Theorem fsuppmapnn0fiubex 14012

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 7432	. . . . . . 7 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
4	3	sseq2d 4012	. . . . . 6 ⊢ (𝑚 = 0 → ((𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ (𝑓 supp 𝑍) ⊆ (0...0)))
5	4	ralbidv 3168	. . . . 5 ⊢ (𝑚 = 0 → (∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0)))
6	5	adantl 480	. . . 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 4517	. . . . . 6 ⊢ ∀𝑓 ∈ ∅ (𝑓 supp 𝑍) ⊆ (0...0)
8		raleq 3312	. . . . . 6 ⊢ (∅ = 𝑀 → (∀𝑓 ∈ ∅ (𝑓 supp 𝑍) ⊆ (0...0) ↔ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0)))
9	7, 8	mpbii 232	. . . . 5 ⊢ (∅ = 𝑀 → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0))
10		0ss 4401	. . . . . . 7 ⊢ ∅ ⊆ (0...0)
11		sseq1 4005	. . . . . . 7 ⊢ ((𝑓 supp 𝑍) = ∅ → ((𝑓 supp 𝑍) ⊆ (0...0) ↔ ∅ ⊆ (0...0)))
12	10, 11	mpbiri 257	. . . . . 6 ⊢ ((𝑓 supp 𝑍) = ∅ → (𝑓 supp 𝑍) ⊆ (0...0))
13	12	ralimi 3073	. . . . 5 ⊢ (∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅ → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0))
14	9, 13	jaoi 855	. . . 4 ⊢ ((∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...0))
15	2, 6, 14	rspcedvd 3610	. . 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 483	. . . . . 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 981	. . . . . . . . . 10 ⊢ (¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ↔ (¬ ∅ = 𝑀 ∧ ¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅))
20		oveq1 7431	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑓 supp 𝑍) = (𝑔 supp 𝑍))
21	20	eqeq1d 2728	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ((𝑓 supp 𝑍) = ∅ ↔ (𝑔 supp 𝑍) = ∅))
22	21	cbvralvw 3225	. . . . . . . . . . . 12 ⊢ (∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅ ↔ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅)
23	22	notbii 319	. . . . . . . . . . 11 ⊢ (¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅ ↔ ¬ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅)
24	23	anbi2i 621	. . . . . . . . . 10 ⊢ ((¬ ∅ = 𝑀 ∧ ¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ↔ (¬ ∅ = 𝑀 ∧ ¬ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅))
25	19, 24	bitri 274	. . . . . . . . 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 2931	. . . . . . . . . . . 12 ⊢ ((𝑔 supp 𝑍) ≠ ∅ ↔ ¬ (𝑔 supp 𝑍) = ∅)
28	27	bicomi 223	. . . . . . . . . . 11 ⊢ (¬ (𝑔 supp 𝑍) = ∅ ↔ (𝑔 supp 𝑍) ≠ ∅)
29	28	rexbii 3084	. . . . . . . . . 10 ⊢ (∃𝑔 ∈ 𝑀 ¬ (𝑔 supp 𝑍) = ∅ ↔ ∃𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
30	26, 29	sylbb1 236	. . . . . . . . 9 ⊢ (¬ ∀𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∅ → ∃𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
31	25, 30	simplbiim 503	. . . . . . . 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 5075	. . . . . . 7 ⊢ (∃𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅ ↔ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
34	32, 33	sylib 217	. . . . . 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 510	. . . . 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 7431	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝑔 supp 𝑍) = (𝑓 supp 𝑍))
37	36	cbviunv 5048	. . . . . 6 ⊢ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍)
38		eqid 2726	. . . . . 6 ⊢ sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ) = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )
39	37, 38	fsuppmapnn0fiublem 14010	. . . . 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 1910	. . . . . . . . . 10 ⊢ Ⅎ𝑓∅ = 𝑀
42		nfra1 3272	. . . . . . . . . 10 ⊢ Ⅎ𝑓∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅
43	41, 42	nfor 1900	. . . . . . . . 9 ⊢ Ⅎ𝑓(∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅)
44	43	nfn 1853	. . . . . . . 8 ⊢ Ⅎ𝑓 ¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅)
45		nfv 1910	. . . . . . . 8 ⊢ Ⅎ𝑓(𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)
46	44, 45	nfan 1895	. . . . . . 7 ⊢ Ⅎ𝑓(¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉))
47		nfra1 3272	. . . . . . 7 ⊢ Ⅎ𝑓∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍
48	46, 47	nfan 1895	. . . . . 6 ⊢ Ⅎ𝑓((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍)
49		nfv 1910	. . . . . 6 ⊢ Ⅎ𝑓 𝑚 = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )
50	48, 49	nfan 1895	. . . . 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 7432	. . . . . . 7 ⊢ (𝑚 = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ) → (0...𝑚) = (0...sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )))
52	51	sseq2d 4012	. . . . . 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 480	. . . . 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 3261	. . . 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 2931	. . . . . . . . . . . 12 ⊢ ((𝑓 supp 𝑍) ≠ ∅ ↔ ¬ (𝑓 supp 𝑍) = ∅)
57	56	bicomi 223	. . . . . . . . . . 11 ⊢ (¬ (𝑓 supp 𝑍) = ∅ ↔ (𝑓 supp 𝑍) ≠ ∅)
58	57	rexbii 3084	. . . . . . . . . 10 ⊢ (∃𝑓 ∈ 𝑀 ¬ (𝑓 supp 𝑍) = ∅ ↔ ∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅)
59	55, 58	sylbb1 236	. . . . . . . . 9 ⊢ (¬ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅ → ∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅)
60	19, 59	simplbiim 503	. . . . . . . 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 5075	. . . . . . . 8 ⊢ (∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅ ↔ ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅)
63	20	cbviunv 5048	. . . . . . . . 9 ⊢ ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍)
64	63	neeq1i 2995	. . . . . . . 8 ⊢ (∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅ ↔ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
65	62, 64	bitri 274	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅ ↔ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) ≠ ∅)
66	61, 65	sylib 217	. . . . . 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 510	. . . . 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 14011	. . . . 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 3610	. . 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 418	. 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 205   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051  ∃wrex 3060   ⊆ wss 3947  ∅c0 4325  ∪ ciun 5001   class class class wbr 5153  (class class class)co 7424   supp csupp 8174   ↑_m cmap 8855  Fincfn 8974   finSupp cfsupp 9405  supcsup 9483  ℝcr 11157  0cc0 11158   < clt 11298  ℕ₀cn0 12524  ...cfz 13538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-supp 8175  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fsupp 9406  df-sup 9485  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-n0 12525  df-z 12611  df-uz 12875  df-fz 13539

This theorem is referenced by:  fsuppmapnn0fiub0  14013