Theorem stoweidlem39 43470

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 4331	. . . . . 6 ⊢ ((𝐷 ⊆ ∪ 𝑟 ∧ 𝐷 ≠ ∅) → ∪ 𝑟 ≠ ∅)
5		unieq 4847	. . . . . . . 8 ⊢ (𝑟 = ∅ → ∪ 𝑟 = ∪ ∅)
6		uni0 4866	. . . . . . . 8 ⊢ ∪ ∅ = ∅
7	5, 6	eqtrdi 2795	. . . . . . 7 ⊢ (𝑟 = ∅ → ∪ 𝑟 = ∅)
8	7	necon3i 2975	. . . . . 6 ⊢ (∪ 𝑟 ≠ ∅ → 𝑟 ≠ ∅)
9	3, 4, 8	3syl 18	. . . . 5 ⊢ (𝜑 → 𝑟 ≠ ∅)
10	9	neneqd 2947	. . . 4 ⊢ (𝜑 → ¬ 𝑟 = ∅)
11		stoweidlem39.7	. . . . . 6 ⊢ (𝜑 → 𝑟 ∈ (𝒫 𝑊 ∩ Fin))
12		elinel2 4126	. . . . . 6 ⊢ (𝑟 ∈ (𝒫 𝑊 ∩ Fin) → 𝑟 ∈ Fin)
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝑟 ∈ Fin)
14		fz1f1o 15350	. . . . 5 ⊢ (𝑟 ∈ Fin → (𝑟 = ∅ ∨ ((♯‘𝑟) ∈ ℕ ∧ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟)))
15		pm2.53 847	. . . . 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 7263	. . . . . 6 ⊢ (𝑚 = (♯‘𝑟) → (1...𝑚) = (1...(♯‘𝑟)))
19	18	f1oeq2d 6696	. . . . 5 ⊢ (𝑚 = (♯‘𝑟) → (𝑣:(1...𝑚)–1-1-onto→𝑟 ↔ 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟))
20	19	exbidv 1925	. . . 4 ⊢ (𝑚 = (♯‘𝑟) → (∃𝑣 𝑣:(1...𝑚)–1-1-onto→𝑟 ↔ ∃𝑣 𝑣:(1...(♯‘𝑟))–1-1-onto→𝑟))
21	20	rspcev 3552	. . 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 6700	. . . . . . . 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 763	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝜑)
26		elinel1 4125	. . . . . . . . 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 6602	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑣:(1...𝑚)⟶𝑊)
30	1	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐷 ⊆ ∪ 𝑟)
31		dff1o2 6705	. . . . . . . . . 10 ⊢ (𝑣:(1...𝑚)–1-1-onto→𝑟 ↔ (𝑣 Fn (1...𝑚) ∧ Fun ◡𝑣 ∧ ran 𝑣 = 𝑟))
32	31	simp3bi 1145	. . . . . . . . 9 ⊢ (𝑣:(1...𝑚)–1-1-onto→𝑟 → ran 𝑣 = 𝑟)
33	32	unieqd 4850	. . . . . . . 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 3958	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐷 ⊆ ∪ ran 𝑣)
36		stoweidlem39.1	. . . . . . . . 9 ⊢ Ⅎℎ𝜑
37		nfv 1918	. . . . . . . . 9 ⊢ Ⅎℎ 𝑚 ∈ ℕ
38	36, 37	nfan 1903	. . . . . . . 8 ⊢ Ⅎℎ(𝜑 ∧ 𝑚 ∈ ℕ)
39		nfv 1918	. . . . . . . 8 ⊢ Ⅎℎ 𝑣:(1...𝑚)–1-1-onto→𝑟
40	38, 39	nfan 1903	. . . . . . 7 ⊢ Ⅎℎ((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟)
41		stoweidlem39.2	. . . . . . . . 9 ⊢ Ⅎ𝑡𝜑
42		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ
43	41, 42	nfan 1903	. . . . . . . 8 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ)
44		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑣:(1...𝑚)–1-1-onto→𝑟
45	43, 44	nfan 1903	. . . . . . 7 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟)
46		stoweidlem39.3	. . . . . . . . 9 ⊢ Ⅎ𝑤𝜑
47		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑤 𝑚 ∈ ℕ
48	46, 47	nfan 1903	. . . . . . . 8 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑚 ∈ ℕ)
49		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑤 𝑣:(1...𝑚)–1-1-onto→𝑟
50	48, 49	nfan 1903	. . . . . . 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 2738	. . . . . . 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 765	. . . . . . 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 722	. . . . . . 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 488	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝐵 → ¬ (𝑏 ∈ 𝑇 ∧ ¬ 𝑏 ∈ 𝐵))
62	61	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ (𝑏 ∈ 𝑇 ∧ ¬ 𝑏 ∈ 𝐵))
63		eldif 3893	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (𝑇 ∖ 𝐵) ↔ (𝑏 ∈ 𝑇 ∧ ¬ 𝑏 ∈ 𝐵))
64	62, 63	sylnibr 328	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ 𝑏 ∈ (𝑇 ∖ 𝐵))
65		stoweidlem39.4	. . . . . . . . . . . . 13 ⊢ 𝑈 = (𝑇 ∖ 𝐵)
66	65	eleq2i 2830	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ 𝑈 ↔ 𝑏 ∈ (𝑇 ∖ 𝐵))
67	64, 66	sylnibr 328	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ¬ 𝑏 ∈ 𝑈)
68	59, 67	eldifd 3894	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ (𝑇 ∖ 𝑈))
69	68	ralrimiva 3107	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 𝑏 ∈ (𝑇 ∖ 𝑈))
70		dfss3 3905	. . . . . . . . 9 ⊢ (𝐵 ⊆ (𝑇 ∖ 𝑈) ↔ ∀𝑏 ∈ 𝐵 𝑏 ∈ (𝑇 ∖ 𝑈))
71	69, 70	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ (𝑇 ∖ 𝑈))
72	71	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐵 ⊆ (𝑇 ∖ 𝑈))
73		stoweidlem39.12	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ V)
74	73	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑊 ∈ V)
75		stoweidlem39.13	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ V)
76	75	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝐴 ∈ V)
77	13	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑣:(1...𝑚)–1-1-onto→𝑟) → 𝑟 ∈ Fin)
78		mptfi 9048	. . . . . . . 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 9032	. . . . . . . 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 43462	. . . . . 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 1126	. . . . 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 1921	. . 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 3202	. 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 843   ∧ w3a 1085   = wceq 1539  ∃wex 1783  Ⅎwnf 1787   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836   class class class wbr 5070   ↦ cmpt 5153  ^◡ccnv 5579  ran crn 5581  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  Fincfn 8691  0cc0 10802  1c1 10803   < clt 10940   ≤ cle 10941   − cmin 11135   / cdiv 11562  ℕcn 11903  ℝ⁺crp 12659  ...cfz 13168  ♯chash 13972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-fz 13169  df-hash 13973

This theorem is referenced by:  stoweidlem57  43488