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Theorem stoweidlem56 39577
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 2621 . . . . 5 {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))} = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
15 eqid 2621 . . . . 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 39576 . . . 4 (𝜑 → ∃𝑝𝐴 (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))
17 df-rex 2913 . . . 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 208 . . 3 (𝜑 → ∃𝑝(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))))
19 simpl 473 . . . . . . 7 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → 𝜑)
20 simprl 793 . . . . . . 7 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → 𝑝𝐴)
21 simprr3 1109 . . . . . . 7 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))
22 nfv 1840 . . . . . . . . 9 𝑡 𝑝𝐴
23 nfra1 2936 . . . . . . . . 9 𝑡𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)
242, 22, 23nf3an 1828 . . . . . . . 8 𝑡(𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))
2543ad2ant1 1080 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → 𝐽 ∈ Comp)
267sselda 3583 . . . . . . . . . 10 ((𝜑𝑝𝐴) → 𝑝𝐶)
2726, 6syl6eleq 2708 . . . . . . . . 9 ((𝜑𝑝𝐴) → 𝑝 ∈ (𝐽 Cn 𝐾))
28273adant3 1079 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → 𝑝 ∈ (𝐽 Cn 𝐾))
29 simp3 1061 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))
30123ad2ant1 1080 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → 𝑈𝐽)
311, 24, 3, 5, 25, 28, 29, 30stoweidlem28 39549 . . . . . . 7 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))
3219, 20, 21, 31syl3anc 1323 . . . . . 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 1065 . . . . . . . . 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 1066 . . . . . . . . 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 799 . . . . . . . . . 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 1107 . . . . . . . . . . . 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 1108 . . . . . . . . . . . 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 1067 . . . . . . . . . . 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 1240 . . . . . . . . . 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 554 . . . . . . . . 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 1240 . . . . . . . 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 450 . . . . . . 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 1843 . . . . . 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 450 . . . 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 1843 . . 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 1840 . . . . . . 7 𝑡 𝑑 ∈ ℝ+
51 nfv 1840 . . . . . . 7 𝑡 𝑑 < 1
52 nfra1 2936 . . . . . . . . 9 𝑡𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1)
53 nfv 1840 . . . . . . . . 9 𝑡(𝑝𝑍) = 0
54 nfra1 2936 . . . . . . . . 9 𝑡𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)
5552, 53, 54nf3an 1828 . . . . . . . 8 𝑡(∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))
5622, 55nfan 1825 . . . . . . 7 𝑡(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))
5750, 51, 56nf3an 1828 . . . . . 6 𝑡(𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))
582, 57nfan 1825 . . . . 5 𝑡(𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))))
59 nfcv 2761 . . . . 5 𝑡𝑝
60 eqid 2621 . . . . 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 1316 . . . . 5 (((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
6393adant1r 1316 . . . . 5 (((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
6410adantlr 750 . . . . 5 (((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
65 simpr1 1065 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑑 ∈ ℝ+)
66 simpr2 1066 . . . . 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 1120 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑝𝐴)
70 simp3r1 1167 . . . . . 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 1168 . . . . . 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 1169 . . . . . 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 39573 . . . 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 450 . . 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 1859 . 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 384  w3a 1036   = wceq 1480  wex 1701  wnf 1705  wcel 1987  wnfc 2748  wne 2790  wral 2907  wrex 2908  {crab 2911  cdif 3552  wss 3555   cuni 4402   class class class wbr 4613  cmpt 4673  ran crn 5075  cfv 5847  (class class class)co 6604  cr 9879  0cc0 9880  1c1 9881   + caddc 9883   · cmul 9885   < clt 10018  cle 10019  cmin 10210   / cdiv 10628  2c2 11014  +crp 11776  (,)cioo 12117  topGenctg 16019   Cn ccn 20938  Compccmp 21099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-rlim 14154  df-sum 14351  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-pt 16026  df-prds 16029  df-xrs 16083  df-qtop 16088  df-imas 16089  df-xps 16091  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cld 20733  df-cn 20941  df-cnp 20942  df-cmp 21100  df-tx 21275  df-hmeo 21468  df-xms 22035  df-ms 22036  df-tms 22037
This theorem is referenced by:  stoweidlem57  39578
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