1d 7407	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((lcm‘𝑥) lcm 𝑛) = ((lcm‘𝑦) lcm 𝑛))
23	21, 22	eqeq12d 2777	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm‘𝑦) lcm 𝑛)))
24	23	ralbidv 3184	. . . . 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 641	. . . 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 346	. . 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 3961	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ ℤ ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ))
28		raleq 3316	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘))
29		fveq2 6863	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘𝑥) = (lcm‘(𝑦 ∪ {𝑧})))
30	29	breq1d 5109	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘𝑥) ∥ 𝑘 ↔ (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
31	28, 30	imbi12d 346	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 ∥ 𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
32	31	ralbidv 3184	. . . . 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 4114	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∪ {𝑛}) = ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
34	33	fveq2d 6867	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})))
35	29	oveq1d 7407	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘𝑥) lcm 𝑛) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
36	34, 35	eqeq12d 2777	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
37	36	ralbidv 3184	. . . . 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 641	. . . 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 346	. . 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 3961	. . . 4 ⊢ (𝑥 = 𝑌 → (𝑥 ⊆ ℤ ↔ 𝑌 ⊆ ℤ))
41		raleq 3316	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 ↔ ∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘))
42		fveq2 6863	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (lcm‘𝑥) = (lcm‘𝑌))
43	42	breq1d 5109	. . . . . . 7 ⊢ (𝑥 = 𝑌 → ((lcm‘𝑥) ∥ 𝑘 ↔ (lcm‘𝑌) ∥ 𝑘))
44	41, 43	imbi12d 346	. . . . . 6 ⊢ (𝑥 = 𝑌 → ((∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ (∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘 → (lcm‘𝑌) ∥ 𝑘)))
45	44	ralbidv 3184	. . . . 5 ⊢ (𝑥 = 𝑌 → (∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑥 𝑚 ∥ 𝑘 → (lcm‘𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚 ∈ 𝑌 𝑚 ∥ 𝑘 → (lcm‘𝑌) ∥ 𝑘)))
46		uneq1 4114	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → (𝑥 ∪ {𝑛}) = (𝑌 ∪ {𝑛}))
47	46	fveq2d 6867	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(𝑌 ∪ {𝑛})))
48	42	oveq1d 7407	. . . . . . 7 ⊢ (𝑥 = 𝑌 → ((lcm‘𝑥) lcm 𝑛) = ((lcm‘𝑌) lcm 𝑛))
49	47, 48	eqeq12d 2777	. . . . . 6 ⊢ (𝑥 = 𝑌 → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm‘𝑥) lcm 𝑛) ↔ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm‘𝑌) lcm 𝑛)))
50	49	ralbidv 3184	. . . . 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 641	. . . 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 346	. . 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 16651	. . . . . . . 8 ⊢ (lcm‘∅) = 1
54		1dvds 16287	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ → 1 ∥ 𝑘)
55	53, 54	eqbrtrid 5134	. . . . . . 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 3153	. . . 4 ⊢ (∅ ⊆ ℤ → ∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘))
59		uncom 4111	. . . . . . . . . 10 ⊢ (∅ ∪ {𝑛}) = ({𝑛} ∪ ∅)
60		un0 4347	. . . . . . . . . 10 ⊢ ({𝑛} ∪ ∅) = {𝑛}
61	59, 60	eqtri 2784	. . . . . . . . 9 ⊢ (∅ ∪ {𝑛}) = {𝑛}
62	61	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (∅ ∪ {𝑛}) = {𝑛})
63	62	fveq2d 6867	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → (lcm‘(∅ ∪ {𝑛})) = (lcm‘{𝑛}))
64		lcmfsn 16652	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → (lcm‘{𝑛}) = (abs‘𝑛))
65	53	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ → (lcm‘∅) = 1)
66	65	oveq1d 7407	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → ((lcm‘∅) lcm 𝑛) = (1 lcm 𝑛))
67		1z 12598	. . . . . . . . 9 ⊢ 1 ∈ ℤ
68		lcmcom 16610	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (1 lcm 𝑛) = (𝑛 lcm 1))
69	67, 68	mpan 700	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (1 lcm 𝑛) = (𝑛 lcm 1))
70		lcm1 16627	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (𝑛 lcm 1) = (abs‘𝑛))
71	66, 69, 70	3eqtrrd 2801	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → (abs‘𝑛) = ((lcm‘∅) lcm 𝑛))
72	63, 64, 71	3eqtrd 2800	. . . . . 6 ⊢ (𝑛 ∈ ℤ → (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
73	72	adantl 485	. . . . 5 ⊢ ((∅ ⊆ ℤ ∧ 𝑛 ∈ ℤ) → (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
74	73	ralrimiva 3153	. . . 4 ⊢ (∅ ⊆ ℤ → ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
75	58, 74	jca 519	. . 3 ⊢ (∅ ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚 ∥ 𝑘 → (lcm‘∅) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
76		unss 4142	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ)
77		simpl 486	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑦 ⊆ ℤ)
78	76, 77	sylbir 237	. . . . . . 7 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑦 ⊆ ℤ)
79	78	adantl 485	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ ℤ) → 𝑦 ⊆ ℤ)
80		vex 3457	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
81	80	snss 4742	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℤ ↔ {𝑧} ⊆ ℤ)
82		lcmfunsnlem1 16654	. . . . . . . . . . . 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 16657	. . . . . . . . . . . 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 519	. . . . . . . . . . 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 1365	. . . . . . . . . 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 237	. . . . . . . . 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 237	. . . . . . 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 9129	. 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 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  {csn 4581   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392  Fincfn 8923  1c1 11071  ℤcz 12565  abscabs 15244   ∥ cdvds 16269   lcm clcm 16605  lcmclcmf 16606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-inf2 9593  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-pre-sup 11148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-sup 9385  df-inf 9386  df-oi 9455  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12208  df-2 12277  df-3 12278  df-n0 12479  df-z 12566  df-uz 12837  df-rp 12991  df-fz 13510  df-fzo 13657  df-fl 13799  df-mod 13877  df-seq 14012  df-exp 14072  df-hash 14341  df-cj 15109  df-re 15110  df-im 15111  df-sqrt 15245  df-abs 15246  df-clim 15498  df-prod 15917  df-dvds 16270  df-gcd 16512  df-lcm 16607  df-lcmf 16608

This theorem is referenced by:  lcmfdvds  16659  lcmfunsn  16661