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Theorem stoweidlem52 44283
Description: There exists a neighborhood 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 2907 . . 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 12957 . . . . 5 (𝜑𝐷 ∈ ℝ)
98rehalfcld 12400 . . . 4 (𝜑 → (𝐷 / 2) ∈ ℝ)
109rexrd 11205 . . 3 (𝜑 → (𝐷 / 2) ∈ ℝ*)
11 stoweidlem52.9 . . . . 5 (𝜑𝐴𝐶)
12 stoweidlem52.8 . . . . 5 𝐶 = (𝐽 Cn 𝐾)
1311, 12sseqtrdi 3994 . . . 4 (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))
14 stoweidlem52.17 . . . 4 (𝜑𝑃𝐴)
1513, 14sseldd 3945 . . 3 (𝜑𝑃 ∈ (𝐽 Cn 𝐾))
161, 2, 3, 4, 5, 6, 10, 15rfcnpre2 43226 . 2 (𝜑𝑉𝐽)
17 stoweidlem52.15 . . . . . . . 8 (𝜑𝑈𝐽)
18 elssuni 4898 . . . . . . . 8 (𝑈𝐽𝑈 𝐽)
1917, 18syl 17 . . . . . . 7 (𝜑𝑈 𝐽)
2019, 5sseqtrrdi 3995 . . . . . 6 (𝜑𝑈𝑇)
21 stoweidlem52.16 . . . . . 6 (𝜑𝑍𝑈)
2220, 21sseldd 3945 . . . . 5 (𝜑𝑍𝑇)
23 stoweidlem52.19 . . . . . 6 (𝜑 → (𝑃𝑍) = 0)
24 2re 12227 . . . . . . . 8 2 ∈ ℝ
2524a1i 11 . . . . . . 7 (𝜑 → 2 ∈ ℝ)
267rpgt0d 12960 . . . . . . 7 (𝜑 → 0 < 𝐷)
27 2pos 12256 . . . . . . . 8 0 < 2
2827a1i 11 . . . . . . 7 (𝜑 → 0 < 2)
298, 25, 26, 28divgt0d 12090 . . . . . 6 (𝜑 → 0 < (𝐷 / 2))
3023, 29eqbrtrd 5127 . . . . 5 (𝜑 → (𝑃𝑍) < (𝐷 / 2))
31 nfcv 2907 . . . . . 6 𝑡𝑍
32 nfcv 2907 . . . . . 6 𝑡𝑇
332, 31nffv 6852 . . . . . . 7 𝑡(𝑃𝑍)
34 nfcv 2907 . . . . . . 7 𝑡 <
3533, 34, 1nfbr 5152 . . . . . 6 𝑡(𝑃𝑍) < (𝐷 / 2)
36 fveq2 6842 . . . . . . 7 (𝑡 = 𝑍 → (𝑃𝑡) = (𝑃𝑍))
3736breq1d 5115 . . . . . 6 (𝑡 = 𝑍 → ((𝑃𝑡) < (𝐷 / 2) ↔ (𝑃𝑍) < (𝐷 / 2)))
3831, 32, 35, 37elrabf 3641 . . . . 5 (𝑍 ∈ {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)} ↔ (𝑍𝑇 ∧ (𝑃𝑍) < (𝐷 / 2)))
3922, 30, 38sylanbrc 583 . . . 4 (𝜑𝑍 ∈ {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)})
4039, 6eleqtrrdi 2849 . . 3 (𝜑𝑍𝑉)
41 nfrab1 3426 . . . . 5 𝑡{𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)}
426, 41nfcxfr 2905 . . . 4 𝑡𝑉
43 stoweidlem52.1 . . . 4 𝑡𝑈
4411, 14sseldd 3945 . . . . . . . . . . 11 (𝜑𝑃𝐶)
454, 5, 12, 44fcnre 43220 . . . . . . . . . 10 (𝜑𝑃:𝑇⟶ℝ)
4645adantr 481 . . . . . . . . 9 ((𝜑𝑡𝑉) → 𝑃:𝑇⟶ℝ)
476reqabi 3429 . . . . . . . . . . . 12 (𝑡𝑉 ↔ (𝑡𝑇 ∧ (𝑃𝑡) < (𝐷 / 2)))
4847biimpi 215 . . . . . . . . . . 11 (𝑡𝑉 → (𝑡𝑇 ∧ (𝑃𝑡) < (𝐷 / 2)))
4948adantl 482 . . . . . . . . . 10 ((𝜑𝑡𝑉) → (𝑡𝑇 ∧ (𝑃𝑡) < (𝐷 / 2)))
5049simpld 495 . . . . . . . . 9 ((𝜑𝑡𝑉) → 𝑡𝑇)
5146, 50ffvelcdmd 7036 . . . . . . . 8 ((𝜑𝑡𝑉) → (𝑃𝑡) ∈ ℝ)
529adantr 481 . . . . . . . 8 ((𝜑𝑡𝑉) → (𝐷 / 2) ∈ ℝ)
538adantr 481 . . . . . . . 8 ((𝜑𝑡𝑉) → 𝐷 ∈ ℝ)
5449simprd 496 . . . . . . . 8 ((𝜑𝑡𝑉) → (𝑃𝑡) < (𝐷 / 2))
55 halfpos 12383 . . . . . . . . . . 11 (𝐷 ∈ ℝ → (0 < 𝐷 ↔ (𝐷 / 2) < 𝐷))
568, 55syl 17 . . . . . . . . . 10 (𝜑 → (0 < 𝐷 ↔ (𝐷 / 2) < 𝐷))
5726, 56mpbid 231 . . . . . . . . 9 (𝜑 → (𝐷 / 2) < 𝐷)
5857adantr 481 . . . . . . . 8 ((𝜑𝑡𝑉) → (𝐷 / 2) < 𝐷)
5951, 52, 53, 54, 58lttrd 11316 . . . . . . 7 ((𝜑𝑡𝑉) → (𝑃𝑡) < 𝐷)
6059adantr 481 . . . . . 6 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → (𝑃𝑡) < 𝐷)
618ad2antrr 724 . . . . . . 7 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → 𝐷 ∈ ℝ)
6251adantr 481 . . . . . . 7 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → (𝑃𝑡) ∈ ℝ)
63 stoweidlem52.20 . . . . . . . . 9 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))
6463ad2antrr 724 . . . . . . . 8 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))
6550anim1i 615 . . . . . . . . 9 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → (𝑡𝑇 ∧ ¬ 𝑡𝑈))
66 eldif 3920 . . . . . . . . 9 (𝑡 ∈ (𝑇𝑈) ↔ (𝑡𝑇 ∧ ¬ 𝑡𝑈))
6765, 66sylibr 233 . . . . . . . 8 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → 𝑡 ∈ (𝑇𝑈))
68 rsp 3230 . . . . . . . 8 (∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡) → (𝑡 ∈ (𝑇𝑈) → 𝐷 ≤ (𝑃𝑡)))
6964, 67, 68sylc 65 . . . . . . 7 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → 𝐷 ≤ (𝑃𝑡))
7061, 62, 69lensymd 11306 . . . . . 6 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → ¬ (𝑃𝑡) < 𝐷)
7160, 70condan 816 . . . . 5 ((𝜑𝑡𝑉) → 𝑡𝑈)
7271ex 413 . . . 4 (𝜑 → (𝑡𝑉𝑡𝑈))
733, 42, 43, 72ssrd 3949 . . 3 (𝜑𝑉𝑈)
74 nfv 1917 . . . . . . . . 9 𝑡 𝑒 ∈ ℝ+
753, 74nfan 1902 . . . . . . . 8 𝑡(𝜑𝑒 ∈ ℝ+)
76 nfv 1917 . . . . . . . 8 𝑡 𝑦𝐴
7775, 76nfan 1902 . . . . . . 7 𝑡((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴)
78 nfra1 3267 . . . . . . . 8 𝑡𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)
79 nfra1 3267 . . . . . . . 8 𝑡𝑡𝑉 (1 − 𝑒) < (𝑦𝑡)
80 nfra1 3267 . . . . . . . 8 𝑡𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒
8178, 79, 80nf3an 1904 . . . . . . 7 𝑡(∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)
8277, 81nfan 1902 . . . . . 6 𝑡(((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒))
83 eqid 2736 . . . . . 6 (𝑡𝑇 ↦ (1 − (𝑦𝑡))) = (𝑡𝑇 ↦ (1 − (𝑦𝑡)))
84 eqid 2736 . . . . . 6 (𝑡𝑇 ↦ 1) = (𝑡𝑇 ↦ 1)
85 ssrab2 4037 . . . . . . 7 {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)} ⊆ 𝑇
866, 85eqsstri 3978 . . . . . 6 𝑉𝑇
87 simplr 767 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝑦𝐴)
88 simplll 773 . . . . . . 7 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝜑)
8911sselda 3944 . . . . . . . 8 ((𝜑𝑦𝐴) → 𝑦𝐶)
904, 5, 12, 89fcnre 43220 . . . . . . 7 ((𝜑𝑦𝐴) → 𝑦:𝑇⟶ℝ)
9188, 87, 90syl2anc 584 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝑦:𝑇⟶ℝ)
9211sselda 3944 . . . . . . . 8 ((𝜑𝑓𝐴) → 𝑓𝐶)
934, 5, 12, 92fcnre 43220 . . . . . . 7 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
9488, 93sylan 580 . . . . . 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 1163 . . . . . 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 1163 . . . . . 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 580 . . . . . 6 (((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) ∧ 𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)
101 simpllr 774 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝑒 ∈ ℝ+)
102 simpr1 1194 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1))
103 simpr2 1195 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡))
104 simpr3 1196 . . . . . 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 44272 . . . . 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 481 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝐷 ∈ ℝ+)
107 stoweidlem52.14 . . . . . . 7 (𝜑𝐷 < 1)
108107adantr 481 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝐷 < 1)
10914adantr 481 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝑃𝐴)
11045adantr 481 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝑃:𝑇⟶ℝ)
111 stoweidlem52.18 . . . . . . 7 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
112111adantr 481 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
11363adantr 481 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))
11493adantlr 713 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑓𝐴) → 𝑓:𝑇⟶ℝ)
115953adant1r 1177 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
116973adant1r 1177 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
11799adantlr 713 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)
118 simpr 485 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝑒 ∈ ℝ+)
1192, 75, 6, 106, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118stoweidlem49 44280 . . . . 5 ((𝜑𝑒 ∈ ℝ+) → ∃𝑦𝐴 (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒))
120105, 119r19.29a 3159 . . . 4 ((𝜑𝑒 ∈ ℝ+) → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)))
121120ralrimiva 3143 . . 3 (𝜑 → ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)))
12240, 73, 121jca31 515 . 2 (𝜑 → ((𝑍𝑉𝑉𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))))
123 eleq2 2826 . . . . 5 (𝑣 = 𝑉 → (𝑍𝑣𝑍𝑉))
124 sseq1 3969 . . . . 5 (𝑣 = 𝑉 → (𝑣𝑈𝑉𝑈))
125123, 124anbi12d 631 . . . 4 (𝑣 = 𝑉 → ((𝑍𝑣𝑣𝑈) ↔ (𝑍𝑉𝑉𝑈)))
126 nfcv 2907 . . . . . . . 8 𝑡𝑣
127126, 42raleqf 3328 . . . . . . 7 (𝑣 = 𝑉 → (∀𝑡𝑣 (𝑥𝑡) < 𝑒 ↔ ∀𝑡𝑉 (𝑥𝑡) < 𝑒))
1281273anbi2d 1441 . . . . . 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 3175 . . . . 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 3174 . . . 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 631 . . 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 3581 . 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 584 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 205  wa 396  w3a 1087   = wceq 1541  wnf 1785  wcel 2106  wnfc 2887  wral 3064  wrex 3073  {crab 3407  cdif 3907  wss 3910   cuni 4865   class class class wbr 5105  cmpt 5188  ran crn 5634  wf 6492  cfv 6496  (class class class)co 7357  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  2c2 12208  +crp 12915  (,)cioo 13264  topGenctg 17319   Cn ccn 22575
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-ioo 13268  df-fl 13697  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578
This theorem is referenced by:  stoweidlem56  44287
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