Theorem fsuppmapnn0fiubex 13354

Description: If all functions of a finite set of functions over the nonnegative integers are finitely supported, then the support of all these functions is contained in a finite set of sequential integers starting at 0. (Contributed by AV, 2-Oct-2019.)

Assertion

Ref	Expression
fsuppmapnn0fiubex	⊢ ((𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → (∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚)))

Distinct variable groups:   𝑓,𝑀,𝑚   𝑅,𝑓,𝑚   𝑓,𝑉,𝑚   𝑓,𝑍,𝑚

Proof of Theorem fsuppmapnn0fiubex

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nn0 11906	. . . . 5 ⊢ 0 ∈ ℕ0
2	1	a1i 11	. . . 4 ⊢ ((∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → 0 ∈ ℕ0)
3		oveq2 7158	. . . . . . 7 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
4	3	sseq2d 3999	. . . . . 6 ⊢ (𝑚 = 0 → ((𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ (𝑓 supp 𝑍) ⊆ (0...0)))
5	4	ralbidv 3197	. . . . 5 ⊢ (𝑚 = 0 → (∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0)))
6	5	adantl 484	. . . 4 ⊢ (((∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ 𝑚 = 0) → (∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0)))
7		ral0 4456	. . . . . 6 ⊢ ∀𝑓 ∈ ∅ (𝑓 supp 𝑍) ⊆ (0...0)
8		raleq 3406	. . . . . 6 ⊢ (∅ = 𝑀 → (∀𝑓 ∈ ∅ (𝑓 supp 𝑍) ⊆ (0...0) ↔ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0)))
9	7, 8	mpbii 235	. . . . 5 ⊢ (∅ = 𝑀 → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0))
10		0ss 4350	. . . . . . 7 ⊢ ∅ ⊆ (0...0)
11		sseq1 3992	. . . . . . 7 ⊢ ((𝑓 supp 𝑍) = ∅ → ((𝑓 supp 𝑍) ⊆ (0...0) ↔ ∅ ⊆ (0...0)))
12	10, 11	mpbiri 260	. . . . . 6 ⊢ ((𝑓 supp 𝑍) = ∅ → (𝑓 supp 𝑍) ⊆ (0...0))
13	12	ralimi 3160	. . . . 5 ⊢ (∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅ → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0))
14	9, 13	jaoi 853	. . . 4 ⊢ ((∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0))
15	2, 6, 14	rspcedvd 3626	. . 3 ⊢ ((∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → ∃𝑚 ∈ ℕ0 ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚))
16	15	2a1d 26	. 2 ⊢ ((∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → ((𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → (∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚))))
17		simplr 767	. . . . 5 ⊢ (((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍) → (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉))
18		simpr 487	. . . . . 6 ⊢ (((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍) → ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍)
19		ioran 980	. . . . . . . . . 10 ⊢ (¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ↔ (¬ ∅ = 𝑀 ∧ ¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅))
20		oveq1 7157	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑓 supp 𝑍) = (𝑔 supp 𝑍))
21	20	eqeq1d 2823	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ((𝑓 supp 𝑍) = ∅ ↔ (𝑔 supp 𝑍) = ∅))
22	21	cbvralvw 3450	. . . . . . . . . . . 12 ⊢ (∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅ ↔ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅)
23	22	notbii 322	. . . . . . . . . . 11 ⊢ (¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅ ↔ ¬ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅)
24	23	anbi2i 624	. . . . . . . . . 10 ⊢ ((¬ ∅ = 𝑀 ∧ ¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ↔ (¬ ∅ = 𝑀 ∧ ¬ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅))
25	19, 24	bitri 277	. . . . . . . . 9 ⊢ (¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ↔ (¬ ∅ = 𝑀 ∧ ¬ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅))
26		rexnal 3238	. . . . . . . . . 10 ⊢ (∃𝑔 ∈ 𝑀 ¬ (𝑔 supp 𝑍) = ∅ ↔ ¬ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅)
27		df-ne 3017	. . . . . . . . . . . 12 ⊢ ((𝑔 supp 𝑍) ≠ ∅ ↔ ¬ (𝑔 supp 𝑍) = ∅)
28	27	bicomi 226	. . . . . . . . . . 11 ⊢ (¬ (𝑔 supp 𝑍) = ∅ ↔ (𝑔 supp 𝑍) ≠ ∅)
29	28	rexbii 3247	. . . . . . . . . 10 ⊢ (∃𝑔 ∈ 𝑀 ¬ (𝑔 supp 𝑍) = ∅ ↔ ∃𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
30	26, 29	sylbb1 239	. . . . . . . . 9 ⊢ (¬ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅ → ∃𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
31	25, 30	simplbiim 507	. . . . . . . 8 ⊢ (¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → ∃𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
32	31	ad2antrr 724	. . . . . . 7 ⊢ (((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍) → ∃𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
33		iunn0 4982	. . . . . . 7 ⊢ (∃𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅ ↔ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
34	32, 33	sylib 220	. . . . . 6 ⊢ (((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍) → ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
35	18, 34	jca 514	. . . . 5 ⊢ (((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍) → (∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 ∧ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅))
36		oveq1 7157	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝑔 supp 𝑍) = (𝑓 supp 𝑍))
37	36	cbviunv 4958	. . . . . 6 ⊢ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍)
38		eqid 2821	. . . . . 6 ⊢ sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ) = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )
39	37, 38	fsuppmapnn0fiublem 13352	. . . . 5 ⊢ ((𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → ((∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 ∧ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅) → sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ) ∈ ℕ0))
40	17, 35, 39	sylc 65	. . . 4 ⊢ (((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍) → sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ) ∈ ℕ0)
41		nfv 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑓∅ = 𝑀
42		nfra1 3219	. . . . . . . . . 10 ⊢ Ⅎ𝑓∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅
43	41, 42	nfor 1901	. . . . . . . . 9 ⊢ Ⅎ𝑓(∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅)
44	43	nfn 1853	. . . . . . . 8 ⊢ Ⅎ𝑓 ¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅)
45		nfv 1911	. . . . . . . 8 ⊢ Ⅎ𝑓(𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)
46	44, 45	nfan 1896	. . . . . . 7 ⊢ Ⅎ𝑓(¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉))
47		nfra1 3219	. . . . . . 7 ⊢ Ⅎ𝑓∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍
48	46, 47	nfan 1896	. . . . . 6 ⊢ Ⅎ𝑓((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍)
49		nfv 1911	. . . . . 6 ⊢ Ⅎ𝑓 𝑚 = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )
50	48, 49	nfan 1896	. . . . 5 ⊢ Ⅎ𝑓(((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍) ∧ 𝑚 = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ))
51		oveq2 7158	. . . . . . 7 ⊢ (𝑚 = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ) → (0...𝑚) = (0...sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )))
52	51	sseq2d 3999	. . . . . 6 ⊢ (𝑚 = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ) → ((𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ (𝑓 supp 𝑍) ⊆ (0...sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ))))
53	52	adantl 484	. . . . 5 ⊢ ((((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍) ∧ 𝑚 = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )) → ((𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ (𝑓 supp 𝑍) ⊆ (0...sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ))))
54	50, 53	ralbid 3231	. . . 4 ⊢ ((((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍) ∧ 𝑚 = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )) → (∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ))))
55		rexnal 3238	. . . . . . . . . 10 ⊢ (∃𝑓 ∈ 𝑀 ¬ (𝑓 supp 𝑍) = ∅ ↔ ¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅)
56		df-ne 3017	. . . . . . . . . . . 12 ⊢ ((𝑓 supp 𝑍) ≠ ∅ ↔ ¬ (𝑓 supp 𝑍) = ∅)
57	56	bicomi 226	. . . . . . . . . . 11 ⊢ (¬ (𝑓 supp 𝑍) = ∅ ↔ (𝑓 supp 𝑍) ≠ ∅)
58	57	rexbii 3247	. . . . . . . . . 10 ⊢ (∃𝑓 ∈ 𝑀 ¬ (𝑓 supp 𝑍) = ∅ ↔ ∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅)
59	55, 58	sylbb1 239	. . . . . . . . 9 ⊢ (¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅ → ∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅)
60	19, 59	simplbiim 507	. . . . . . . 8 ⊢ (¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → ∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅)
61	60	ad2antrr 724	. . . . . . 7 ⊢ (((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍) → ∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅)
62		iunn0 4982	. . . . . . . 8 ⊢ (∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅ ↔ ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅)
63	20	cbviunv 4958	. . . . . . . . 9 ⊢ ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍)
64	63	neeq1i 3080	. . . . . . . 8 ⊢ (∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅ ↔ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
65	62, 64	bitri 277	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅ ↔ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
66	61, 65	sylib 220	. . . . . 6 ⊢ (((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍) → ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
67	18, 66	jca 514	. . . . 5 ⊢ (((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍) → (∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 ∧ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅))
68	37, 38	fsuppmapnn0fiub 13353	. . . . 5 ⊢ ((𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → ((∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 ∧ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅) → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ))))
69	17, 67, 68	sylc 65	. . . 4 ⊢ (((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍) → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )))
70	40, 54, 69	rspcedvd 3626	. . 3 ⊢ (((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍) → ∃𝑚 ∈ ℕ0 ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚))
71	70	exp31 422	. 2 ⊢ (¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → ((𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → (∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚))))
72	16, 71	pm2.61i 184	1 ⊢ ((𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → (∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139   ⊆ wss 3936  ∅c0 4291  ∪ ciun 4912   class class class wbr 5059  (class class class)co 7150   supp csupp 7824   ↑_m cmap 8400  Fincfn 8503   finSupp cfsupp 8827  supcsup 8898  ℝcr 10530  0cc0 10531   < clt 10669  ℕ₀cn0 11891  ...cfz 12886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-sup 8900  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12887

This theorem is referenced by:  fsuppmapnn0fiub0  13355