Theorem fsuppmapnn0fiubex 13949

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 12447	. . . . 5 ⊢ 0 ∈ ℕ0
2	1	a1i 11	. . . 4 ⊢ ((∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → 0 ∈ ℕ0)
3		oveq2 7370	. . . . . . 7 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
4	3	sseq2d 3955	. . . . . 6 ⊢ (𝑚 = 0 → ((𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ (𝑓 supp 𝑍) ⊆ (0...0)))
5	4	ralbidv 3161	. . . . 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 4439	. . . . . 6 ⊢ ∀𝑓 ∈ ∅ (𝑓 supp 𝑍) ⊆ (0...0)
8		raleq 3293	. . . . . 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 4341	. . . . . . 7 ⊢ ∅ ⊆ (0...0)
11		sseq1 3948	. . . . . . 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 3075	. . . . 5 ⊢ (∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅ → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0))
14	9, 13	jaoi 858	. . . 4 ⊢ ((∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0))
15	2, 6, 14	rspcedvd 3567	. . 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 986	. . . . . . . . . 10 ⊢ (¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ↔ (¬ ∅ = 𝑀 ∧ ¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅))
20		oveq1 7369	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑓 supp 𝑍) = (𝑔 supp 𝑍))
21	20	eqeq1d 2739	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ((𝑓 supp 𝑍) = ∅ ↔ (𝑔 supp 𝑍) = ∅))
22	21	cbvralvw 3216	. . . . . . . . . . . 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 624	. . . . . . . . . 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 3090	. . . . . . . . . 10 ⊢ (∃𝑔 ∈ 𝑀 ¬ (𝑔 supp 𝑍) = ∅ ↔ ¬ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅)
27		df-ne 2934	. . . . . . . . . . . 12 ⊢ ((𝑔 supp 𝑍) ≠ ∅ ↔ ¬ (𝑔 supp 𝑍) = ∅)
28	27	bicomi 224	. . . . . . . . . . 11 ⊢ (¬ (𝑔 supp 𝑍) = ∅ ↔ (𝑔 supp 𝑍) ≠ ∅)
29	28	rexbii 3085	. . . . . . . . . 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 727	. . . . . . 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 5010	. . . . . . 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 7369	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝑔 supp 𝑍) = (𝑓 supp 𝑍))
37	36	cbviunv 4982	. . . . . 6 ⊢ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍)
38		eqid 2737	. . . . . 6 ⊢ sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ) = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )
39	37, 38	fsuppmapnn0fiublem 13947	. . . . 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 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑓∅ = 𝑀
42		nfra1 3262	. . . . . . . . . 10 ⊢ Ⅎ𝑓∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅
43	41, 42	nfor 1906	. . . . . . . . 9 ⊢ Ⅎ𝑓(∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅)
44	43	nfn 1859	. . . . . . . 8 ⊢ Ⅎ𝑓 ¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅)
45		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑓(𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)
46	44, 45	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑓(¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉))
47		nfra1 3262	. . . . . . 7 ⊢ Ⅎ𝑓∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍
48	46, 47	nfan 1901	. . . . . 6 ⊢ Ⅎ𝑓((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍)
49		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑓 𝑚 = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )
50	48, 49	nfan 1901	. . . . 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 7370	. . . . . . 7 ⊢ (𝑚 = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ) → (0...𝑚) = (0...sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )))
52	51	sseq2d 3955	. . . . . 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 3251	. . . 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 3090	. . . . . . . . . 10 ⊢ (∃𝑓 ∈ 𝑀 ¬ (𝑓 supp 𝑍) = ∅ ↔ ¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅)
56		df-ne 2934	. . . . . . . . . . . 12 ⊢ ((𝑓 supp 𝑍) ≠ ∅ ↔ ¬ (𝑓 supp 𝑍) = ∅)
57	56	bicomi 224	. . . . . . . . . . 11 ⊢ (¬ (𝑓 supp 𝑍) = ∅ ↔ (𝑓 supp 𝑍) ≠ ∅)
58	57	rexbii 3085	. . . . . . . . . 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 727	. . . . . . 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 5010	. . . . . . . 8 ⊢ (∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅ ↔ ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅)
63	20	cbviunv 4982	. . . . . . . . 9 ⊢ ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍)
64	63	neeq1i 2997	. . . . . . . 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 13948	. . . . 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 3567	. . 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 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890  ∅c0 4274  ∪ ciun 4934   class class class wbr 5086  (class class class)co 7362   supp csupp 8105   ↑_m cmap 8768  Fincfn 8888   finSupp cfsupp 9269  supcsup 9348  ℝcr 11032  0cc0 11033   < clt 11174  ℕ₀cn0 12432  ...cfz 13456

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-supp 8106  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-er 8638  df-map 8770  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-fsupp 9270  df-sup 9350  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-n0 12433  df-z 12520  df-uz 12784  df-fz 13457

This theorem is referenced by:  fsuppmapnn0fiub0  13950