Theorem fsuppmapnn0fiubex 14015

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 12521	. . . . 5 ⊢ 0 ∈ ℕ0
2	1	a1i 11	. . . 4 ⊢ ((∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) → 0 ∈ ℕ0)
3		oveq2 7418	. . . . . . 7 ⊢ (𝑚 = 0 → (0...𝑚) = (0...0))
4	3	sseq2d 3996	. . . . . 6 ⊢ (𝑚 = 0 → ((𝑓 supp 𝑍) ⊆ (0...𝑚) ↔ (𝑓 supp 𝑍) ⊆ (0...0)))
5	4	ralbidv 3164	. . . . 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 4493	. . . . . 6 ⊢ ∀𝑓 ∈ ∅ (𝑓 supp 𝑍) ⊆ (0...0)
8		raleq 3306	. . . . . 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 4380	. . . . . . 7 ⊢ ∅ ⊆ (0...0)
11		sseq1 3989	. . . . . . 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 3074	. . . . 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 3608	. . 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 768	. . . . 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 7417	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑓 supp 𝑍) = (𝑔 supp 𝑍))
21	20	eqeq1d 2738	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ((𝑓 supp 𝑍) = ∅ ↔ (𝑔 supp 𝑍) = ∅))
22	21	cbvralvw 3224	. . . . . . . . . . . 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 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 3084	. . . . . . . . . 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 5048	. . . . . . 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 7417	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝑔 supp 𝑍) = (𝑓 supp 𝑍))
37	36	cbviunv 5021	. . . . . 6 ⊢ ∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍) = ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍)
38		eqid 2736	. . . . . 6 ⊢ sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ) = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )
39	37, 38	fsuppmapnn0fiublem 14013	. . . . 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 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑓∅ = 𝑀
42		nfra1 3270	. . . . . . . . . 10 ⊢ Ⅎ𝑓∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅
43	41, 42	nfor 1904	. . . . . . . . 9 ⊢ Ⅎ𝑓(∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅)
44	43	nfn 1857	. . . . . . . 8 ⊢ Ⅎ𝑓 ¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅)
45		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑓(𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)
46	44, 45	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑓(¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉))
47		nfra1 3270	. . . . . . 7 ⊢ Ⅎ𝑓∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍
48	46, 47	nfan 1899	. . . . . 6 ⊢ Ⅎ𝑓((¬ (∅ = 𝑀 ∨ ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) = ∅) ∧ (𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉)) ∧ ∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍)
49		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑓 𝑚 = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )
50	48, 49	nfan 1899	. . . . 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 7418	. . . . . . 7 ⊢ (𝑚 = sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < ) → (0...𝑚) = (0...sup(∪ 𝑔 ∈ 𝑀 (𝑔 supp 𝑍), ℝ, < )))
52	51	sseq2d 3996	. . . . . 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 3259	. . . 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 3084	. . . . . . . . . 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 5048	. . . . . . . 8 ⊢ (∃𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅ ↔ ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ≠ ∅)
63	20	cbviunv 5021	. . . . . . . . 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 14014	. . . . 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 3608	. . 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 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061   ⊆ wss 3931  ∅c0 4313  ∪ ciun 4972   class class class wbr 5124  (class class class)co 7410   supp csupp 8164   ↑_m cmap 8845  Fincfn 8964   finSupp cfsupp 9378  supcsup 9457  ℝcr 11133  0cc0 11134   < clt 11274  ℕ₀cn0 12506  ...cfz 13529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-supp 8165  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9379  df-sup 9459  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-n0 12507  df-z 12594  df-uz 12858  df-fz 13530

This theorem is referenced by:  fsuppmapnn0fiub0  14016