Theorem stoweidlem39 46496

Description: This lemma is used to prove that there exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91: assuming that 𝑟 is a finite subset of 𝑊, 𝑥 indexes a finite set of functions in the subalgebra (of the Stone Weierstrass theorem), such that for all i ranging in the finite indexing set, 0 ≤ x_i ≤ 1, x_i < ε / m on V(t_i), and x_i > 1 - ε / m on 𝐵. Here 𝐷 is used to represent A in the paper's Lemma 2 (because 𝐴 is used for the subalgebra), 𝑀 is used to represent m in the paper, 𝐸 is used to represent ε, and v_i is used to represent V(t_i). 𝑊 is just a local definition, used to shorten statements. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem39.1	⊢ Ⅎℎ𝜑
stoweidlem39.2	⊢ Ⅎ𝑡𝜑
stoweidlem39.3	⊢ Ⅎ𝑤𝜑
stoweidlem39.4	⊢ 𝑈 = (𝑇 ∖ 𝐵)
stoweidlem39.5	⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
stoweidlem39.6	⊢ 𝑊 = {𝑤 ∈ 𝐽 ∣ ∀𝑒 ∈ ℝ+ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))}
stoweidlem39.7	⊢ (𝜑 → 𝑟 ∈ (𝒫 𝑊 ∩ Fin))
stoweidlem39.8	⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑟)
stoweidlem39.9	⊢ (𝜑 → 𝐷 ≠ ∅)
stoweidlem39.10	⊢ (𝜑 → 𝐸 ∈ ℝ+)
stoweidlem39.11	⊢ (𝜑 → 𝐵 ⊆ 𝑇)
stoweidlem39.12	⊢ (𝜑 → 𝑊 ∈ V)
stoweidlem39.13	⊢ (𝜑 → 𝐴 ∈ V)

Assertion

Ref	Expression
stoweidlem39	⊢ (𝜑 → ∃𝑚 ∈ ℕ ∃𝑣(𝑣:(1...𝑚)⟶𝑊 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥‘𝑖)‘𝑡)))))

Distinct variable groups:   𝑒,ℎ,𝑚,𝑡,𝑤   𝐴,𝑒,ℎ,𝑡,𝑤   𝑒,𝐸,ℎ,𝑡,𝑤   𝑇,𝑒,ℎ,𝑤   𝑈,𝑒,ℎ,𝑤   ℎ,𝑖,𝑟,𝑣,𝑥,𝑚,𝑡,𝑤   𝐴,𝑖,𝑥   𝑖,𝐸,𝑥   𝑇,𝑖,𝑥   𝑈,𝑖,𝑥   𝜑,𝑖,𝑚,𝑣   𝑤,𝑌,𝑥   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑤,𝑡,𝑒,ℎ,𝑟)   𝐴(𝑣,𝑚,𝑟)   𝐵(𝑤,𝑣,𝑡,𝑒,ℎ,𝑖,𝑚,𝑟)   𝐷(𝑥,𝑤,𝑣,𝑡,𝑒,ℎ,𝑖,𝑚,𝑟)   𝑇(𝑣,𝑡,𝑚,𝑟)   𝑈(𝑣,𝑡,𝑚,𝑟)   𝐸(𝑣,𝑚,𝑟)   𝐽(𝑥,𝑤,𝑣,𝑡,𝑒,ℎ,𝑖,𝑚,𝑟)   𝑊(𝑥,𝑤,𝑣,𝑡,𝑒,ℎ,𝑖,𝑚,𝑟)   𝑌(𝑣,𝑡,𝑒,ℎ,𝑖,𝑚,𝑟)

Proof of Theorem stoweidlem39

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem39.8	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑟)
2		stoweidlem39.9	. . . . . . 7 ⊢ (𝜑 → 𝐷 ≠ ∅)
3	1, 2	jca 517	. . . . . 6 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑟 ∧ 𝐷 ≠ ∅))
4		ssn0 4335	. . . . . 6 ⊢ ((𝐷 ⊆ ∪ 𝑟 ∧ 𝐷 ≠ ∅) → ∪ 𝑟 ≠ ∅)
5		unieq 4852	. . . . . . . 8 ⊢ (𝑟 = ∅ → ∪ 𝑟 = ∪ ∅)
6		uni0 4869	. . . . . . . 8 ⊢ ∪ ∅ = ∅
7	5, 6	eqtrdi 2792	. . . . . . 7 ⊢ (𝑟 = ∅ → ∪ 𝑟 = ∅)
8	7	necon3i 2968	. . . . . 6 ⊢ (∪ 𝑟 ≠ ∅ → 𝑟 ≠ ∅)
9	3, 4, 8	3syl 18	. . . . 5 ⊢ (𝜑 → 𝑟 ≠ ∅)
10	9	neneqd 2941	. . . 4 ⊢ (𝜑 → ¬ 𝑟 = ∅)
11		stoweidlem39.7	. . . . . 6 ⊢ (𝜑 → 𝑟 ∈ (𝒫 𝑊 ∩ Fin))
12		elinel2 4134	. . . . . 6 ⊢ (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟 ∈ Fin)
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝑟 ∈ Fin)
14		fz1f1o 15667	. . . . 5 ⊢ (𝑟 ∈ Fin → (𝑟 = ∅ ∨ ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟)))
15		pm2.53 858	. . . . 5 ⊢ ((𝑟 = ∅ ∨ ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟)) → (¬ 𝑟 = ∅ → ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟)))
16	13, 14, 15	3syl 18	. . . 4 ⊢ (𝜑 → (¬ 𝑟 = ∅ → ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟)))
17	10, 16	mpd 15	. . 3 ⊢ (𝜑 → ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟))
18		oveq2 7368	. . . . . 6 ⊢ (𝑚 = (♯‘𝑟) → (1...𝑚) = (1...(♯‘𝑟)))
19	18	f1oeq2d 6767	. . . . 5 ⊢ (𝑚 = (♯‘𝑟) → (𝑣:(1...𝑚)–1-1-onto→𝑟 ↔ 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟))
20	19	exbidv 1929	. . . 4 ⊢ (𝑚 = (♯‘𝑟) → (∃𝑣 𝑣:(1...𝑚)–1-1-onto→𝑟 ↔ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟))
21	20	rspcev 3562	. . 3 ⊢ (((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟) → ∃𝑚 ∈ ℕ ∃𝑣 𝑣:(1...𝑚)–1-1-onto→𝑟)
22	17, 21	syl 17	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℕ ∃𝑣 𝑣:(1...𝑚)–1-1-onto→𝑟)
23		f1of 6771	. . . . . . . 8 ⊢ (𝑣:(1...𝑚)–1-1-onto→𝑟 → 𝑣:(1...𝑚)⟶𝑟)
24	23	adantl 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑣:(1...𝑚)⟶𝑟)
25		simpll 773	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝜑)
26		elinel1 4133	. . . . . . . . 9 ⊢ (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟 ∈ 𝒫 𝑊)
27	26	elpwid 4541	. . . . . . . 8 ⊢ (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟 ⊆ 𝑊)
28	25, 11, 27	3syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑟 ⊆ 𝑊)
29	24, 28	fssd 6676	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑣:(1...𝑚)⟶𝑊)
30	1	ad2antrr 733	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐷 ⊆ ∪ 𝑟)
31		dff1o2 6776	. . . . . . . . . 10 ⊢ (𝑣:(1...𝑚)–1-1-onto→𝑟 ↔ (𝑣 Fn (1...𝑚) ∧ Fun ◡𝑣 ∧ ran 𝑣 = 𝑟))
32	31	simp3bi 1154	. . . . . . . . 9 ⊢ (𝑣:(1...𝑚)–1-1-onto→𝑟 → ran 𝑣 = 𝑟)
33	32	unieqd 4854	. . . . . . . 8 ⊢ (𝑣:(1...𝑚)–1-1-onto→𝑟 → ∪ ran 𝑣 = ∪ 𝑟)
34	33	adantl 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → ∪ ran 𝑣 = ∪ 𝑟)
35	30, 34	sseqtrrd 3954	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐷 ⊆ ∪ ran 𝑣)
36		stoweidlem39.1	. . . . . . . . 9 ⊢ Ⅎℎ𝜑
37		nfv 1922	. . . . . . . . 9 ⊢ Ⅎℎ 𝑚 ∈ ℕ
38	36, 37	nfan 1907	. . . . . . . 8 ⊢ Ⅎℎ(𝜑 ∧ 𝑚 ∈ ℕ)
39		nfv 1922	. . . . . . . 8 ⊢ Ⅎℎ 𝑣:(1...𝑚)–1-1-onto→𝑟
40	38, 39	nfan 1907	. . . . . . 7 ⊢ Ⅎℎ((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟)
41		stoweidlem39.2	. . . . . . . . 9 ⊢ Ⅎ𝑡𝜑
42		nfv 1922	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ
43	41, 42	nfan 1907	. . . . . . . 8 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ)
44		nfv 1922	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑣:(1...𝑚)–1-1-onto→𝑟
45	43, 44	nfan 1907	. . . . . . 7 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟)
46		stoweidlem39.3	. . . . . . . . 9 ⊢ Ⅎ𝑤𝜑
47		nfv 1922	. . . . . . . . 9 ⊢ Ⅎ𝑤 𝑚 ∈ ℕ
48	46, 47	nfan 1907	. . . . . . . 8 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑚 ∈ ℕ)
49		nfv 1922	. . . . . . . 8 ⊢ Ⅎ𝑤 𝑣:(1...𝑚)–1-1-onto→𝑟
50	48, 49	nfan 1907	. . . . . . 7 ⊢ Ⅎ𝑤((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟)
51		stoweidlem39.5	. . . . . . 7 ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
52		stoweidlem39.6	. . . . . . 7 ⊢ 𝑊 = {𝑤 ∈ 𝐽 ∣ ∀𝑒 ∈ ℝ+ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))}
53		eqid 2741	. . . . . . 7 ⊢ (𝑤 ∈ 𝑟 ↦ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑚)) < (ℎ‘𝑡))}) = (𝑤 ∈ 𝑟 ↦ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑚)) < (ℎ‘𝑡))})
54		simplr 775	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑚 ∈ ℕ)
55		simpr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑣:(1...𝑚)–1-1-onto→𝑟)
56		stoweidlem39.10	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
57	56	ad2antrr 733	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐸 ∈ ℝ+)
58		stoweidlem39.11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
59	58	sselda 3917	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝑇)
60		notnot 142	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ 𝐵 → ¬ ¬ 𝑏 ∈ 𝐵)
61	60	intnand 490	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝐵 → ¬ (𝑏 ∈ 𝑇 ∧ ¬ 𝑏 ∈ 𝐵))
62	61	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ (𝑏 ∈ 𝑇 ∧ ¬ 𝑏 ∈ 𝐵))
63		eldif 3895	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝑇 ∖ 𝐵) ↔ (𝑏 ∈ 𝑇 ∧ ¬ 𝑏 ∈ 𝐵))
64	62, 63	sylnibr 331	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ 𝑏 ∈ (𝑇 ∖ 𝐵))
65		stoweidlem39.4	. . . . . . . . . . . . 13 ⊢ 𝑈 = (𝑇 ∖ 𝐵)
66	65	eleq2i 2833	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ 𝑈 ↔ 𝑏 ∈ (𝑇 ∖ 𝐵))
67	64, 66	sylnibr 331	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ 𝑏 ∈ 𝑈)
68	59, 67	eldifd 3896	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ (𝑇 ∖ 𝑈))
69	68	ralrimiva 3133	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 𝑏 ∈ (𝑇 ∖ 𝑈))
70		dfss3 3906	. . . . . . . . 9 ⊢ (𝐵 ⊆ (𝑇 ∖ 𝑈) ↔ ∀𝑏 ∈ 𝐵 𝑏 ∈ (𝑇 ∖ 𝑈))
71	69, 70	sylibr 236	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ (𝑇 ∖ 𝑈))
72	71	ad2antrr 733	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐵 ⊆ (𝑇 ∖ 𝑈))
73		stoweidlem39.12	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ V)
74	73	ad2antrr 733	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑊 ∈ V)
75		stoweidlem39.13	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ V)
76	75	ad2antrr 733	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐴 ∈ V)
77	13	ad2antrr 733	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑟 ∈ Fin)
78		mptfi 9255	. . . . . . . 8 ⊢ (𝑟 ∈ Fin → (𝑤 ∈ 𝑟 ↦ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑚)) < (ℎ‘𝑡))}) ∈ Fin)
79		rnfi 9244	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑟 ↦ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑚)) < (ℎ‘𝑡))}) ∈ Fin → ran (𝑤 ∈ 𝑟 ↦ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑚)) < (ℎ‘𝑡))}) ∈ Fin)
80	77, 78, 79	3syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → ran (𝑤 ∈ 𝑟 ↦ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑚)) < (ℎ‘𝑡))}) ∈ Fin)
81	40, 45, 50, 51, 52, 53, 28, 54, 55, 57, 72, 74, 76, 80	stoweidlem31 46488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥‘𝑖)‘𝑡))))
82	29, 35, 81	3jca 1135	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → (𝑣:(1...𝑚)⟶𝑊 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥‘𝑖)‘𝑡)))))
83	82	ex 414	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑣:(1...𝑚)–1-1-onto→𝑟 → (𝑣:(1...𝑚)⟶𝑊 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥‘𝑖)‘𝑡))))))
84	83	eximdv 1925	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∃𝑣 𝑣:(1...𝑚)–1-1-onto→𝑟 → ∃𝑣(𝑣:(1...𝑚)⟶𝑊 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥‘𝑖)‘𝑡))))))
85	84	reximdva 3154	. 2 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ∃𝑣 𝑣:(1...𝑚)–1-1-onto→𝑟 → ∃𝑚 ∈ ℕ ∃𝑣(𝑣:(1...𝑚)⟶𝑊 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥‘𝑖)‘𝑡))))))
86	22, 85	mpd 15	1 ⊢ (𝜑 → ∃𝑚 ∈ ℕ ∃𝑣(𝑣:(1...𝑚)⟶𝑊 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥‘𝑖)‘𝑡)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∨ wo 854   ∧ w3a 1093   = wceq 1548  ∃wex 1787  Ⅎwnf 1791   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055  ∃wrex 3065  {crab 3393  Vcvv 3433   ∖ cdif 3882   ∩ cin 3884   ⊆ wss 3885  ∅c0 4264  𝒫 cpw 4532  ∪ cuni 4841   class class class wbr 5075   ↦ cmpt 5156  ^◡ccnv 5620  ran crn 5622  Fun wfun 6483   Fn wfn 6484  ⟶wf 6485  –1-1-onto→wf1o 6488  ‘cfv 6489  (class class class)co 7360  Fincfn 8887  0cc0 11033  1c1 11034   < clt 11174   ≤ cle 11175   − cmin 11372   / cdiv 11802  ℕcn 12169  ℝ⁺crp 12937  ...cfz 13456  ♯chash 14287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  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 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-n0 12433  df-z 12520  df-uz 12784  df-rp 12938  df-fz 13457  df-hash 14288

This theorem is referenced by:  stoweidlem57  46514