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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  wnf 1699  wcel 1977  wnfc 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  cr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953  cle 9954  cmin 10145   / cdiv 10563  2c2 10947  +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
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