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Theorem stoweidlem28 42536
Description: There exists a δ as in Lemma 1 [BrosowskiDeutsh] p. 90: 0 < delta < 1 and p >= delta on 𝑇𝑈. Here 𝑑 is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem28.1 𝑡𝑈
stoweidlem28.2 𝑡𝜑
stoweidlem28.3 𝐾 = (topGen‘ran (,))
stoweidlem28.4 𝑇 = 𝐽
stoweidlem28.5 (𝜑𝐽 ∈ Comp)
stoweidlem28.6 (𝜑𝑃 ∈ (𝐽 Cn 𝐾))
stoweidlem28.7 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡))
stoweidlem28.8 (𝜑𝑈𝐽)
Assertion
Ref Expression
stoweidlem28 (𝜑 → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑃𝑡)))
Distinct variable groups:   𝑡,𝑑,𝑃   𝑇,𝑑,𝑡   𝑈,𝑑   𝑡,𝐽
Allowed substitution hints:   𝜑(𝑡,𝑑)   𝑈(𝑡)   𝐽(𝑑)   𝐾(𝑡,𝑑)

Proof of Theorem stoweidlem28
Dummy variables 𝑐 𝑥 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 halfre 11846 . . . . 5 (1 / 2) ∈ ℝ
2 halfgt0 11848 . . . . 5 0 < (1 / 2)
31, 2elrpii 12387 . . . 4 (1 / 2) ∈ ℝ+
43a1i 11 . . 3 ((𝜑 ∧ (𝑇𝑈) = ∅) → (1 / 2) ∈ ℝ+)
5 halflt1 11850 . . . 4 (1 / 2) < 1
65a1i 11 . . 3 ((𝜑 ∧ (𝑇𝑈) = ∅) → (1 / 2) < 1)
7 nfcv 2982 . . . . . . 7 𝑡𝑇
8 stoweidlem28.1 . . . . . . 7 𝑡𝑈
97, 8nfdif 4088 . . . . . 6 𝑡(𝑇𝑈)
109nfeq1 2997 . . . . 5 𝑡(𝑇𝑈) = ∅
1110rzalf 41506 . . . 4 ((𝑇𝑈) = ∅ → ∀𝑡 ∈ (𝑇𝑈)(1 / 2) ≤ (𝑃𝑡))
1211adantl 485 . . 3 ((𝜑 ∧ (𝑇𝑈) = ∅) → ∀𝑡 ∈ (𝑇𝑈)(1 / 2) ≤ (𝑃𝑡))
13 ovex 7179 . . . 4 (1 / 2) ∈ V
14 eleq1 2903 . . . . 5 (𝑑 = (1 / 2) → (𝑑 ∈ ℝ+ ↔ (1 / 2) ∈ ℝ+))
15 breq1 5056 . . . . 5 (𝑑 = (1 / 2) → (𝑑 < 1 ↔ (1 / 2) < 1))
16 breq1 5056 . . . . . 6 (𝑑 = (1 / 2) → (𝑑 ≤ (𝑃𝑡) ↔ (1 / 2) ≤ (𝑃𝑡)))
1716ralbidv 3192 . . . . 5 (𝑑 = (1 / 2) → (∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑃𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)(1 / 2) ≤ (𝑃𝑡)))
1814, 15, 173anbi123d 1433 . . . 4 (𝑑 = (1 / 2) → ((𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑃𝑡)) ↔ ((1 / 2) ∈ ℝ+ ∧ (1 / 2) < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 / 2) ≤ (𝑃𝑡))))
1913, 18spcev 3593 . . 3 (((1 / 2) ∈ ℝ+ ∧ (1 / 2) < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 / 2) ≤ (𝑃𝑡)) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑃𝑡)))
204, 6, 12, 19syl3anc 1368 . 2 ((𝜑 ∧ (𝑇𝑈) = ∅) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑃𝑡)))
21 simplll 774 . . . 4 ((((𝜑 ∧ ¬ (𝑇𝑈) = ∅) ∧ 𝑥 ∈ (𝑇𝑈)) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) → 𝜑)
22 simplr 768 . . . 4 ((((𝜑 ∧ ¬ (𝑇𝑈) = ∅) ∧ 𝑥 ∈ (𝑇𝑈)) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) → 𝑥 ∈ (𝑇𝑈))
23 simpr 488 . . . 4 ((((𝜑 ∧ ¬ (𝑇𝑈) = ∅) ∧ 𝑥 ∈ (𝑇𝑈)) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) → ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡))
24 stoweidlem28.3 . . . . . . . . . . 11 𝐾 = (topGen‘ran (,))
25 stoweidlem28.4 . . . . . . . . . . 11 𝑇 = 𝐽
26 eqid 2824 . . . . . . . . . . 11 (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾)
27 stoweidlem28.6 . . . . . . . . . . 11 (𝜑𝑃 ∈ (𝐽 Cn 𝐾))
2824, 25, 26, 27fcnre 41514 . . . . . . . . . 10 (𝜑𝑃:𝑇⟶ℝ)
2928adantr 484 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑇𝑈)) → 𝑃:𝑇⟶ℝ)
30 eldifi 4089 . . . . . . . . . 10 (𝑥 ∈ (𝑇𝑈) → 𝑥𝑇)
3130adantl 485 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝑇𝑈)) → 𝑥𝑇)
3229, 31ffvelrnd 6841 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑇𝑈)) → (𝑃𝑥) ∈ ℝ)
33 stoweidlem28.7 . . . . . . . . 9 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡))
34 nfcv 2982 . . . . . . . . . . . 12 𝑥(𝑇𝑈)
35 nfv 1916 . . . . . . . . . . . 12 𝑥0 < (𝑃𝑡)
36 nfv 1916 . . . . . . . . . . . 12 𝑡0 < (𝑃𝑥)
37 fveq2 6659 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → (𝑃𝑡) = (𝑃𝑥))
3837breq2d 5065 . . . . . . . . . . . 12 (𝑡 = 𝑥 → (0 < (𝑃𝑡) ↔ 0 < (𝑃𝑥)))
399, 34, 35, 36, 38cbvralfw 3421 . . . . . . . . . . 11 (∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡) ↔ ∀𝑥 ∈ (𝑇𝑈)0 < (𝑃𝑥))
4039biimpi 219 . . . . . . . . . 10 (∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡) → ∀𝑥 ∈ (𝑇𝑈)0 < (𝑃𝑥))
4140r19.21bi 3203 . . . . . . . . 9 ((∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡) ∧ 𝑥 ∈ (𝑇𝑈)) → 0 < (𝑃𝑥))
4233, 41sylan 583 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑇𝑈)) → 0 < (𝑃𝑥))
4332, 42elrpd 12423 . . . . . . 7 ((𝜑𝑥 ∈ (𝑇𝑈)) → (𝑃𝑥) ∈ ℝ+)
44433adant3 1129 . . . . . 6 ((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) → (𝑃𝑥) ∈ ℝ+)
45 stoweidlem28.2 . . . . . . . 8 𝑡𝜑
469nfcri 2972 . . . . . . . 8 𝑡 𝑥 ∈ (𝑇𝑈)
47 nfra1 3214 . . . . . . . 8 𝑡𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)
4845, 46, 47nf3an 1903 . . . . . . 7 𝑡(𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡))
49 rspa 3201 . . . . . . . . . 10 ((∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡) ∧ 𝑡 ∈ (𝑇𝑈)) → ((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡))
50493ad2antl3 1184 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇𝑈)) → ((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡))
51 simpl2 1189 . . . . . . . . . 10 (((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇𝑈)) → 𝑥 ∈ (𝑇𝑈))
52 fvres 6678 . . . . . . . . . 10 (𝑥 ∈ (𝑇𝑈) → ((𝑃 ↾ (𝑇𝑈))‘𝑥) = (𝑃𝑥))
5351, 52syl 17 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇𝑈)) → ((𝑃 ↾ (𝑇𝑈))‘𝑥) = (𝑃𝑥))
54 fvres 6678 . . . . . . . . . 10 (𝑡 ∈ (𝑇𝑈) → ((𝑃 ↾ (𝑇𝑈))‘𝑡) = (𝑃𝑡))
5554adantl 485 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇𝑈)) → ((𝑃 ↾ (𝑇𝑈))‘𝑡) = (𝑃𝑡))
5650, 53, 553brtr3d 5084 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇𝑈)) → (𝑃𝑥) ≤ (𝑃𝑡))
5756ex 416 . . . . . . 7 ((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) → (𝑡 ∈ (𝑇𝑈) → (𝑃𝑥) ≤ (𝑃𝑡)))
5848, 57ralrimi 3211 . . . . . 6 ((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) → ∀𝑡 ∈ (𝑇𝑈)(𝑃𝑥) ≤ (𝑃𝑡))
59 eleq1 2903 . . . . . . . . 9 (𝑐 = (𝑃𝑥) → (𝑐 ∈ ℝ+ ↔ (𝑃𝑥) ∈ ℝ+))
60 breq1 5056 . . . . . . . . . 10 (𝑐 = (𝑃𝑥) → (𝑐 ≤ (𝑃𝑡) ↔ (𝑃𝑥) ≤ (𝑃𝑡)))
6160ralbidv 3192 . . . . . . . . 9 (𝑐 = (𝑃𝑥) → (∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)(𝑃𝑥) ≤ (𝑃𝑡)))
6259, 61anbi12d 633 . . . . . . . 8 (𝑐 = (𝑃𝑥) → ((𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡)) ↔ ((𝑃𝑥) ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑃𝑥) ≤ (𝑃𝑡))))
6362spcegv 3583 . . . . . . 7 ((𝑃𝑥) ∈ ℝ+ → (((𝑃𝑥) ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑃𝑥) ≤ (𝑃𝑡)) → ∃𝑐(𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡))))
6444, 63syl 17 . . . . . 6 ((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) → (((𝑃𝑥) ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑃𝑥) ≤ (𝑃𝑡)) → ∃𝑐(𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡))))
6544, 58, 64mp2and 698 . . . . 5 ((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) → ∃𝑐(𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡)))
66 simpl1 1188 . . . . . 6 (((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) ∧ (𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡))) → 𝜑)
67 simprl 770 . . . . . 6 (((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) ∧ (𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡))) → 𝑐 ∈ ℝ+)
68 simprr 772 . . . . . 6 (((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) ∧ (𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡))) → ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡))
69 nfv 1916 . . . . . . . 8 𝑡 𝑐 ∈ ℝ+
70 nfra1 3214 . . . . . . . 8 𝑡𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡)
7145, 69, 70nf3an 1903 . . . . . . 7 𝑡(𝜑𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡))
72 eqid 2824 . . . . . . 7 if(𝑐 ≤ (1 / 2), 𝑐, (1 / 2)) = if(𝑐 ≤ (1 / 2), 𝑐, (1 / 2))
73283ad2ant1 1130 . . . . . . 7 ((𝜑𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡)) → 𝑃:𝑇⟶ℝ)
74 difssd 4095 . . . . . . 7 ((𝜑𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡)) → (𝑇𝑈) ⊆ 𝑇)
75 simp2 1134 . . . . . . 7 ((𝜑𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡)) → 𝑐 ∈ ℝ+)
76 simp3 1135 . . . . . . 7 ((𝜑𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡)) → ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡))
7771, 72, 73, 74, 75, 76stoweidlem5 42513 . . . . . 6 ((𝜑𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡)) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑃𝑡)))
7866, 67, 68, 77syl3anc 1368 . . . . 5 (((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) ∧ (𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇𝑈)𝑐 ≤ (𝑃𝑡))) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑃𝑡)))
7965, 78exlimddv 1937 . . . 4 ((𝜑𝑥 ∈ (𝑇𝑈) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑃𝑡)))
8021, 22, 23, 79syl3anc 1368 . . 3 ((((𝜑 ∧ ¬ (𝑇𝑈) = ∅) ∧ 𝑥 ∈ (𝑇𝑈)) ∧ ∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑃𝑡)))
81 eqid 2824 . . . . . 6 (𝐽t (𝑇𝑈)) = (𝐽t (𝑇𝑈))
82 stoweidlem28.5 . . . . . . . 8 (𝜑𝐽 ∈ Comp)
83 stoweidlem28.8 . . . . . . . . 9 (𝜑𝑈𝐽)
84 cmptop 21998 . . . . . . . . . . 11 (𝐽 ∈ Comp → 𝐽 ∈ Top)
8582, 84syl 17 . . . . . . . . . 10 (𝜑𝐽 ∈ Top)
86 elssuni 4855 . . . . . . . . . . . 12 (𝑈𝐽𝑈 𝐽)
8783, 86syl 17 . . . . . . . . . . 11 (𝜑𝑈 𝐽)
8887, 25sseqtrrdi 4004 . . . . . . . . . 10 (𝜑𝑈𝑇)
8925isopn2 21635 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑈𝑇) → (𝑈𝐽 ↔ (𝑇𝑈) ∈ (Clsd‘𝐽)))
9085, 88, 89syl2anc 587 . . . . . . . . 9 (𝜑 → (𝑈𝐽 ↔ (𝑇𝑈) ∈ (Clsd‘𝐽)))
9183, 90mpbid 235 . . . . . . . 8 (𝜑 → (𝑇𝑈) ∈ (Clsd‘𝐽))
92 cmpcld 22005 . . . . . . . 8 ((𝐽 ∈ Comp ∧ (𝑇𝑈) ∈ (Clsd‘𝐽)) → (𝐽t (𝑇𝑈)) ∈ Comp)
9382, 91, 92syl2anc 587 . . . . . . 7 (𝜑 → (𝐽t (𝑇𝑈)) ∈ Comp)
9493adantr 484 . . . . . 6 ((𝜑 ∧ ¬ (𝑇𝑈) = ∅) → (𝐽t (𝑇𝑈)) ∈ Comp)
9527adantr 484 . . . . . . 7 ((𝜑 ∧ ¬ (𝑇𝑈) = ∅) → 𝑃 ∈ (𝐽 Cn 𝐾))
96 difssd 4095 . . . . . . 7 ((𝜑 ∧ ¬ (𝑇𝑈) = ∅) → (𝑇𝑈) ⊆ 𝑇)
9725cnrest 21888 . . . . . . 7 ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ (𝑇𝑈) ⊆ 𝑇) → (𝑃 ↾ (𝑇𝑈)) ∈ ((𝐽t (𝑇𝑈)) Cn 𝐾))
9895, 96, 97syl2anc 587 . . . . . 6 ((𝜑 ∧ ¬ (𝑇𝑈) = ∅) → (𝑃 ↾ (𝑇𝑈)) ∈ ((𝐽t (𝑇𝑈)) Cn 𝐾))
99 difssd 4095 . . . . . . . . . 10 (𝜑 → (𝑇𝑈) ⊆ 𝑇)
10025restuni 21765 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑇𝑈) ⊆ 𝑇) → (𝑇𝑈) = (𝐽t (𝑇𝑈)))
10185, 99, 100syl2anc 587 . . . . . . . . 9 (𝜑 → (𝑇𝑈) = (𝐽t (𝑇𝑈)))
102101neeq1d 3073 . . . . . . . 8 (𝜑 → ((𝑇𝑈) ≠ ∅ ↔ (𝐽t (𝑇𝑈)) ≠ ∅))
103 df-ne 3015 . . . . . . . 8 ((𝑇𝑈) ≠ ∅ ↔ ¬ (𝑇𝑈) = ∅)
104102, 103bitr3di 289 . . . . . . 7 (𝜑 → ( (𝐽t (𝑇𝑈)) ≠ ∅ ↔ ¬ (𝑇𝑈) = ∅))
105104biimpar 481 . . . . . 6 ((𝜑 ∧ ¬ (𝑇𝑈) = ∅) → (𝐽t (𝑇𝑈)) ≠ ∅)
10681, 24, 94, 98, 105evth2 23563 . . . . 5 ((𝜑 ∧ ¬ (𝑇𝑈) = ∅) → ∃𝑥 (𝐽t (𝑇𝑈))∀𝑠 (𝐽t (𝑇𝑈))((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑠))
107 nfcv 2982 . . . . . . 7 𝑠 (𝐽t (𝑇𝑈))
108 nfcv 2982 . . . . . . . . 9 𝑡𝐽
109 nfcv 2982 . . . . . . . . 9 𝑡t
110108, 109, 9nfov 7176 . . . . . . . 8 𝑡(𝐽t (𝑇𝑈))
111110nfuni 4832 . . . . . . 7 𝑡 (𝐽t (𝑇𝑈))
112 nfcv 2982 . . . . . . . . . 10 𝑡𝑃
113112, 9nfres 5843 . . . . . . . . 9 𝑡(𝑃 ↾ (𝑇𝑈))
114 nfcv 2982 . . . . . . . . 9 𝑡𝑥
115113, 114nffv 6669 . . . . . . . 8 𝑡((𝑃 ↾ (𝑇𝑈))‘𝑥)
116 nfcv 2982 . . . . . . . 8 𝑡
117 nfcv 2982 . . . . . . . . 9 𝑡𝑠
118113, 117nffv 6669 . . . . . . . 8 𝑡((𝑃 ↾ (𝑇𝑈))‘𝑠)
119115, 116, 118nfbr 5100 . . . . . . 7 𝑡((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑠)
120 nfv 1916 . . . . . . 7 𝑠((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)
121 fveq2 6659 . . . . . . . 8 (𝑠 = 𝑡 → ((𝑃 ↾ (𝑇𝑈))‘𝑠) = ((𝑃 ↾ (𝑇𝑈))‘𝑡))
122121breq2d 5065 . . . . . . 7 (𝑠 = 𝑡 → (((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑠) ↔ ((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)))
123107, 111, 119, 120, 122cbvralfw 3421 . . . . . 6 (∀𝑠 (𝐽t (𝑇𝑈))((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑠) ↔ ∀𝑡 (𝐽t (𝑇𝑈))((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡))
124123rexbii 3242 . . . . 5 (∃𝑥 (𝐽t (𝑇𝑈))∀𝑠 (𝐽t (𝑇𝑈))((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑠) ↔ ∃𝑥 (𝐽t (𝑇𝑈))∀𝑡 (𝐽t (𝑇𝑈))((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡))
125106, 124sylib 221 . . . 4 ((𝜑 ∧ ¬ (𝑇𝑈) = ∅) → ∃𝑥 (𝐽t (𝑇𝑈))∀𝑡 (𝐽t (𝑇𝑈))((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡))
1269, 111raleqf 3389 . . . . . . 7 ((𝑇𝑈) = (𝐽t (𝑇𝑈)) → (∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡) ↔ ∀𝑡 (𝐽t (𝑇𝑈))((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)))
127126rexeqbi1dv 3396 . . . . . 6 ((𝑇𝑈) = (𝐽t (𝑇𝑈)) → (∃𝑥 ∈ (𝑇𝑈)∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡) ↔ ∃𝑥 (𝐽t (𝑇𝑈))∀𝑡 (𝐽t (𝑇𝑈))((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)))
128101, 127syl 17 . . . . 5 (𝜑 → (∃𝑥 ∈ (𝑇𝑈)∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡) ↔ ∃𝑥 (𝐽t (𝑇𝑈))∀𝑡 (𝐽t (𝑇𝑈))((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)))
129128adantr 484 . . . 4 ((𝜑 ∧ ¬ (𝑇𝑈) = ∅) → (∃𝑥 ∈ (𝑇𝑈)∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡) ↔ ∃𝑥 (𝐽t (𝑇𝑈))∀𝑡 (𝐽t (𝑇𝑈))((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡)))
130125, 129mpbird 260 . . 3 ((𝜑 ∧ ¬ (𝑇𝑈) = ∅) → ∃𝑥 ∈ (𝑇𝑈)∀𝑡 ∈ (𝑇𝑈)((𝑃 ↾ (𝑇𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇𝑈))‘𝑡))
13180, 130r19.29a 3282 . 2 ((𝜑 ∧ ¬ (𝑇𝑈) = ∅) → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑃𝑡)))
13220, 131pm2.61dan 812 1 (𝜑 → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑃𝑡)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wnf 1785  wcel 2115  wnfc 2962  wne 3014  wral 3133  wrex 3134  cdif 3916  wss 3919  c0 4276  ifcif 4450   cuni 4825   class class class wbr 5053  ran crn 5544  cres 5545  wf 6340  cfv 6344  (class class class)co 7146  cr 10530  0cc0 10531  1c1 10532   < clt 10669  cle 10670   / cdiv 11291  2c2 11687  +crp 12384  (,)cioo 12733  t crest 16692  topGenctg 16709  Topctop 21496  Clsdccld 21619   Cn ccn 21827  Compccmp 21989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609  ax-mulf 10611
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-int 4864  df-iun 4908  df-iin 4909  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-se 5503  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-isom 6353  df-riota 7104  df-ov 7149  df-oprab 7150  df-mpo 7151  df-of 7400  df-om 7572  df-1st 7681  df-2nd 7682  df-supp 7823  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8827  df-fi 8868  df-sup 8899  df-inf 8900  df-oi 8967  df-card 9361  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11695  df-3 11696  df-4 11697  df-5 11698  df-6 11699  df-7 11700  df-8 11701  df-9 11702  df-n0 11893  df-z 11977  df-dec 12094  df-uz 12239  df-q 12344  df-rp 12385  df-xneg 12502  df-xadd 12503  df-xmul 12504  df-ioo 12737  df-icc 12740  df-fz 12893  df-fzo 13036  df-seq 13372  df-exp 13433  df-hash 13694  df-cj 14456  df-re 14457  df-im 14458  df-sqrt 14592  df-abs 14593  df-struct 16483  df-ndx 16484  df-slot 16485  df-base 16487  df-sets 16488  df-ress 16489  df-plusg 16576  df-mulr 16577  df-starv 16578  df-sca 16579  df-vsca 16580  df-ip 16581  df-tset 16582  df-ple 16583  df-ds 16585  df-unif 16586  df-hom 16587  df-cco 16588  df-rest 16694  df-topn 16695  df-0g 16713  df-gsum 16714  df-topgen 16715  df-pt 16716  df-prds 16719  df-xrs 16773  df-qtop 16778  df-imas 16779  df-xps 16781  df-mre 16855  df-mrc 16856  df-acs 16858  df-mgm 17850  df-sgrp 17899  df-mnd 17910  df-submnd 17955  df-mulg 18223  df-cntz 18445  df-cmn 18906  df-psmet 20532  df-xmet 20533  df-met 20534  df-bl 20535  df-mopn 20536  df-cnfld 20541  df-top 21497  df-topon 21514  df-topsp 21536  df-bases 21549  df-cld 21622  df-cn 21830  df-cnp 21831  df-cmp 21990  df-tx 22165  df-hmeo 22358  df-xms 22925  df-ms 22926  df-tms 22927
This theorem is referenced by:  stoweidlem56  42564
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