Theorem fsuppmapnn0fiubex 14030

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 12539	. . . . 5 ⊢ 0 ∈ ℕ0
2	1	a1i 11	. . . 4 ⊢ ((∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → 0 ∈ ℕ0)
3		oveq2 7439	. . . . . . 7 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
4	3	sseq2d 4028	. . . . . 6 ⊢ (𝑚 = 0 → ((𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ (𝑓 supp 𝑍) ⊆ (0...0)))
5	4	ralbidv 3176	. . . . 5 ⊢ (𝑚 = 0 → (∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0)))
6	5	adantl 481	. . . 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 4519	. . . . . 6 ⊢ ∀𝑓 ∈ ∅ (𝑓 supp 𝑍) ⊆ (0...0)
8		raleq 3321	. . . . . 6 ⊢ (∅ = 𝑀 → (∀𝑓 ∈ ∅ (𝑓 supp 𝑍) ⊆ (0...0) ↔ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0)))
9	7, 8	mpbii 233	. . . . 5 ⊢ (∅ = 𝑀 → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0))
10		0ss 4406	. . . . . . 7 ⊢ ∅ ⊆ (0...0)
11		sseq1 4021	. . . . . . 7 ⊢ ((𝑓 supp 𝑍) = ∅ → ((𝑓 supp 𝑍) ⊆ (0...0) ↔ ∅ ⊆ (0...0)))
12	10, 11	mpbiri 258	. . . . . 6 ⊢ ((𝑓 supp 𝑍) = ∅ → (𝑓 supp 𝑍) ⊆ (0...0))
13	12	ralimi 3081	. . . . 5 ⊢ (∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅ → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0))
14	9, 13	jaoi 857	. . . 4 ⊢ ((∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0))
15	2, 6, 14	rspcedvd 3624	. . 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 769	. . . . 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 484	. . . . . 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 985	. . . . . . . . . 10 ⊢ (¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ↔ (¬ ∅ = 𝑀 ∧ ¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅))
20		oveq1 7438	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑓 supp 𝑍) = (𝑔 supp 𝑍))
21	20	eqeq1d 2737	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ((𝑓 supp 𝑍) = ∅ ↔ (𝑔 supp 𝑍) = ∅))
22	21	cbvralvw 3235	. . . . . . . . . . . 12 ⊢ (∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅ ↔ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅)
23	22	notbii 320	. . . . . . . . . . 11 ⊢ (¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅ ↔ ¬ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅)
24	23	anbi2i 623	. . . . . . . . . 10 ⊢ ((¬ ∅ = 𝑀 ∧ ¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ↔ (¬ ∅ = 𝑀 ∧ ¬ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅))
25	19, 24	bitri 275	. . . . . . . . 9 ⊢ (¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ↔ (¬ ∅ = 𝑀 ∧ ¬ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅))
26		rexnal 3098	. . . . . . . . . 10 ⊢ (∃𝑔 ∈ 𝑀 ¬ (𝑔 supp 𝑍) = ∅ ↔ ¬ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅)
27		df-ne 2939	. . . . . . . . . . . 12 ⊢ ((𝑔 supp 𝑍) ≠ ∅ ↔ ¬ (𝑔 supp 𝑍) = ∅)
28	27	bicomi 224	. . . . . . . . . . 11 ⊢ (¬ (𝑔 supp 𝑍) = ∅ ↔ (𝑔 supp 𝑍) ≠ ∅)
29	28	rexbii 3092	. . . . . . . . . 10 ⊢ (∃𝑔 ∈ 𝑀 ¬ (𝑔 supp 𝑍) = ∅ ↔ ∃𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
30	26, 29	sylbb1 237	. . . . . . . . 9 ⊢ (¬ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅ → ∃𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
31	25, 30	simplbiim 504	. . . . . . . 8 ⊢ (¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → ∃𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
32	31	ad2antrr 726	. . . . . . 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 5072	. . . . . . 7 ⊢ (∃𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅ ↔ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
34	32, 33	sylib 218	. . . . . 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 511	. . . . 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 7438	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝑔 supp 𝑍) = (𝑓 supp 𝑍))
37	36	cbviunv 5045	. . . . . 6 ⊢ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍)
38		eqid 2735	. . . . . 6 ⊢ sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ) = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )
39	37, 38	fsuppmapnn0fiublem 14028	. . . . 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 1912	. . . . . . . . . 10 ⊢ Ⅎ𝑓∅ = 𝑀
42		nfra1 3282	. . . . . . . . . 10 ⊢ Ⅎ𝑓∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅
43	41, 42	nfor 1902	. . . . . . . . 9 ⊢ Ⅎ𝑓(∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅)
44	43	nfn 1855	. . . . . . . 8 ⊢ Ⅎ𝑓 ¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅)
45		nfv 1912	. . . . . . . 8 ⊢ Ⅎ𝑓(𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)
46	44, 45	nfan 1897	. . . . . . 7 ⊢ Ⅎ𝑓(¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉))
47		nfra1 3282	. . . . . . 7 ⊢ Ⅎ𝑓∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍
48	46, 47	nfan 1897	. . . . . 6 ⊢ Ⅎ𝑓((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍)
49		nfv 1912	. . . . . 6 ⊢ Ⅎ𝑓 𝑚 = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )
50	48, 49	nfan 1897	. . . . 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 7439	. . . . . . 7 ⊢ (𝑚 = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ) → (0...𝑚) = (0...sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )))
52	51	sseq2d 4028	. . . . . 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 481	. . . . 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 3271	. . . 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 3098	. . . . . . . . . 10 ⊢ (∃𝑓 ∈ 𝑀 ¬ (𝑓 supp 𝑍) = ∅ ↔ ¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅)
56		df-ne 2939	. . . . . . . . . . . 12 ⊢ ((𝑓 supp 𝑍) ≠ ∅ ↔ ¬ (𝑓 supp 𝑍) = ∅)
57	56	bicomi 224	. . . . . . . . . . 11 ⊢ (¬ (𝑓 supp 𝑍) = ∅ ↔ (𝑓 supp 𝑍) ≠ ∅)
58	57	rexbii 3092	. . . . . . . . . 10 ⊢ (∃𝑓 ∈ 𝑀 ¬ (𝑓 supp 𝑍) = ∅ ↔ ∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅)
59	55, 58	sylbb1 237	. . . . . . . . 9 ⊢ (¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅ → ∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅)
60	19, 59	simplbiim 504	. . . . . . . 8 ⊢ (¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → ∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅)
61	60	ad2antrr 726	. . . . . . 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 5072	. . . . . . . 8 ⊢ (∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅ ↔ ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅)
63	20	cbviunv 5045	. . . . . . . . 9 ⊢ ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍)
64	63	neeq1i 3003	. . . . . . . 8 ⊢ (∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅ ↔ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
65	62, 64	bitri 275	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅ ↔ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
66	61, 65	sylib 218	. . . . . 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 511	. . . . 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 14029	. . . . 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 3624	. . 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 419	. 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 182	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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963  ∅c0 4339  ∪ ciun 4996   class class class wbr 5148  (class class class)co 7431   supp csupp 8184   ↑_m cmap 8865  Fincfn 8984   finSupp cfsupp 9399  supcsup 9478  ℝcr 11152  0cc0 11153   < clt 11293  ℕ₀cn0 12524  ...cfz 13544

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545

This theorem is referenced by:  fsuppmapnn0fiub0  14031