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Theorem stoweidlem56 43852
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 2736 . . . . 5 {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))} = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
15 eqid 2736 . . . . 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 43851 . . . 4 (𝜑 → ∃𝑝𝐴 (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))
17 df-rex 3071 . . . 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 217 . . 3 (𝜑 → ∃𝑝(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))))
19 simpl 483 . . . . . . 7 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → 𝜑)
20 simprl 768 . . . . . . 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 1916 . . . . . . . . 9 𝑡 𝑝𝐴
23 nfra1 3263 . . . . . . . . 9 𝑡𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)
242, 22, 23nf3an 1903 . . . . . . . 8 𝑡(𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))
2543ad2ant1 1132 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → 𝐽 ∈ Comp)
267sselda 3930 . . . . . . . . . 10 ((𝜑𝑝𝐴) → 𝑝𝐶)
2726, 6eleqtrdi 2847 . . . . . . . . 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 43824 . . . . . . 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 774 . . . . . . . . . 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 481 . . . . . . . . . . 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 481 . . . . . . . . . . 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 512 . . . . . . . . 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 413 . . . . . . 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 1919 . . . . . 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 413 . . . 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 1919 . . 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 1916 . . . . . . 7 𝑡 𝑑 ∈ ℝ+
51 nfv 1916 . . . . . . 7 𝑡 𝑑 < 1
52 nfra1 3263 . . . . . . . . 9 𝑡𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1)
53 nfv 1916 . . . . . . . . 9 𝑡(𝑝𝑍) = 0
54 nfra1 3263 . . . . . . . . 9 𝑡𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)
5552, 53, 54nf3an 1903 . . . . . . . 8 𝑡(∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))
5622, 55nfan 1901 . . . . . . 7 𝑡(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))
5750, 51, 56nf3an 1903 . . . . . 6 𝑡(𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))
582, 57nfan 1901 . . . . 5 𝑡(𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))))
59 nfcv 2904 . . . . 5 𝑡𝑝
60 eqid 2736 . . . . 5 {𝑡𝑇 ∣ (𝑝𝑡) < (𝑑 / 2)} = {𝑡𝑇 ∣ (𝑝𝑡) < (𝑑 / 2)}
617adantr 481 . . . . 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 712 . . . . 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 481 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑈𝐽)
6813adantr 481 . . . . 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 482 . . . . 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 482 . . . . 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 482 . . . . 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 43848 . . . 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 413 . . 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 1936 . 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 396  w3a 1086   = wceq 1540  wex 1780  wnf 1784  wcel 2105  wnfc 2884  wne 2940  wral 3061  wrex 3070  {crab 3403  cdif 3893  wss 3896   cuni 4849   class class class wbr 5086  cmpt 5169  ran crn 5608  cfv 6465  (class class class)co 7316  cr 10949  0cc0 10950  1c1 10951   + caddc 10953   · cmul 10955   < clt 11088  cle 11089  cmin 11284   / cdiv 11711  2c2 12107  +crp 12809  (,)cioo 13158  topGenctg 17222   Cn ccn 22455  Compccmp 22617
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629  ax-inf2 9476  ax-cnex 11006  ax-resscn 11007  ax-1cn 11008  ax-icn 11009  ax-addcl 11010  ax-addrcl 11011  ax-mulcl 11012  ax-mulrcl 11013  ax-mulcom 11014  ax-addass 11015  ax-mulass 11016  ax-distr 11017  ax-i2m1 11018  ax-1ne0 11019  ax-1rid 11020  ax-rnegex 11021  ax-rrecex 11022  ax-cnre 11023  ax-pre-lttri 11024  ax-pre-lttrn 11025  ax-pre-ltadd 11026  ax-pre-mulgt0 11027  ax-pre-sup 11028  ax-mulf 11030
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4850  df-int 4892  df-iun 4938  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5170  df-tr 5204  df-id 5506  df-eprel 5512  df-po 5520  df-so 5521  df-fr 5562  df-se 5563  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-isom 6474  df-riota 7273  df-ov 7319  df-oprab 7320  df-mpo 7321  df-of 7574  df-om 7759  df-1st 7877  df-2nd 7878  df-supp 8026  df-frecs 8145  df-wrecs 8176  df-recs 8250  df-rdg 8289  df-1o 8345  df-2o 8346  df-er 8547  df-map 8666  df-pm 8667  df-ixp 8735  df-en 8783  df-dom 8784  df-sdom 8785  df-fin 8786  df-fsupp 9205  df-fi 9246  df-sup 9277  df-inf 9278  df-oi 9345  df-card 9774  df-pnf 11090  df-mnf 11091  df-xr 11092  df-ltxr 11093  df-le 11094  df-sub 11286  df-neg 11287  df-div 11712  df-nn 12053  df-2 12115  df-3 12116  df-4 12117  df-5 12118  df-6 12119  df-7 12120  df-8 12121  df-9 12122  df-n0 12313  df-z 12399  df-dec 12517  df-uz 12662  df-q 12768  df-rp 12810  df-xneg 12927  df-xadd 12928  df-xmul 12929  df-ioo 13162  df-ico 13164  df-icc 13165  df-fz 13319  df-fzo 13462  df-fl 13591  df-seq 13801  df-exp 13862  df-hash 14124  df-cj 14886  df-re 14887  df-im 14888  df-sqrt 15022  df-abs 15023  df-clim 15273  df-rlim 15274  df-sum 15474  df-struct 16922  df-sets 16939  df-slot 16957  df-ndx 16969  df-base 16987  df-ress 17016  df-plusg 17049  df-mulr 17050  df-starv 17051  df-sca 17052  df-vsca 17053  df-ip 17054  df-tset 17055  df-ple 17056  df-ds 17058  df-unif 17059  df-hom 17060  df-cco 17061  df-rest 17207  df-topn 17208  df-0g 17226  df-gsum 17227  df-topgen 17228  df-pt 17229  df-prds 17232  df-xrs 17287  df-qtop 17292  df-imas 17293  df-xps 17295  df-mre 17369  df-mrc 17370  df-acs 17372  df-mgm 18400  df-sgrp 18449  df-mnd 18460  df-submnd 18505  df-mulg 18774  df-cntz 18996  df-cmn 19460  df-psmet 20669  df-xmet 20670  df-met 20671  df-bl 20672  df-mopn 20673  df-cnfld 20678  df-top 22123  df-topon 22140  df-topsp 22162  df-bases 22176  df-cld 22250  df-cn 22458  df-cnp 22459  df-cmp 22618  df-tx 22793  df-hmeo 22986  df-xms 23553  df-ms 23554  df-tms 23555
This theorem is referenced by:  stoweidlem57  43853
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