Theorem stoweidlem39 47048

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 521	. . . . . 6 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑟 ∧ 𝐷 ≠ ∅))
4		ssn0 4355	. . . . . 6 ⊢ ((𝐷 ⊆ ∪ 𝑟 ∧ 𝐷 ≠ ∅) → ∪ 𝑟 ≠ ∅)
5		unieq 4878	. . . . . . . 8 ⊢ (𝑟 = ∅ → ∪ 𝑟 = ∪ ∅)
6		uni0 4896	. . . . . . . 8 ⊢ ∪ ∅ = ∅
7	5, 6	eqtrdi 2812	. . . . . . 7 ⊢ (𝑟 = ∅ → ∪ 𝑟 = ∅)
8	7	necon3i 2988	. . . . . 6 ⊢ (∪ 𝑟 ≠ ∅ → 𝑟 ≠ ∅)
9	3, 4, 8	3syl 19	. . . . 5 ⊢ (𝜑 → 𝑟 ≠ ∅)
10	9	neneqd 2961	. . . 4 ⊢ (𝜑 → ¬ 𝑟 = ∅)
11		stoweidlem39.7	. . . . . 6 ⊢ (𝜑 → 𝑟 ∈ (𝒫 𝑊 ∩ Fin))
12		elinel2 4148	. . . . . 6 ⊢ (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟 ∈ Fin)
13	11, 12	syl 18	. . . . 5 ⊢ (𝜑 → 𝑟 ∈ Fin)
14		fz1f1o 15876	. . . . 5 ⊢ (𝑟 ∈ Fin → (𝑟 = ∅ ∨ ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟)))
15		pm2.53 865	. . . . 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 19	. . . 4 ⊢ (𝜑 → (¬ 𝑟 = ∅ → ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟)))
17	10, 16	mpd 16	. . 3 ⊢ (𝜑 → ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟))
18		oveq2 7428	. . . . . 6 ⊢ (𝑚 = (♯‘𝑟) → (1...𝑚) = (1...(♯‘𝑟)))
19	18	f1oeq2d 6820	. . . . 5 ⊢ (𝑚 = (♯‘𝑟) → (𝑣:(1...𝑚)–1-1-onto→𝑟 ↔ 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟))
20	19	exbidv 1954	. . . 4 ⊢ (𝑚 = (♯‘𝑟) → (∃𝑣 𝑣:(1...𝑚)–1-1-onto→𝑟 ↔ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟))
21	20	rspcev 3577	. . 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 18	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℕ ∃𝑣 𝑣:(1...𝑚)–1-1-onto→𝑟)
23		f1of 6824	. . . . . . . 8 ⊢ (𝑣:(1...𝑚)–1-1-onto→𝑟 → 𝑣:(1...𝑚)⟶𝑟)
24	23	adantl 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑣:(1...𝑚)⟶𝑟)
25		simpll 779	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝜑)
26		elinel1 4147	. . . . . . . . 9 ⊢ (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟 ∈ 𝒫 𝑊)
27	26	elpwid 4566	. . . . . . . 8 ⊢ (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟 ⊆ 𝑊)
28	25, 11, 27	3syl 19	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑟 ⊆ 𝑊)
29	24, 28	fssd 6727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑣:(1...𝑚)⟶𝑊)
30	1	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐷 ⊆ ∪ 𝑟)
31		dff1o2 6830	. . . . . . . . . 10 ⊢ (𝑣:(1...𝑚)–1-1-onto→𝑟 ↔ (𝑣 Fn (1...𝑚) ∧ Fun ◡𝑣 ∧ ran 𝑣 = 𝑟))
32	31	simp3bi 1165	. . . . . . . . 9 ⊢ (𝑣:(1...𝑚)–1-1-onto→𝑟 → ran 𝑣 = 𝑟)
33	32	unieqd 4880	. . . . . . . 8 ⊢ (𝑣:(1...𝑚)–1-1-onto→𝑟 → ∪ ran 𝑣 = ∪ 𝑟)
34	33	adantl 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → ∪ ran 𝑣 = ∪ 𝑟)
35	30, 34	sseqtrrd 3968	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐷 ⊆ ∪ ran 𝑣)
36		stoweidlem39.1	. . . . . . . . 9 ⊢ Ⅎℎ𝜑
37		nfv 1947	. . . . . . . . 9 ⊢ Ⅎℎ 𝑚 ∈ ℕ
38	36, 37	nfan 1932	. . . . . . . 8 ⊢ Ⅎℎ(𝜑 ∧ 𝑚 ∈ ℕ)
39		nfv 1947	. . . . . . . 8 ⊢ Ⅎℎ 𝑣:(1...𝑚)–1-1-onto→𝑟
40	38, 39	nfan 1932	. . . . . . 7 ⊢ Ⅎℎ((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟)
41		stoweidlem39.2	. . . . . . . . 9 ⊢ Ⅎ𝑡𝜑
42		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ
43	41, 42	nfan 1932	. . . . . . . 8 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ)
44		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑣:(1...𝑚)–1-1-onto→𝑟
45	43, 44	nfan 1932	. . . . . . 7 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟)
46		stoweidlem39.3	. . . . . . . . 9 ⊢ Ⅎ𝑤𝜑
47		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑤 𝑚 ∈ ℕ
48	46, 47	nfan 1932	. . . . . . . 8 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑚 ∈ ℕ)
49		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑤 𝑣:(1...𝑚)–1-1-onto→𝑟
50	48, 49	nfan 1932	. . . . . . 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 2761	. . . . . . 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 781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑚 ∈ ℕ)
55		simpr 490	. . . . . . 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 739	. . . . . . 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 143	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ 𝐵 → ¬ ¬ 𝑏 ∈ 𝐵)
61	60	intnand 494	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝐵 → ¬ (𝑏 ∈ 𝑇 ∧ ¬ 𝑏 ∈ 𝐵))
62	61	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ (𝑏 ∈ 𝑇 ∧ ¬ 𝑏 ∈ 𝐵))
63		eldif 3909	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝑇 ∖ 𝐵) ↔ (𝑏 ∈ 𝑇 ∧ ¬ 𝑏 ∈ 𝐵))
64	62, 63	sylnibr 332	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ 𝑏 ∈ (𝑇 ∖ 𝐵))
65		stoweidlem39.4	. . . . . . . . . . . . 13 ⊢ 𝑈 = (𝑇 ∖ 𝐵)
66	65	eleq2i 2853	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ 𝑈 ↔ 𝑏 ∈ (𝑇 ∖ 𝐵))
67	64, 66	sylnibr 332	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ 𝑏 ∈ 𝑈)
68	59, 67	eldifd 3910	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ (𝑇 ∖ 𝑈))
69	68	ralrimiva 3155	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 𝑏 ∈ (𝑇 ∖ 𝑈))
70		dfss3 3920	. . . . . . . . 9 ⊢ (𝐵 ⊆ (𝑇 ∖ 𝑈) ↔ ∀𝑏 ∈ 𝐵 𝑏 ∈ (𝑇 ∖ 𝑈))
71	69, 70	sylibr 237	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ (𝑇 ∖ 𝑈))
72	71	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐵 ⊆ (𝑇 ∖ 𝑈))
73		stoweidlem39.12	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ V)
74	73	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑊 ∈ V)
75		stoweidlem39.13	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ V)
76	75	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐴 ∈ V)
77	13	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑟 ∈ Fin)
78		mptfi 9340	. . . . . . . 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 9329	. . . . . . . 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 19	. . . . . . 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 47040	. . . . . 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 1146	. . . . 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 418	. . . 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 1950	. . 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 3176	. 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 16	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)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812  Ⅎwnf 1816   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  {crab 3413  Vcvv 3451   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ∪ cuni 4867   class class class wbr 5103   ↦ cmpt 5186  ^◡ccnv 5650  ran crn 5652  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420  Fincfn 8973  0cc0 11200  1c1 11201   < clt 11343   ≤ cle 11344   − cmin 11541   / cdiv 11973  ℕcn 12335  ℝ⁺crp 13120  ...cfz 13639  ♯chash 14474

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-n0 12607  df-z 12694  df-uz 12966  df-rp 13121  df-fz 13640  df-hash 14475

This theorem is used by:  stoweidlem57  47066