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Theorem stoweidlem56 40774
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 2758 . . . . 5 {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))} = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
15 eqid 2758 . . . . 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 40773 . . . 4 (𝜑 → ∃𝑝𝐴 (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))
17 df-rex 3054 . . . 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 474 . . . . . . 7 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → 𝜑)
20 simprl 811 . . . . . . 7 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → 𝑝𝐴)
21 simprr3 1277 . . . . . . 7 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))
22 nfv 1990 . . . . . . . . 9 𝑡 𝑝𝐴
23 nfra1 3077 . . . . . . . . 9 𝑡𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)
242, 22, 23nf3an 1978 . . . . . . . 8 𝑡(𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))
2543ad2ant1 1128 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → 𝐽 ∈ Comp)
267sselda 3742 . . . . . . . . . 10 ((𝜑𝑝𝐴) → 𝑝𝐶)
2726, 6syl6eleq 2847 . . . . . . . . 9 ((𝜑𝑝𝐴) → 𝑝 ∈ (𝐽 Cn 𝐾))
28273adant3 1127 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → 𝑝 ∈ (𝐽 Cn 𝐾))
29 simp3 1133 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡))
30123ad2ant1 1128 . . . . . . . 8 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → 𝑈𝐽)
311, 24, 3, 5, 25, 28, 29, 30stoweidlem28 40746 . . . . . . 7 ((𝜑𝑝𝐴 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))
3219, 20, 21, 31syl3anc 1477 . . . . . 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 1234 . . . . . . . . 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 1236 . . . . . . . . 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 819 . . . . . . . . . 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 1273 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → ∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1))
3736adantr 472 . . . . . . . . . . 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 1275 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))) → (𝑝𝑍) = 0)
3938adantr 472 . . . . . . . . . . 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 1238 . . . . . . . . . . 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 1123 . . . . . . . . . 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 555 . . . . . . . . 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 1123 . . . . . . . 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 449 . . . . . . 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 1993 . . . . . 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 449 . . . 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 1993 . . 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 1990 . . . . . . 7 𝑡 𝑑 ∈ ℝ+
51 nfv 1990 . . . . . . 7 𝑡 𝑑 < 1
52 nfra1 3077 . . . . . . . . 9 𝑡𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1)
53 nfv 1990 . . . . . . . . 9 𝑡(𝑝𝑍) = 0
54 nfra1 3077 . . . . . . . . 9 𝑡𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)
5552, 53, 54nf3an 1978 . . . . . . . 8 𝑡(∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))
5622, 55nfan 1975 . . . . . . 7 𝑡(𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))
5750, 51, 56nf3an 1978 . . . . . 6 𝑡(𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))
582, 57nfan 1975 . . . . 5 𝑡(𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))))
59 nfcv 2900 . . . . 5 𝑡𝑝
60 eqid 2758 . . . . 5 {𝑡𝑇 ∣ (𝑝𝑡) < (𝑑 / 2)} = {𝑡𝑇 ∣ (𝑝𝑡) < (𝑑 / 2)}
617adantr 472 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝐴𝐶)
6283adant1r 1188 . . . . 5 (((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
6393adant1r 1188 . . . . 5 (((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
6410adantlr 753 . . . . 5 (((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
65 simpr1 1234 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑑 ∈ ℝ+)
66 simpr2 1236 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑑 < 1)
6712adantr 472 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑈𝐽)
6813adantr 472 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑍𝑈)
69 simpr3l 1299 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → 𝑝𝐴)
70 simp3r1 1366 . . . . . 6 ((𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))) → ∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1))
7170adantl 473 . . . . 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 1367 . . . . . 6 ((𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))) → (𝑝𝑍) = 0)
7372adantl 473 . . . . 5 ((𝜑 ∧ (𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))))) → (𝑝𝑍) = 0)
74 simp3r3 1368 . . . . . 6 ((𝑑 ∈ ℝ+𝑑 < 1 ∧ (𝑝𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡)))) → ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑝𝑡))
7574adantl 473 . . . . 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 40770 . . . 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 449 . . 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 2009 . 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 383  w3a 1072   = wceq 1630  wex 1851  wnf 1855  wcel 2137  wnfc 2887  wne 2930  wral 3048  wrex 3049  {crab 3052  cdif 3710  wss 3713   cuni 4586   class class class wbr 4802  cmpt 4879  ran crn 5265  cfv 6047  (class class class)co 6811  cr 10125  0cc0 10126  1c1 10127   + caddc 10129   · cmul 10131   < clt 10264  cle 10265  cmin 10456   / cdiv 10874  2c2 11260  +crp 12023  (,)cioo 12366  topGenctg 16298   Cn ccn 21228  Compccmp 21389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203  ax-pre-sup 10204  ax-mulf 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-iin 4673  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-of 7060  df-om 7229  df-1st 7331  df-2nd 7332  df-supp 7462  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-2o 7728  df-oadd 7731  df-er 7909  df-map 8023  df-pm 8024  df-ixp 8073  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-fsupp 8439  df-fi 8480  df-sup 8511  df-inf 8512  df-oi 8578  df-card 8953  df-cda 9180  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-div 10875  df-nn 11211  df-2 11269  df-3 11270  df-4 11271  df-5 11272  df-6 11273  df-7 11274  df-8 11275  df-9 11276  df-n0 11483  df-z 11568  df-dec 11684  df-uz 11878  df-q 11980  df-rp 12024  df-xneg 12137  df-xadd 12138  df-xmul 12139  df-ioo 12370  df-ico 12372  df-icc 12373  df-fz 12518  df-fzo 12658  df-fl 12785  df-seq 12994  df-exp 13053  df-hash 13310  df-cj 14036  df-re 14037  df-im 14038  df-sqrt 14172  df-abs 14173  df-clim 14416  df-rlim 14417  df-sum 14614  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-sets 16064  df-ress 16065  df-plusg 16154  df-mulr 16155  df-starv 16156  df-sca 16157  df-vsca 16158  df-ip 16159  df-tset 16160  df-ple 16161  df-ds 16164  df-unif 16165  df-hom 16166  df-cco 16167  df-rest 16283  df-topn 16284  df-0g 16302  df-gsum 16303  df-topgen 16304  df-pt 16305  df-prds 16308  df-xrs 16362  df-qtop 16367  df-imas 16368  df-xps 16370  df-mre 16446  df-mrc 16447  df-acs 16449  df-mgm 17441  df-sgrp 17483  df-mnd 17494  df-submnd 17535  df-mulg 17740  df-cntz 17948  df-cmn 18393  df-psmet 19938  df-xmet 19939  df-met 19940  df-bl 19941  df-mopn 19942  df-cnfld 19947  df-top 20899  df-topon 20916  df-topsp 20937  df-bases 20950  df-cld 21023  df-cn 21231  df-cnp 21232  df-cmp 21390  df-tx 21565  df-hmeo 21758  df-xms 22324  df-ms 22325  df-tms 22326
This theorem is referenced by:  stoweidlem57  40775
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