Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem52 Structured version   Visualization version   GIF version

Theorem stoweidlem52 38945
 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 2751 . . 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 11748 . . . . 5 (𝜑𝐷 ∈ ℝ)
98rehalfcld 11156 . . . 4 (𝜑 → (𝐷 / 2) ∈ ℝ)
109rexrd 9968 . . 3 (𝜑 → (𝐷 / 2) ∈ ℝ*)
11 stoweidlem52.9 . . . . 5 (𝜑𝐴𝐶)
12 stoweidlem52.8 . . . . 5 𝐶 = (𝐽 Cn 𝐾)
1311, 12syl6sseq 3614 . . . 4 (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))
14 stoweidlem52.17 . . . 4 (𝜑𝑃𝐴)
1513, 14sseldd 3569 . . 3 (𝜑𝑃 ∈ (𝐽 Cn 𝐾))
161, 2, 3, 4, 5, 6, 10, 15rfcnpre2 38213 . 2 (𝜑𝑉𝐽)
17 stoweidlem52.15 . . . . . . . 8 (𝜑𝑈𝐽)
18 elssuni 4403 . . . . . . . 8 (𝑈𝐽𝑈 𝐽)
1917, 18syl 17 . . . . . . 7 (𝜑𝑈 𝐽)
2019, 5syl6sseqr 3615 . . . . . 6 (𝜑𝑈𝑇)
21 stoweidlem52.16 . . . . . 6 (𝜑𝑍𝑈)
2220, 21sseldd 3569 . . . . 5 (𝜑𝑍𝑇)
23 stoweidlem52.19 . . . . . 6 (𝜑 → (𝑃𝑍) = 0)
24 2re 10967 . . . . . . . 8 2 ∈ ℝ
2524a1i 11 . . . . . . 7 (𝜑 → 2 ∈ ℝ)
267rpgt0d 11751 . . . . . . 7 (𝜑 → 0 < 𝐷)
27 2pos 10989 . . . . . . . 8 0 < 2
2827a1i 11 . . . . . . 7 (𝜑 → 0 < 2)
298, 25, 26, 28divgt0d 10838 . . . . . 6 (𝜑 → 0 < (𝐷 / 2))
3023, 29eqbrtrd 4605 . . . . 5 (𝜑 → (𝑃𝑍) < (𝐷 / 2))
31 nfcv 2751 . . . . . 6 𝑡𝑍
32 nfcv 2751 . . . . . 6 𝑡𝑇
332, 31nffv 6110 . . . . . . 7 𝑡(𝑃𝑍)
34 nfcv 2751 . . . . . . 7 𝑡 <
3533, 34, 1nfbr 4629 . . . . . 6 𝑡(𝑃𝑍) < (𝐷 / 2)
36 fveq2 6103 . . . . . . 7 (𝑡 = 𝑍 → (𝑃𝑡) = (𝑃𝑍))
3736breq1d 4593 . . . . . 6 (𝑡 = 𝑍 → ((𝑃𝑡) < (𝐷 / 2) ↔ (𝑃𝑍) < (𝐷 / 2)))
3831, 32, 35, 37elrabf 3329 . . . . 5 (𝑍 ∈ {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)} ↔ (𝑍𝑇 ∧ (𝑃𝑍) < (𝐷 / 2)))
3922, 30, 38sylanbrc 695 . . . 4 (𝜑𝑍 ∈ {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)})
4039, 6syl6eleqr 2699 . . 3 (𝜑𝑍𝑉)
41 nfrab1 3099 . . . . 5 𝑡{𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)}
426, 41nfcxfr 2749 . . . 4 𝑡𝑉
43 stoweidlem52.1 . . . 4 𝑡𝑈
4411, 14sseldd 3569 . . . . . . . . . . 11 (𝜑𝑃𝐶)
454, 5, 12, 44fcnre 38207 . . . . . . . . . 10 (𝜑𝑃:𝑇⟶ℝ)
4645adantr 480 . . . . . . . . 9 ((𝜑𝑡𝑉) → 𝑃:𝑇⟶ℝ)
476rabeq2i 3170 . . . . . . . . . . . 12 (𝑡𝑉 ↔ (𝑡𝑇 ∧ (𝑃𝑡) < (𝐷 / 2)))
4847biimpi 205 . . . . . . . . . . 11 (𝑡𝑉 → (𝑡𝑇 ∧ (𝑃𝑡) < (𝐷 / 2)))
4948adantl 481 . . . . . . . . . 10 ((𝜑𝑡𝑉) → (𝑡𝑇 ∧ (𝑃𝑡) < (𝐷 / 2)))
5049simpld 474 . . . . . . . . 9 ((𝜑𝑡𝑉) → 𝑡𝑇)
5146, 50ffvelrnd 6268 . . . . . . . 8 ((𝜑𝑡𝑉) → (𝑃𝑡) ∈ ℝ)
529adantr 480 . . . . . . . 8 ((𝜑𝑡𝑉) → (𝐷 / 2) ∈ ℝ)
538adantr 480 . . . . . . . 8 ((𝜑𝑡𝑉) → 𝐷 ∈ ℝ)
5449simprd 478 . . . . . . . 8 ((𝜑𝑡𝑉) → (𝑃𝑡) < (𝐷 / 2))
55 halfpos 11139 . . . . . . . . . . 11 (𝐷 ∈ ℝ → (0 < 𝐷 ↔ (𝐷 / 2) < 𝐷))
568, 55syl 17 . . . . . . . . . 10 (𝜑 → (0 < 𝐷 ↔ (𝐷 / 2) < 𝐷))
5726, 56mpbid 221 . . . . . . . . 9 (𝜑 → (𝐷 / 2) < 𝐷)
5857adantr 480 . . . . . . . 8 ((𝜑𝑡𝑉) → (𝐷 / 2) < 𝐷)
5951, 52, 53, 54, 58lttrd 10077 . . . . . . 7 ((𝜑𝑡𝑉) → (𝑃𝑡) < 𝐷)
6059adantr 480 . . . . . 6 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → (𝑃𝑡) < 𝐷)
618ad2antrr 758 . . . . . . 7 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → 𝐷 ∈ ℝ)
6251adantr 480 . . . . . . 7 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → (𝑃𝑡) ∈ ℝ)
63 stoweidlem52.20 . . . . . . . . 9 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))
6463ad2antrr 758 . . . . . . . 8 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))
6550anim1i 590 . . . . . . . . 9 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → (𝑡𝑇 ∧ ¬ 𝑡𝑈))
66 eldif 3550 . . . . . . . . 9 (𝑡 ∈ (𝑇𝑈) ↔ (𝑡𝑇 ∧ ¬ 𝑡𝑈))
6765, 66sylibr 223 . . . . . . . 8 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → 𝑡 ∈ (𝑇𝑈))
68 rsp 2913 . . . . . . . 8 (∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡) → (𝑡 ∈ (𝑇𝑈) → 𝐷 ≤ (𝑃𝑡)))
6964, 67, 68sylc 63 . . . . . . 7 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → 𝐷 ≤ (𝑃𝑡))
7061, 62, 69lensymd 10067 . . . . . 6 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → ¬ (𝑃𝑡) < 𝐷)
7160, 70condan 831 . . . . 5 ((𝜑𝑡𝑉) → 𝑡𝑈)
7271ex 449 . . . 4 (𝜑 → (𝑡𝑉𝑡𝑈))
733, 42, 43, 72ssrd 3573 . . 3 (𝜑𝑉𝑈)
74 nfv 1830 . . . . . . . . 9 𝑡 𝑒 ∈ ℝ+
753, 74nfan 1816 . . . . . . . 8 𝑡(𝜑𝑒 ∈ ℝ+)
76 nfv 1830 . . . . . . . 8 𝑡 𝑦𝐴
7775, 76nfan 1816 . . . . . . 7 𝑡((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴)
78 nfra1 2925 . . . . . . . 8 𝑡𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)
79 nfra1 2925 . . . . . . . 8 𝑡𝑡𝑉 (1 − 𝑒) < (𝑦𝑡)
80 nfra1 2925 . . . . . . . 8 𝑡𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒
8178, 79, 80nf3an 1819 . . . . . . 7 𝑡(∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)
8277, 81nfan 1816 . . . . . 6 𝑡(((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒))
83 eqid 2610 . . . . . 6 (𝑡𝑇 ↦ (1 − (𝑦𝑡))) = (𝑡𝑇 ↦ (1 − (𝑦𝑡)))
84 eqid 2610 . . . . . 6 (𝑡𝑇 ↦ 1) = (𝑡𝑇 ↦ 1)
85 ssrab2 3650 . . . . . . 7 {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)} ⊆ 𝑇
866, 85eqsstri 3598 . . . . . 6 𝑉𝑇
87 simplr 788 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝑦𝐴)
88 simplll 794 . . . . . . 7 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝜑)
8911sselda 3568 . . . . . . . 8 ((𝜑𝑦𝐴) → 𝑦𝐶)
904, 5, 12, 89fcnre 38207 . . . . . . 7 ((𝜑𝑦𝐴) → 𝑦:𝑇⟶ℝ)
9188, 87, 90syl2anc 691 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝑦:𝑇⟶ℝ)
9211sselda 3568 . . . . . . . 8 ((𝜑𝑓𝐴) → 𝑓𝐶)
934, 5, 12, 92fcnre 38207 . . . . . . 7 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
9488, 93sylan 487 . . . . . 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 1351 . . . . . 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 1351 . . . . . 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 487 . . . . . 6 (((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) ∧ 𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)
101 simpllr 795 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝑒 ∈ ℝ+)
102 simpr1 1060 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1))
103 simpr2 1061 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡))
104 simpr3 1062 . . . . . 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 38934 . . . . 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 480 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝐷 ∈ ℝ+)
107 stoweidlem52.14 . . . . . . 7 (𝜑𝐷 < 1)
108107adantr 480 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝐷 < 1)
10914adantr 480 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝑃𝐴)
11045adantr 480 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝑃:𝑇⟶ℝ)
111 stoweidlem52.18 . . . . . . 7 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
112111adantr 480 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
11363adantr 480 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))
11493adantlr 747 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑓𝐴) → 𝑓:𝑇⟶ℝ)
115953adant1r 1311 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
116973adant1r 1311 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
11799adantlr 747 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)
118 simpr 476 . . . . . 6 ((𝜑𝑒 ∈ ℝ+) → 𝑒 ∈ ℝ+)
1192, 75, 6, 106, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118stoweidlem49 38942 . . . . 5 ((𝜑𝑒 ∈ ℝ+) → ∃𝑦𝐴 (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒))
120105, 119r19.29a 3060 . . . 4 ((𝜑𝑒 ∈ ℝ+) → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)))
121120ralrimiva 2949 . . 3 (𝜑 → ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)))
12240, 73, 121jca31 555 . 2 (𝜑 → ((𝑍𝑉𝑉𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))))
123 eleq2 2677 . . . . 5 (𝑣 = 𝑉 → (𝑍𝑣𝑍𝑉))
124 sseq1 3589 . . . . 5 (𝑣 = 𝑉 → (𝑣𝑈𝑉𝑈))
125123, 124anbi12d 743 . . . 4 (𝑣 = 𝑉 → ((𝑍𝑣𝑣𝑈) ↔ (𝑍𝑉𝑉𝑈)))
126 nfcv 2751 . . . . . . . 8 𝑡𝑣
127126, 42raleqf 3111 . . . . . . 7 (𝑣 = 𝑉 → (∀𝑡𝑣 (𝑥𝑡) < 𝑒 ↔ ∀𝑡𝑉 (𝑥𝑡) < 𝑒))
1281273anbi2d 1396 . . . . . 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 3034 . . . . 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 2969 . . . 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 743 . . 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 3282 . 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 691 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  2c2 10947  ℝ+crp 11708  (,)cioo 12046  topGenctg 15921   Cn ccn 20838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-ioo 12050  df-fl 12455  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cn 20841 This theorem is referenced by:  stoweidlem56  38949
 Copyright terms: Public domain W3C validator