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Theorem stoweidlem52 43050
 Description: There exists a neighborood V as in Lemma 1 of [BrosowskiDeutsh] p. 90. Here Z is used to represent t0 in the paper, and v is used to represent V in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem52.1 𝑡𝑈
stoweidlem52.2 𝑡𝜑
stoweidlem52.3 𝑡𝑃
stoweidlem52.4 𝐾 = (topGen‘ran (,))
stoweidlem52.5 𝑉 = {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)}
stoweidlem52.7 𝑇 = 𝐽
stoweidlem52.8 𝐶 = (𝐽 Cn 𝐾)
stoweidlem52.9 (𝜑𝐴𝐶)
stoweidlem52.10 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
stoweidlem52.11 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stoweidlem52.12 ((𝜑𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)
stoweidlem52.13 (𝜑𝐷 ∈ ℝ+)
stoweidlem52.14 (𝜑𝐷 < 1)
stoweidlem52.15 (𝜑𝑈𝐽)
stoweidlem52.16 (𝜑𝑍𝑈)
stoweidlem52.17 (𝜑𝑃𝐴)
stoweidlem52.18 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
stoweidlem52.19 (𝜑 → (𝑃𝑍) = 0)
stoweidlem52.20 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))
Assertion
Ref Expression
stoweidlem52 (𝜑 → ∃𝑣𝐽 ((𝑍𝑣𝑣𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑣 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))))
Distinct variable groups:   𝑒,𝑎,𝑡   𝐴,𝑎,𝑡   𝐷,𝑎,𝑡   𝑇,𝑎,𝑡   𝑈,𝑎   𝑉,𝑎,𝑒   𝜑,𝑎,𝑒   𝑒,𝑓,𝑔,𝑡   𝑣,𝑒,𝑥,𝑡   𝐴,𝑓,𝑔   𝐷,𝑓,𝑔   𝑃,𝑓,𝑔   𝑇,𝑓,𝑔   𝑈,𝑓,𝑔   𝑓,𝑉,𝑔   𝜑,𝑓,𝑔   𝑡,𝑍,𝑣   𝑣,𝐴   𝑣,𝐽   𝑣,𝑇,𝑥   𝑣,𝑈,𝑥   𝑣,𝑉,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑣,𝑡)   𝐴(𝑒)   𝐶(𝑥,𝑣,𝑡,𝑒,𝑓,𝑔,𝑎)   𝐷(𝑥,𝑣,𝑒)   𝑃(𝑥,𝑣,𝑡,𝑒,𝑎)   𝑇(𝑒)   𝑈(𝑡,𝑒)   𝐽(𝑥,𝑡,𝑒,𝑓,𝑔,𝑎)   𝐾(𝑥,𝑣,𝑡,𝑒,𝑓,𝑔,𝑎)   𝑉(𝑡)   𝑍(𝑥,𝑒,𝑓,𝑔,𝑎)

Proof of Theorem stoweidlem52
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2920 . . 3 𝑡(𝐷 / 2)
2 stoweidlem52.3 . . 3 𝑡𝑃
3 stoweidlem52.2 . . 3 𝑡𝜑
4 stoweidlem52.4 . . 3 𝐾 = (topGen‘ran (,))
5 stoweidlem52.7 . . 3 𝑇 = 𝐽
6 stoweidlem52.5 . . 3 𝑉 = {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)}
7 stoweidlem52.13 . . . . . 6 (𝜑𝐷 ∈ ℝ+)
87rpred 12462 . . . . 5 (𝜑𝐷 ∈ ℝ)
98rehalfcld 11911 . . . 4 (𝜑 → (𝐷 / 2) ∈ ℝ)
109rexrd 10719 . . 3 (𝜑 → (𝐷 / 2) ∈ ℝ*)
11 stoweidlem52.9 . . . . 5 (𝜑𝐴𝐶)
12 stoweidlem52.8 . . . . 5 𝐶 = (𝐽 Cn 𝐾)
1311, 12sseqtrdi 3943 . . . 4 (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))
14 stoweidlem52.17 . . . 4 (𝜑𝑃𝐴)
1513, 14sseldd 3894 . . 3 (𝜑𝑃 ∈ (𝐽 Cn 𝐾))
161, 2, 3, 4, 5, 6, 10, 15rfcnpre2 42023 . 2 (𝜑𝑉𝐽)
17 stoweidlem52.15 . . . . . . . 8 (𝜑𝑈𝐽)
18 elssuni 4828 . . . . . . . 8 (𝑈𝐽𝑈 𝐽)
1917, 18syl 17 . . . . . . 7 (𝜑𝑈 𝐽)
2019, 5sseqtrrdi 3944 . . . . . 6 (𝜑𝑈𝑇)
21 stoweidlem52.16 . . . . . 6 (𝜑𝑍𝑈)
2220, 21sseldd 3894 . . . . 5 (𝜑𝑍𝑇)
23 stoweidlem52.19 . . . . . 6 (𝜑 → (𝑃𝑍) = 0)
24 2re 11738 . . . . . . . 8 2 ∈ ℝ
2524a1i 11 . . . . . . 7 (𝜑 → 2 ∈ ℝ)
267rpgt0d 12465 . . . . . . 7 (𝜑 → 0 < 𝐷)
27 2pos 11767 . . . . . . . 8 0 < 2
2827a1i 11 . . . . . . 7 (𝜑 → 0 < 2)
298, 25, 26, 28divgt0d 11603 . . . . . 6 (𝜑 → 0 < (𝐷 / 2))
3023, 29eqbrtrd 5052 . . . . 5 (𝜑 → (𝑃𝑍) < (𝐷 / 2))
31 nfcv 2920 . . . . . 6 𝑡𝑍
32 nfcv 2920 . . . . . 6 𝑡𝑇
332, 31nffv 6666 . . . . . . 7 𝑡(𝑃𝑍)
34 nfcv 2920 . . . . . . 7 𝑡 <
3533, 34, 1nfbr 5077 . . . . . 6 𝑡(𝑃𝑍) < (𝐷 / 2)
36 fveq2 6656 . . . . . . 7 (𝑡 = 𝑍 → (𝑃𝑡) = (𝑃𝑍))
3736breq1d 5040 . . . . . 6 (𝑡 = 𝑍 → ((𝑃𝑡) < (𝐷 / 2) ↔ (𝑃𝑍) < (𝐷 / 2)))
3831, 32, 35, 37elrabf 3599 . . . . 5 (𝑍 ∈ {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)} ↔ (𝑍𝑇 ∧ (𝑃𝑍) < (𝐷 / 2)))
3922, 30, 38sylanbrc 587 . . . 4 (𝜑𝑍 ∈ {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)})
4039, 6eleqtrrdi 2864 . . 3 (𝜑𝑍𝑉)
41 nfrab1 3303 . . . . 5 𝑡{𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)}
426, 41nfcxfr 2918 . . . 4 𝑡𝑉
43 stoweidlem52.1 . . . 4 𝑡𝑈
4411, 14sseldd 3894 . . . . . . . . . . 11 (𝜑𝑃𝐶)
454, 5, 12, 44fcnre 42017 . . . . . . . . . 10 (𝜑𝑃:𝑇⟶ℝ)
4645adantr 485 . . . . . . . . 9 ((𝜑𝑡𝑉) → 𝑃:𝑇⟶ℝ)
476rabeq2i 3401 . . . . . . . . . . . 12 (𝑡𝑉 ↔ (𝑡𝑇 ∧ (𝑃𝑡) < (𝐷 / 2)))
4847biimpi 219 . . . . . . . . . . 11 (𝑡𝑉 → (𝑡𝑇 ∧ (𝑃𝑡) < (𝐷 / 2)))
4948adantl 486 . . . . . . . . . 10 ((𝜑𝑡𝑉) → (𝑡𝑇 ∧ (𝑃𝑡) < (𝐷 / 2)))
5049simpld 499 . . . . . . . . 9 ((𝜑𝑡𝑉) → 𝑡𝑇)
5146, 50ffvelrnd 6841 . . . . . . . 8 ((𝜑𝑡𝑉) → (𝑃𝑡) ∈ ℝ)
529adantr 485 . . . . . . . 8 ((𝜑𝑡𝑉) → (𝐷 / 2) ∈ ℝ)
538adantr 485 . . . . . . . 8 ((𝜑𝑡𝑉) → 𝐷 ∈ ℝ)
5449simprd 500 . . . . . . . 8 ((𝜑𝑡𝑉) → (𝑃𝑡) < (𝐷 / 2))
55 halfpos 11894 . . . . . . . . . . 11 (𝐷 ∈ ℝ → (0 < 𝐷 ↔ (𝐷 / 2) < 𝐷))
568, 55syl 17 . . . . . . . . . 10 (𝜑 → (0 < 𝐷 ↔ (𝐷 / 2) < 𝐷))
5726, 56mpbid 235 . . . . . . . . 9 (𝜑 → (𝐷 / 2) < 𝐷)
5857adantr 485 . . . . . . . 8 ((𝜑𝑡𝑉) → (𝐷 / 2) < 𝐷)
5951, 52, 53, 54, 58lttrd 10829 . . . . . . 7 ((𝜑𝑡𝑉) → (𝑃𝑡) < 𝐷)
6059adantr 485 . . . . . 6 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → (𝑃𝑡) < 𝐷)
618ad2antrr 726 . . . . . . 7 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → 𝐷 ∈ ℝ)
6251adantr 485 . . . . . . 7 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → (𝑃𝑡) ∈ ℝ)
63 stoweidlem52.20 . . . . . . . . 9 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))
6463ad2antrr 726 . . . . . . . 8 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))
6550anim1i 618 . . . . . . . . 9 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → (𝑡𝑇 ∧ ¬ 𝑡𝑈))
66 eldif 3869 . . . . . . . . 9 (𝑡 ∈ (𝑇𝑈) ↔ (𝑡𝑇 ∧ ¬ 𝑡𝑈))
6765, 66sylibr 237 . . . . . . . 8 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → 𝑡 ∈ (𝑇𝑈))
68 rsp 3135 . . . . . . . 8 (∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡) → (𝑡 ∈ (𝑇𝑈) → 𝐷 ≤ (𝑃𝑡)))
6964, 67, 68sylc 65 . . . . . . 7 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → 𝐷 ≤ (𝑃𝑡))
7061, 62, 69lensymd 10819 . . . . . 6 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → ¬ (𝑃𝑡) < 𝐷)
7160, 70condan 818 . . . . 5 ((𝜑𝑡𝑉) → 𝑡𝑈)
7271ex 417 . . . 4 (𝜑 → (𝑡𝑉𝑡𝑈))
733, 42, 43, 72ssrd 3898 . . 3 (𝜑𝑉𝑈)
74 nfv 1916 . . . . . . . . 9 𝑡 𝑒 ∈ ℝ+
753, 74nfan 1901 . . . . . . . 8 𝑡(𝜑𝑒 ∈ ℝ+)
76 nfv 1916 . . . . . . . 8 𝑡 𝑦𝐴
7775, 76nfan 1901 . . . . . . 7 𝑡((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴)
78 nfra1 3148 . . . . . . . 8 𝑡𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)
79 nfra1 3148 . . . . . . . 8 𝑡𝑡𝑉 (1 − 𝑒) < (𝑦𝑡)
80 nfra1 3148 . . . . . . . 8 𝑡𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒
8178, 79, 80nf3an 1903 . . . . . . 7 𝑡(∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)
8277, 81nfan 1901 . . . . . 6 𝑡(((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒))
83 eqid 2759 . . . . . 6 (𝑡𝑇 ↦ (1 − (𝑦𝑡))) = (𝑡𝑇 ↦ (1 − (𝑦𝑡)))
84 eqid 2759 . . . . . 6 (𝑡𝑇 ↦ 1) = (𝑡𝑇 ↦ 1)
85 ssrab2 3985 . . . . . . 7 {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)} ⊆ 𝑇
866, 85eqsstri 3927 . . . . . 6 𝑉𝑇
87 simplr 769 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝑦𝐴)
88 simplll 775 . . . . . . 7 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝜑)
8911sselda 3893 . . . . . . . 8 ((𝜑𝑦𝐴) → 𝑦𝐶)
904, 5, 12, 89fcnre 42017 . . . . . . 7 ((𝜑𝑦𝐴) → 𝑦:𝑇⟶ℝ)
9188, 87, 90syl2anc 588 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝑦:𝑇⟶ℝ)
9211sselda 3893 . . . . . . . 8 ((𝜑𝑓𝐴) → 𝑓𝐶)
934, 5, 12, 92fcnre 42017 . . . . . . 7 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
9488, 93sylan 584 . . . . . 6 (((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) ∧ 𝑓𝐴) → 𝑓:𝑇⟶ℝ)
95 stoweidlem52.10 . . . . . . 7 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
9688, 95syl3an1 1161 . . . . . 6 (((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
97 stoweidlem52.11 . . . . . . 7 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
9888, 97syl3an1 1161 . . . . . 6 (((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
99 stoweidlem52.12 . . . . . . 7 ((𝜑𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)
10088, 99sylan 584 . . . . . 6 (((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) ∧ 𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)
101 simpllr 776 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝑒 ∈ ℝ+)
102 simpr1 1192 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1))
103 simpr2 1193 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡))
104 simpr3 1194 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)
10582, 83, 84, 86, 87, 91, 94, 96, 98, 100, 101, 102, 103, 104stoweidlem41 43039 . . . . 5 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)))
1067adantr 485 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝐷 ∈ ℝ+)
107 stoweidlem52.14 . . . . . . 7 (𝜑𝐷 < 1)
108107adantr 485 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝐷 < 1)
10914adantr 485 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝑃𝐴)
11045adantr 485 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝑃:𝑇⟶ℝ)
111 stoweidlem52.18 . . . . . . 7 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
112111adantr 485 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
11363adantr 485 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))
11493adantlr 715 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑓𝐴) → 𝑓:𝑇⟶ℝ)
115953adant1r 1175 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
116973adant1r 1175 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
11799adantlr 715 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)
118 simpr 489 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝑒 ∈ ℝ+)
1192, 75, 6, 106, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118stoweidlem49 43047 . . . . 5 ((𝜑𝑒 ∈ ℝ+) → ∃𝑦𝐴 (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒))
120105, 119r19.29a 3214 . . . 4 ((𝜑𝑒 ∈ ℝ+) → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)))
121120ralrimiva 3114 . . 3 (𝜑 → ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)))
12240, 73, 121jca31 519 . 2 (𝜑 → ((𝑍𝑉𝑉𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))))
123 eleq2 2841 . . . . 5 (𝑣 = 𝑉 → (𝑍𝑣𝑍𝑉))
124 sseq1 3918 . . . . 5 (𝑣 = 𝑉 → (𝑣𝑈𝑉𝑈))
125123, 124anbi12d 634 . . . 4 (𝑣 = 𝑉 → ((𝑍𝑣𝑣𝑈) ↔ (𝑍𝑉𝑉𝑈)))
126 nfcv 2920 . . . . . . . 8 𝑡𝑣
127126, 42raleqf 3316 . . . . . . 7 (𝑣 = 𝑉 → (∀𝑡𝑣 (𝑥𝑡) < 𝑒 ↔ ∀𝑡𝑉 (𝑥𝑡) < 𝑒))
1281273anbi2d 1439 . . . . . 6 (𝑣 = 𝑉 → ((∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑣 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)) ↔ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))))
129128rexbidv 3222 . . . . 5 (𝑣 = 𝑉 → (∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑣 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)) ↔ ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))))
130129ralbidv 3127 . . . 4 (𝑣 = 𝑉 → (∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑣 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)) ↔ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))))
131125, 130anbi12d 634 . . 3 (𝑣 = 𝑉 → (((𝑍𝑣𝑣𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑣 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))) ↔ ((𝑍𝑉𝑉𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)))))
132131rspcev 3542 . 2 ((𝑉𝐽 ∧ ((𝑍𝑉𝑉𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)))) → ∃𝑣𝐽 ((𝑍𝑣𝑣𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑣 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))))
13316, 122, 132syl2anc 588 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:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2112  Ⅎwnfc 2900  ∀wral 3071  ∃wrex 3072  {crab 3075   ∖ cdif 3856   ⊆ wss 3859  ∪ cuni 4796   class class class wbr 5030   ↦ cmpt 5110  ran crn 5523  ⟶wf 6329  ‘cfv 6333  (class class class)co 7148  ℝcr 10564  0cc0 10565  1c1 10566   + caddc 10568   · cmul 10570   < clt 10703   ≤ cle 10704   − cmin 10898   / cdiv 11325  2c2 11719  ℝ+crp 12420  (,)cioo 12769  topGenctg 16759   Cn ccn 21914 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457  ax-cnex 10621  ax-resscn 10622  ax-1cn 10623  ax-icn 10624  ax-addcl 10625  ax-addrcl 10626  ax-mulcl 10627  ax-mulrcl 10628  ax-mulcom 10629  ax-addass 10630  ax-mulass 10631  ax-distr 10632  ax-i2m1 10633  ax-1ne0 10634  ax-1rid 10635  ax-rnegex 10636  ax-rrecex 10637  ax-cnre 10638  ax-pre-lttri 10639  ax-pre-lttrn 10640  ax-pre-ltadd 10641  ax-pre-mulgt0 10642  ax-pre-sup 10643 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-tp 4525  df-op 4527  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7578  df-1st 7691  df-2nd 7692  df-wrecs 7955  df-recs 8016  df-rdg 8054  df-er 8297  df-map 8416  df-pm 8417  df-en 8526  df-dom 8527  df-sdom 8528  df-sup 8929  df-inf 8930  df-pnf 10705  df-mnf 10706  df-xr 10707  df-ltxr 10708  df-le 10709  df-sub 10900  df-neg 10901  df-div 11326  df-nn 11665  df-2 11727  df-3 11728  df-n0 11925  df-z 12011  df-uz 12273  df-q 12379  df-rp 12421  df-ioo 12773  df-fl 13201  df-seq 13409  df-exp 13470  df-cj 14496  df-re 14497  df-im 14498  df-sqrt 14632  df-abs 14633  df-clim 14883  df-rlim 14884  df-topgen 16765  df-top 21584  df-topon 21601  df-bases 21636  df-cn 21917 This theorem is referenced by:  stoweidlem56  43054
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