Theorem stoweidlem39 46225

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 511	. . . . . 6 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑟 ∧ 𝐷 ≠ ∅))
4		ssn0 4354	. . . . . 6 ⊢ ((𝐷 ⊆ ∪ 𝑟 ∧ 𝐷 ≠ ∅) → ∪ 𝑟 ≠ ∅)
5		unieq 4872	. . . . . . . 8 ⊢ (𝑟 = ∅ → ∪ 𝑟 = ∪ ∅)
6		uni0 4889	. . . . . . . 8 ⊢ ∪ ∅ = ∅
7	5, 6	eqtrdi 2785	. . . . . . 7 ⊢ (𝑟 = ∅ → ∪ 𝑟 = ∅)
8	7	necon3i 2962	. . . . . 6 ⊢ (∪ 𝑟 ≠ ∅ → 𝑟 ≠ ∅)
9	3, 4, 8	3syl 18	. . . . 5 ⊢ (𝜑 → 𝑟 ≠ ∅)
10	9	neneqd 2935	. . . 4 ⊢ (𝜑 → ¬ 𝑟 = ∅)
11		stoweidlem39.7	. . . . . 6 ⊢ (𝜑 → 𝑟 ∈ (𝒫 𝑊 ∩ Fin))
12		elinel2 4152	. . . . . 6 ⊢ (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟 ∈ Fin)
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝑟 ∈ Fin)
14		fz1f1o 15631	. . . . 5 ⊢ (𝑟 ∈ Fin → (𝑟 = ∅ ∨ ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟)))
15		pm2.53 851	. . . . 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 7364	. . . . . 6 ⊢ (𝑚 = (♯‘𝑟) → (1...𝑚) = (1...(♯‘𝑟)))
19	18	f1oeq2d 6768	. . . . 5 ⊢ (𝑚 = (♯‘𝑟) → (𝑣:(1...𝑚)–1-1-onto→𝑟 ↔ 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟))
20	19	exbidv 1922	. . . 4 ⊢ (𝑚 = (♯‘𝑟) → (∃𝑣 𝑣:(1...𝑚)–1-1-onto→𝑟 ↔ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟))
21	20	rspcev 3574	. . 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 6772	. . . . . . . 8 ⊢ (𝑣:(1...𝑚)–1-1-onto→𝑟 → 𝑣:(1...𝑚)⟶𝑟)
24	23	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑣:(1...𝑚)⟶𝑟)
25		simpll 766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝜑)
26		elinel1 4151	. . . . . . . . 9 ⊢ (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟 ∈ 𝒫 𝑊)
27	26	elpwid 4561	. . . . . . . 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 6677	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑣:(1...𝑚)⟶𝑊)
30	1	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐷 ⊆ ∪ 𝑟)
31		dff1o2 6777	. . . . . . . . . 10 ⊢ (𝑣:(1...𝑚)–1-1-onto→𝑟 ↔ (𝑣 Fn (1...𝑚) ∧ Fun ◡𝑣 ∧ ran 𝑣 = 𝑟))
32	31	simp3bi 1147	. . . . . . . . 9 ⊢ (𝑣:(1...𝑚)–1-1-onto→𝑟 → ran 𝑣 = 𝑟)
33	32	unieqd 4874	. . . . . . . 8 ⊢ (𝑣:(1...𝑚)–1-1-onto→𝑟 → ∪ ran 𝑣 = ∪ 𝑟)
34	33	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → ∪ ran 𝑣 = ∪ 𝑟)
35	30, 34	sseqtrrd 3969	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐷 ⊆ ∪ ran 𝑣)
36		stoweidlem39.1	. . . . . . . . 9 ⊢ Ⅎℎ𝜑
37		nfv 1915	. . . . . . . . 9 ⊢ Ⅎℎ 𝑚 ∈ ℕ
38	36, 37	nfan 1900	. . . . . . . 8 ⊢ Ⅎℎ(𝜑 ∧ 𝑚 ∈ ℕ)
39		nfv 1915	. . . . . . . 8 ⊢ Ⅎℎ 𝑣:(1...𝑚)–1-1-onto→𝑟
40	38, 39	nfan 1900	. . . . . . 7 ⊢ Ⅎℎ((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟)
41		stoweidlem39.2	. . . . . . . . 9 ⊢ Ⅎ𝑡𝜑
42		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ
43	41, 42	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ)
44		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑣:(1...𝑚)–1-1-onto→𝑟
45	43, 44	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟)
46		stoweidlem39.3	. . . . . . . . 9 ⊢ Ⅎ𝑤𝜑
47		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑤 𝑚 ∈ ℕ
48	46, 47	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑚 ∈ ℕ)
49		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑤 𝑣:(1...𝑚)–1-1-onto→𝑟
50	48, 49	nfan 1900	. . . . . . 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 2734	. . . . . . 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 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑚 ∈ ℕ)
55		simpr 484	. . . . . . 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 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐸 ∈ ℝ+)
58		stoweidlem39.11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
59	58	sselda 3931	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝑇)
60		notnot 142	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ 𝐵 → ¬ ¬ 𝑏 ∈ 𝐵)
61	60	intnand 488	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝐵 → ¬ (𝑏 ∈ 𝑇 ∧ ¬ 𝑏 ∈ 𝐵))
62	61	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ (𝑏 ∈ 𝑇 ∧ ¬ 𝑏 ∈ 𝐵))
63		eldif 3909	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝑇 ∖ 𝐵) ↔ (𝑏 ∈ 𝑇 ∧ ¬ 𝑏 ∈ 𝐵))
64	62, 63	sylnibr 329	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ 𝑏 ∈ (𝑇 ∖ 𝐵))
65		stoweidlem39.4	. . . . . . . . . . . . 13 ⊢ 𝑈 = (𝑇 ∖ 𝐵)
66	65	eleq2i 2826	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ 𝑈 ↔ 𝑏 ∈ (𝑇 ∖ 𝐵))
67	64, 66	sylnibr 329	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ 𝑏 ∈ 𝑈)
68	59, 67	eldifd 3910	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ (𝑇 ∖ 𝑈))
69	68	ralrimiva 3126	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 𝑏 ∈ (𝑇 ∖ 𝑈))
70		dfss3 3920	. . . . . . . . 9 ⊢ (𝐵 ⊆ (𝑇 ∖ 𝑈) ↔ ∀𝑏 ∈ 𝐵 𝑏 ∈ (𝑇 ∖ 𝑈))
71	69, 70	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ (𝑇 ∖ 𝑈))
72	71	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐵 ⊆ (𝑇 ∖ 𝑈))
73		stoweidlem39.12	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ V)
74	73	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑊 ∈ V)
75		stoweidlem39.13	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ V)
76	75	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐴 ∈ V)
77	13	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑟 ∈ Fin)
78		mptfi 9249	. . . . . . . 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 9238	. . . . . . . 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 46217	. . . . . 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 1128	. . . . 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 412	. . . 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 1918	. . 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 3147	. 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 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541  ∃wex 1780  Ⅎwnf 1784   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  {crab 3397  Vcvv 3438   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4283  𝒫 cpw 4552  ∪ cuni 4861   class class class wbr 5096   ↦ cmpt 5177  ^◡ccnv 5621  ran crn 5623  Fun wfun 6484   Fn wfn 6485  ⟶wf 6486  –1-1-onto→wf1o 6489  ‘cfv 6490  (class class class)co 7356  Fincfn 8881  0cc0 11024  1c1 11025   < clt 11164   ≤ cle 11165   − cmin 11362   / cdiv 11792  ℕcn 12143  ℝ⁺crp 12903  ...cfz 13421  ♯chash 14251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-n0 12400  df-z 12487  df-uz 12750  df-rp 12904  df-fz 13422  df-hash 14252

This theorem is referenced by:  stoweidlem57  46243