Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem56 Structured version   Visualization version   GIF version

Theorem stoweidlem56 46012
Description: This theorem proves Lemma 1 in [BrosowskiDeutsh] p. 90. Here 𝑍 is used to represent t0 in the paper, 𝑣 is used to represent 𝑉 in the paper, and 𝑒 is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem56.1 𝑡𝑈
stoweidlem56.2 𝑡𝜑
stoweidlem56.3 𝐾 = (topGen‘ran (,))
stoweidlem56.4 (𝜑𝐽 ∈ Comp)
stoweidlem56.5 𝑇 = 𝐽
stoweidlem56.6 𝐶 = (𝐽 Cn 𝐾)
stoweidlem56.7 (𝜑𝐴𝐶)
stoweidlem56.8 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
stoweidlem56.9 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stoweidlem56.10 ((𝜑𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
stoweidlem56.11 ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
stoweidlem56.12 (𝜑𝑈𝐽)
stoweidlem56.13 (𝜑𝑍𝑈)
Assertion
Ref Expression
stoweidlem56 (𝜑 → ∃𝑣𝐽 ((𝑍𝑣𝑣𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑣 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))))
Distinct variable groups:   𝑒,𝑓,𝑔,𝑡,𝐴   𝑣,𝑒,𝑥,𝑡,𝐴   𝑦,𝑒,𝑓,𝑡,𝐴   𝑔,𝐽,𝑡   𝑇,𝑒,𝑓,𝑔,𝑡   𝑈,𝑒,𝑓,𝑔   𝑒,𝑍,𝑓,𝑔,𝑡   𝜑,𝑒,𝑓,𝑔   𝑓,𝑞,𝑔,𝑡,𝐴,𝑟   𝑦,𝑞,𝑇   𝑈,𝑞,𝑦   𝑍,𝑞,𝑦   𝜑,𝑞,𝑦,𝑟   𝑇,𝑟   𝑈,𝑟   𝜑,𝑟   𝑡,𝐾   𝑣,𝐽   𝑣,𝑇,𝑥   𝑣,𝑈,𝑥   𝑣,𝑍
Allowed substitution hints:   𝜑(𝑥,𝑣,𝑡)   𝐶(𝑥,𝑦,𝑣,𝑡,𝑒,𝑓,𝑔,𝑟,𝑞)   𝑈(𝑡)   𝐽(𝑥,𝑦,𝑒,𝑓,𝑟,𝑞)   𝐾(𝑥,𝑦,𝑣,𝑒,𝑓,𝑔,𝑟,𝑞)   𝑍(𝑥,𝑟)

Proof of Theorem stoweidlem56
Dummy variables 𝑑 𝑝 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 stoweidlem56.1 . . . . 5 𝑡𝑈
2 stoweidlem56.2 . . . . 5 𝑡𝜑
3 stoweidlem56.3 . . . . 5 𝐾 = (topGen‘ran (,))
4 stoweidlem56.4 . . . . 5 (𝜑𝐽 ∈ Comp)
5 stoweidlem56.5 . . . . 5 𝑇 = 𝐽
6 stoweidlem56.6 . . . . 5 𝐶 = (𝐽 Cn 𝐾)
7 stoweidlem56.7 . . . . 5 (𝜑𝐴𝐶)
8 stoweidlem56.8 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
9 stoweidlem56.9 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
10 stoweidlem56.10 . . . . 5 ((𝜑𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
11 stoweidlem56.11 . . . . 5 ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
12 stoweidlem56.12 . . . . 5 (𝜑𝑈𝐽)
13 stoweidlem56.13 . . . . 5 (𝜑𝑍𝑈)
14 eqid 2735 . . . . 5 {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))} = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
15 eqid 2735 . . . . 5 {𝑤𝐽 ∣ ∃ ∈ {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} = {𝑤𝐽 ∣ ∃ ∈ {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15stoweidlem55 46011 . . . 4 (𝜑 → ∃𝑝𝐴 (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))
17 df-rex 3069 . . . 4 (∃𝑝𝐴 (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) ↔ ∃𝑝(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))))
1816, 17sylib 218 . . 3 (𝜑 → ∃𝑝(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))))
19 simpl 482 . . . . . . 7 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → 𝜑)
20 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → 𝑝𝐴)
21 simprr3 1222 . . . . . . 7 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))
22 nfv 1912 . . . . . . . . 9 𝑡 𝑝𝐴
23 nfra1 3282 . . . . . . . . 9 𝑡𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)
242, 22, 23nf3an 1899 . . . . . . . 8 𝑡(𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))
2543ad2ant1 1132 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → 𝐽 ∈ Comp)
267sselda 3995 . . . . . . . . . 10 ((𝜑𝑝𝐴) → 𝑝𝐶)
2726, 6eleqtrdi 2849 . . . . . . . . 9 ((𝜑𝑝𝐴) → 𝑝 ∈ (𝐽 Cn 𝐾))
28273adant3 1131 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → 𝑝 ∈ (𝐽 Cn 𝐾))
29 simp3 1137 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))
30123ad2ant1 1132 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → 𝑈𝐽)
311, 24, 3, 5, 25, 28, 29, 30stoweidlem28 45984 . . . . . . 7 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))
3219, 20, 21, 31syl3anc 1370 . . . . . 6 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))
33 simpr1 1193 . . . . . . . . 9 (((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))) → 𝑑 ∈ ℝ+)
34 simpr2 1194 . . . . . . . . 9 (((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))) → 𝑑 < 1)
35 simplrl 777 . . . . . . . . . 10 (((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))) → 𝑝𝐴)
36 simprr1 1220 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → ∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1))
3736adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))) → ∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1))
38 simprr2 1221 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → (𝑝𝑍) = 0)
3938adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))) → (𝑝𝑍) = 0)
40 simpr3 1195 . . . . . . . . . . 11 (((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))) → ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))
4137, 39, 403jca 1127 . . . . . . . . . 10 (((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))) → (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))
4235, 41jca 511 . . . . . . . . 9 (((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))) → (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))
4333, 34, 423jca 1127 . . . . . . . 8 (((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))) → (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))))
4443ex 412 . . . . . . 7 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → ((𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)) → (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))))
4544eximdv 1915 . . . . . 6 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → (∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))))
4632, 45mpd 15 . . . . 5 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))))
4746ex 412 . . . 4 (𝜑 → ((𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))))
4847eximdv 1915 . . 3 (𝜑 → (∃𝑝(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))) → ∃𝑝𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))))
4918, 48mpd 15 . 2 (𝜑 → ∃𝑝𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))))
50 nfv 1912 . . . . . . 7 𝑡 𝑑 ∈ ℝ+
51 nfv 1912 . . . . . . 7 𝑡 𝑑 < 1
52 nfra1 3282 . . . . . . . . 9 𝑡𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1)
53 nfv 1912 . . . . . . . . 9 𝑡(𝑝𝑍) = 0
54 nfra1 3282 . . . . . . . . 9 𝑡𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)
5552, 53, 54nf3an 1899 . . . . . . . 8 𝑡(∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))
5622, 55nfan 1897 . . . . . . 7 𝑡(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))
5750, 51, 56nf3an 1899 . . . . . 6 𝑡(𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))
582, 57nfan 1897 . . . . 5 𝑡(𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))))
59 nfcv 2903 . . . . 5 𝑡𝑝
60 eqid 2735 . . . . 5 {𝑡𝑇 ∣ (𝑝𝑡) < (𝑑 / 2)} = {𝑡𝑇 ∣ (𝑝𝑡) < (𝑑 / 2)}
617adantr 480 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝐴𝐶)
6283adant1r 1176 . . . . 5 (((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
6393adant1r 1176 . . . . 5 (((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
6410adantlr 715 . . . . 5 (((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
65 simpr1 1193 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑑 ∈ ℝ+)
66 simpr2 1194 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑑 < 1)
6712adantr 480 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑈𝐽)
6813adantr 480 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑍𝑈)
69 simpr3l 1233 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑝𝐴)
70 simp3r1 1280 . . . . . 6 ((𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))) → ∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1))
7170adantl 481 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → ∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1))
72 simp3r2 1281 . . . . . 6 ((𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))) → (𝑝𝑍) = 0)
7372adantl 481 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → (𝑝𝑍) = 0)
74 simp3r3 1282 . . . . . 6 ((𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))) → ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))
7574adantl 481 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))
761, 58, 59, 3, 60, 5, 6, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 75stoweidlem52 46008 . . . 4 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → ∃𝑣𝐽 ((𝑍𝑣𝑣𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑣 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))))
7776ex 412 . . 3 (𝜑 → ((𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))) → ∃𝑣𝐽 ((𝑍𝑣𝑣𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑣 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)))))
7877exlimdvv 1932 . 2 (𝜑 → (∃𝑝𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))) → ∃𝑣𝐽 ((𝑍𝑣𝑣𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑣 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)))))
7949, 78mpd 15 1 (𝜑 → ∃𝑣𝐽 ((𝑍𝑣𝑣𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑣 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wex 1776  wnf 1780  wcel 2106  wnfc 2888  wne 2938  wral 3059  wrex 3068  {crab 3433  cdif 3960  wss 3963   cuni 4912   class class class wbr 5148  cmpt 5231  ran crn 5690  cfv 6563  (class class class)co 7431  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  2c2 12319  +crp 13032  (,)cioo 13384  topGenctg 17484   Cn ccn 23248  Compccmp 23410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-rlim 15522  df-sum 15720  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-cn 23251  df-cnp 23252  df-cmp 23411  df-tx 23586  df-hmeo 23779  df-xms 24346  df-ms 24347  df-tms 24348
This theorem is referenced by:  stoweidlem57  46013
  Copyright terms: Public domain W3C validator