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

Theorem stoweidlem52 40587
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 2793 . . 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 11910 . . . . 5 (𝜑𝐷 ∈ ℝ)
98rehalfcld 11317 . . . 4 (𝜑 → (𝐷 / 2) ∈ ℝ)
109rexrd 10127 . . 3 (𝜑 → (𝐷 / 2) ∈ ℝ*)
11 stoweidlem52.9 . . . . 5 (𝜑𝐴𝐶)
12 stoweidlem52.8 . . . . 5 𝐶 = (𝐽 Cn 𝐾)
1311, 12syl6sseq 3684 . . . 4 (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))
14 stoweidlem52.17 . . . 4 (𝜑𝑃𝐴)
1513, 14sseldd 3637 . . 3 (𝜑𝑃 ∈ (𝐽 Cn 𝐾))
161, 2, 3, 4, 5, 6, 10, 15rfcnpre2 39504 . 2 (𝜑𝑉𝐽)
17 stoweidlem52.15 . . . . . . . 8 (𝜑𝑈𝐽)
18 elssuni 4499 . . . . . . . 8 (𝑈𝐽𝑈 𝐽)
1917, 18syl 17 . . . . . . 7 (𝜑𝑈 𝐽)
2019, 5syl6sseqr 3685 . . . . . 6 (𝜑𝑈𝑇)
21 stoweidlem52.16 . . . . . 6 (𝜑𝑍𝑈)
2220, 21sseldd 3637 . . . . 5 (𝜑𝑍𝑇)
23 stoweidlem52.19 . . . . . 6 (𝜑 → (𝑃𝑍) = 0)
24 2re 11128 . . . . . . . 8 2 ∈ ℝ
2524a1i 11 . . . . . . 7 (𝜑 → 2 ∈ ℝ)
267rpgt0d 11913 . . . . . . 7 (𝜑 → 0 < 𝐷)
27 2pos 11150 . . . . . . . 8 0 < 2
2827a1i 11 . . . . . . 7 (𝜑 → 0 < 2)
298, 25, 26, 28divgt0d 10997 . . . . . 6 (𝜑 → 0 < (𝐷 / 2))
3023, 29eqbrtrd 4707 . . . . 5 (𝜑 → (𝑃𝑍) < (𝐷 / 2))
31 nfcv 2793 . . . . . 6 𝑡𝑍
32 nfcv 2793 . . . . . 6 𝑡𝑇
332, 31nffv 6236 . . . . . . 7 𝑡(𝑃𝑍)
34 nfcv 2793 . . . . . . 7 𝑡 <
3533, 34, 1nfbr 4732 . . . . . 6 𝑡(𝑃𝑍) < (𝐷 / 2)
36 fveq2 6229 . . . . . . 7 (𝑡 = 𝑍 → (𝑃𝑡) = (𝑃𝑍))
3736breq1d 4695 . . . . . 6 (𝑡 = 𝑍 → ((𝑃𝑡) < (𝐷 / 2) ↔ (𝑃𝑍) < (𝐷 / 2)))
3831, 32, 35, 37elrabf 3392 . . . . 5 (𝑍 ∈ {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)} ↔ (𝑍𝑇 ∧ (𝑃𝑍) < (𝐷 / 2)))
3922, 30, 38sylanbrc 699 . . . 4 (𝜑𝑍 ∈ {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)})
4039, 6syl6eleqr 2741 . . 3 (𝜑𝑍𝑉)
41 nfrab1 3152 . . . . 5 𝑡{𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)}
426, 41nfcxfr 2791 . . . 4 𝑡𝑉
43 stoweidlem52.1 . . . 4 𝑡𝑈
4411, 14sseldd 3637 . . . . . . . . . . 11 (𝜑𝑃𝐶)
454, 5, 12, 44fcnre 39498 . . . . . . . . . 10 (𝜑𝑃:𝑇⟶ℝ)
4645adantr 480 . . . . . . . . 9 ((𝜑𝑡𝑉) → 𝑃:𝑇⟶ℝ)
476rabeq2i 3228 . . . . . . . . . . . 12 (𝑡𝑉 ↔ (𝑡𝑇 ∧ (𝑃𝑡) < (𝐷 / 2)))
4847biimpi 206 . . . . . . . . . . 11 (𝑡𝑉 → (𝑡𝑇 ∧ (𝑃𝑡) < (𝐷 / 2)))
4948adantl 481 . . . . . . . . . 10 ((𝜑𝑡𝑉) → (𝑡𝑇 ∧ (𝑃𝑡) < (𝐷 / 2)))
5049simpld 474 . . . . . . . . 9 ((𝜑𝑡𝑉) → 𝑡𝑇)
5146, 50ffvelrnd 6400 . . . . . . . 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 11300 . . . . . . . . . . 11 (𝐷 ∈ ℝ → (0 < 𝐷 ↔ (𝐷 / 2) < 𝐷))
568, 55syl 17 . . . . . . . . . 10 (𝜑 → (0 < 𝐷 ↔ (𝐷 / 2) < 𝐷))
5726, 56mpbid 222 . . . . . . . . 9 (𝜑 → (𝐷 / 2) < 𝐷)
5857adantr 480 . . . . . . . 8 ((𝜑𝑡𝑉) → (𝐷 / 2) < 𝐷)
5951, 52, 53, 54, 58lttrd 10236 . . . . . . 7 ((𝜑𝑡𝑉) → (𝑃𝑡) < 𝐷)
6059adantr 480 . . . . . 6 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → (𝑃𝑡) < 𝐷)
618ad2antrr 762 . . . . . . 7 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → 𝐷 ∈ ℝ)
6251adantr 480 . . . . . . 7 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → (𝑃𝑡) ∈ ℝ)
63 stoweidlem52.20 . . . . . . . . 9 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))
6463ad2antrr 762 . . . . . . . 8 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))
6550anim1i 591 . . . . . . . . 9 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → (𝑡𝑇 ∧ ¬ 𝑡𝑈))
66 eldif 3617 . . . . . . . . 9 (𝑡 ∈ (𝑇𝑈) ↔ (𝑡𝑇 ∧ ¬ 𝑡𝑈))
6765, 66sylibr 224 . . . . . . . 8 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → 𝑡 ∈ (𝑇𝑈))
68 rsp 2958 . . . . . . . 8 (∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡) → (𝑡 ∈ (𝑇𝑈) → 𝐷 ≤ (𝑃𝑡)))
6964, 67, 68sylc 65 . . . . . . 7 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → 𝐷 ≤ (𝑃𝑡))
7061, 62, 69lensymd 10226 . . . . . 6 (((𝜑𝑡𝑉) ∧ ¬ 𝑡𝑈) → ¬ (𝑃𝑡) < 𝐷)
7160, 70condan 852 . . . . 5 ((𝜑𝑡𝑉) → 𝑡𝑈)
7271ex 449 . . . 4 (𝜑 → (𝑡𝑉𝑡𝑈))
733, 42, 43, 72ssrd 3641 . . 3 (𝜑𝑉𝑈)
74 nfv 1883 . . . . . . . . 9 𝑡 𝑒 ∈ ℝ+
753, 74nfan 1868 . . . . . . . 8 𝑡(𝜑𝑒 ∈ ℝ+)
76 nfv 1883 . . . . . . . 8 𝑡 𝑦𝐴
7775, 76nfan 1868 . . . . . . 7 𝑡((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴)
78 nfra1 2970 . . . . . . . 8 𝑡𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)
79 nfra1 2970 . . . . . . . 8 𝑡𝑡𝑉 (1 − 𝑒) < (𝑦𝑡)
80 nfra1 2970 . . . . . . . 8 𝑡𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒
8178, 79, 80nf3an 1871 . . . . . . 7 𝑡(∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)
8277, 81nfan 1868 . . . . . 6 𝑡(((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒))
83 eqid 2651 . . . . . 6 (𝑡𝑇 ↦ (1 − (𝑦𝑡))) = (𝑡𝑇 ↦ (1 − (𝑦𝑡)))
84 eqid 2651 . . . . . 6 (𝑡𝑇 ↦ 1) = (𝑡𝑇 ↦ 1)
85 ssrab2 3720 . . . . . . 7 {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)} ⊆ 𝑇
866, 85eqsstri 3668 . . . . . 6 𝑉𝑇
87 simplr 807 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝑦𝐴)
88 simplll 813 . . . . . . 7 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝜑)
8911sselda 3636 . . . . . . . 8 ((𝜑𝑦𝐴) → 𝑦𝐶)
904, 5, 12, 89fcnre 39498 . . . . . . 7 ((𝜑𝑦𝐴) → 𝑦:𝑇⟶ℝ)
9188, 87, 90syl2anc 694 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝑦:𝑇⟶ℝ)
9211sselda 3636 . . . . . . . 8 ((𝜑𝑓𝐴) → 𝑓𝐶)
934, 5, 12, 92fcnre 39498 . . . . . . 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 1399 . . . . . 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 1399 . . . . . 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 815 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → 𝑒 ∈ ℝ+)
102 simpr1 1087 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1))
103 simpr2 1088 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑦𝐴) ∧ (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒)) → ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡))
104 simpr3 1089 . . . . . 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 40576 . . . . 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 751 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑓𝐴) → 𝑓:𝑇⟶ℝ)
115953adant1r 1359 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
116973adant1r 1359 . . . . . 6 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
11799adantlr 751 . . . . . 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 40584 . . . . 5 ((𝜑𝑒 ∈ ℝ+) → ∃𝑦𝐴 (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝑒) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝑒))
120105, 119r19.29a 3107 . . . 4 ((𝜑𝑒 ∈ ℝ+) → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)))
121120ralrimiva 2995 . . 3 (𝜑 → ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡)))
12240, 73, 121jca31 556 . 2 (𝜑 → ((𝑍𝑉𝑉𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))))
123 eleq2 2719 . . . . 5 (𝑣 = 𝑉 → (𝑍𝑣𝑍𝑉))
124 sseq1 3659 . . . . 5 (𝑣 = 𝑉 → (𝑣𝑈𝑉𝑈))
125123, 124anbi12d 747 . . . 4 (𝑣 = 𝑉 → ((𝑍𝑣𝑣𝑈) ↔ (𝑍𝑉𝑉𝑈)))
126 nfcv 2793 . . . . . . . 8 𝑡𝑣
127126, 42raleqf 3164 . . . . . . 7 (𝑣 = 𝑉 → (∀𝑡𝑣 (𝑥𝑡) < 𝑒 ↔ ∀𝑡𝑉 (𝑥𝑡) < 𝑒))
1281273anbi2d 1444 . . . . . 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 3081 . . . . 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 3015 . . . 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 747 . . 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 3340 . 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 694 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 196  wa 383  w3a 1054   = wceq 1523  wnf 1748  wcel 2030  wnfc 2780  wral 2941  wrex 2942  {crab 2945  cdif 3604  wss 3607   cuni 4468   class class class wbr 4685  cmpt 4762  ran crn 5144  wf 5922  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  2c2 11108  +crp 11870  (,)cioo 12213  topGenctg 16145   Cn ccn 21076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-ioo 12217  df-fl 12633  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cn 21079
This theorem is referenced by:  stoweidlem56  40591
  Copyright terms: Public domain W3C validator