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Theorem stoweidlem56 42218
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 2818 . . . . 5 {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))} = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
15 eqid 2818 . . . . 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 42217 . . . 4 (𝜑 → ∃𝑝𝐴 (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))
17 df-rex 3141 . . . 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 219 . . 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 767 . . . . . . 7 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → 𝑝𝐴)
21 simprr3 1215 . . . . . . 7 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))
22 nfv 1906 . . . . . . . . 9 𝑡 𝑝𝐴
23 nfra1 3216 . . . . . . . . 9 𝑡𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)
242, 22, 23nf3an 1893 . . . . . . . 8 𝑡(𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))
2543ad2ant1 1125 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → 𝐽 ∈ Comp)
267sselda 3964 . . . . . . . . . 10 ((𝜑𝑝𝐴) → 𝑝𝐶)
2726, 6eleqtrdi 2920 . . . . . . . . 9 ((𝜑𝑝𝐴) → 𝑝 ∈ (𝐽 Cn 𝐾))
28273adant3 1124 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → 𝑝 ∈ (𝐽 Cn 𝐾))
29 simp3 1130 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))
30123ad2ant1 1125 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → 𝑈𝐽)
311, 24, 3, 5, 25, 28, 29, 30stoweidlem28 42190 . . . . . . 7 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))
3219, 20, 21, 31syl3anc 1363 . . . . . 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 1186 . . . . . . . . 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 1187 . . . . . . . . 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 773 . . . . . . . . . 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 1213 . . . . . . . . . . . 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 1214 . . . . . . . . . . . 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 1188 . . . . . . . . . . 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 1120 . . . . . . . . . 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 1120 . . . . . . . 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 1909 . . . . . 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 1909 . . 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 1906 . . . . . . 7 𝑡 𝑑 ∈ ℝ+
51 nfv 1906 . . . . . . 7 𝑡 𝑑 < 1
52 nfra1 3216 . . . . . . . . 9 𝑡𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1)
53 nfv 1906 . . . . . . . . 9 𝑡(𝑝𝑍) = 0
54 nfra1 3216 . . . . . . . . 9 𝑡𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)
5552, 53, 54nf3an 1893 . . . . . . . 8 𝑡(∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))
5622, 55nfan 1891 . . . . . . 7 𝑡(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))
5750, 51, 56nf3an 1893 . . . . . 6 𝑡(𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))
582, 57nfan 1891 . . . . 5 𝑡(𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))))
59 nfcv 2974 . . . . 5 𝑡𝑝
60 eqid 2818 . . . . 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 1169 . . . . 5 (((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
6393adant1r 1169 . . . . 5 (((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
6410adantlr 711 . . . . 5 (((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
65 simpr1 1186 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑑 ∈ ℝ+)
66 simpr2 1187 . . . . 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 1226 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑝𝐴)
70 simp3r1 1273 . . . . . 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 1274 . . . . . 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 1275 . . . . . 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 42214 . . . 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 1926 . 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 1079   = wceq 1528  wex 1771  wnf 1775  wcel 2105  wnfc 2958  wne 3013  wral 3135  wrex 3136  {crab 3139  cdif 3930  wss 3933   cuni 4830   class class class wbr 5057  cmpt 5137  ran crn 5549  cfv 6348  (class class class)co 7145  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530   < clt 10663  cle 10664  cmin 10858   / cdiv 11285  2c2 11680  +crp 12377  (,)cioo 12726  topGenctg 16699   Cn ccn 21760  Compccmp 21922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-rlim 14834  df-sum 15031  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-starv 16568  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-hom 16577  df-cco 16578  df-rest 16684  df-topn 16685  df-0g 16703  df-gsum 16704  df-topgen 16705  df-pt 16706  df-prds 16709  df-xrs 16763  df-qtop 16768  df-imas 16769  df-xps 16771  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-mulg 18163  df-cntz 18385  df-cmn 18837  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-cnfld 20474  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cld 21555  df-cn 21763  df-cnp 21764  df-cmp 21923  df-tx 22098  df-hmeo 22291  df-xms 22857  df-ms 22858  df-tms 22859
This theorem is referenced by:  stoweidlem57  42219
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