Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem45 Structured version   Visualization version   GIF version

Theorem stoweidlem45 42626
 Description: This lemma proves that, given an appropriate 𝐾 (in another theorem we prove such a 𝐾 exists), there exists a function qn as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91 ( at the top of page 91): 0 <= qn <= 1 , qn < ε on T \ U, and qn > 1 - ε on 𝑉. We use y to represent the final qn in the paper (the one with n large enough), 𝑁 to represent 𝑛 in the paper, 𝐾 to represent 𝑘, 𝐷 to represent δ, 𝐸 to represent ε, and 𝑃 to represent 𝑝. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem45.1 𝑡𝑃
stoweidlem45.2 𝑡𝜑
stoweidlem45.3 𝑉 = {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)}
stoweidlem45.4 𝑄 = (𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
stoweidlem45.5 (𝜑𝑁 ∈ ℕ)
stoweidlem45.6 (𝜑𝐾 ∈ ℕ)
stoweidlem45.7 (𝜑𝐷 ∈ ℝ+)
stoweidlem45.8 (𝜑𝐷 < 1)
stoweidlem45.9 (𝜑𝑃𝐴)
stoweidlem45.10 (𝜑𝑃:𝑇⟶ℝ)
stoweidlem45.11 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
stoweidlem45.12 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))
stoweidlem45.13 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem45.14 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
stoweidlem45.15 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stoweidlem45.16 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
stoweidlem45.17 (𝜑𝐸 ∈ ℝ+)
stoweidlem45.18 (𝜑 → (1 − 𝐸) < (1 − (((𝐾 · 𝐷) / 2)↑𝑁)))
stoweidlem45.19 (𝜑 → (1 / ((𝐾 · 𝐷)↑𝑁)) < 𝐸)
Assertion
Ref Expression
stoweidlem45 (𝜑 → ∃𝑦𝐴 (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸))
Distinct variable groups:   𝑓,𝑔,𝑡,𝐴   𝑓,𝑁,𝑔,𝑡   𝑃,𝑓,𝑔   𝑇,𝑓,𝑔,𝑡   𝜑,𝑓,𝑔   𝑥,𝑡,𝐴   𝑦,𝑡,𝐴   𝑡,𝐾   𝑥,𝑇   𝜑,𝑥   𝑦,𝐸   𝑦,𝑄   𝑦,𝑇   𝑦,𝑈   𝑦,𝑉
Allowed substitution hints:   𝜑(𝑦,𝑡)   𝐷(𝑥,𝑦,𝑡,𝑓,𝑔)   𝑃(𝑥,𝑦,𝑡)   𝑄(𝑥,𝑡,𝑓,𝑔)   𝑈(𝑥,𝑡,𝑓,𝑔)   𝐸(𝑥,𝑡,𝑓,𝑔)   𝐾(𝑥,𝑦,𝑓,𝑔)   𝑁(𝑥,𝑦)   𝑉(𝑥,𝑡,𝑓,𝑔)

Proof of Theorem stoweidlem45
StepHypRef Expression
1 stoweidlem45.1 . . 3 𝑡𝑃
2 stoweidlem45.2 . . 3 𝑡𝜑
3 stoweidlem45.4 . . 3 𝑄 = (𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
4 eqid 2822 . . 3 (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁))) = (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁)))
5 eqid 2822 . . 3 (𝑡𝑇 ↦ 1) = (𝑡𝑇 ↦ 1)
6 eqid 2822 . . 3 (𝑡𝑇 ↦ ((𝑃𝑡)↑𝑁)) = (𝑡𝑇 ↦ ((𝑃𝑡)↑𝑁))
7 stoweidlem45.9 . . 3 (𝜑𝑃𝐴)
8 stoweidlem45.10 . . 3 (𝜑𝑃:𝑇⟶ℝ)
9 stoweidlem45.13 . . 3 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
10 stoweidlem45.14 . . 3 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
11 stoweidlem45.15 . . 3 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
12 stoweidlem45.16 . . 3 ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
13 stoweidlem45.5 . . 3 (𝜑𝑁 ∈ ℕ)
14 stoweidlem45.6 . . . 4 (𝜑𝐾 ∈ ℕ)
1513nnnn0d 11943 . . . 4 (𝜑𝑁 ∈ ℕ0)
1614, 15nnexpcld 13602 . . 3 (𝜑 → (𝐾𝑁) ∈ ℕ)
171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16stoweidlem40 42621 . 2 (𝜑𝑄𝐴)
18 1red 10631 . . . . . . . 8 ((𝜑𝑡𝑇) → 1 ∈ ℝ)
198ffvelrnda 6833 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝑃𝑡) ∈ ℝ)
2015adantr 484 . . . . . . . . 9 ((𝜑𝑡𝑇) → 𝑁 ∈ ℕ0)
2119, 20reexpcld 13523 . . . . . . . 8 ((𝜑𝑡𝑇) → ((𝑃𝑡)↑𝑁) ∈ ℝ)
2218, 21resubcld 11057 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ∈ ℝ)
2314nnnn0d 11943 . . . . . . . . 9 (𝜑𝐾 ∈ ℕ0)
2423, 15nn0expcld 13603 . . . . . . . 8 (𝜑 → (𝐾𝑁) ∈ ℕ0)
2524adantr 484 . . . . . . 7 ((𝜑𝑡𝑇) → (𝐾𝑁) ∈ ℕ0)
26 1m1e0 11697 . . . . . . . 8 (1 − 1) = 0
27 stoweidlem45.11 . . . . . . . . . . . 12 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
2827r19.21bi 3198 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
2928simpld 498 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 0 ≤ (𝑃𝑡))
3028simprd 499 . . . . . . . . . 10 ((𝜑𝑡𝑇) → (𝑃𝑡) ≤ 1)
31 exple1 13536 . . . . . . . . . 10 ((((𝑃𝑡) ∈ ℝ ∧ 0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1) ∧ 𝑁 ∈ ℕ0) → ((𝑃𝑡)↑𝑁) ≤ 1)
3219, 29, 30, 20, 31syl31anc 1370 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝑃𝑡)↑𝑁) ≤ 1)
3321, 18, 18, 32lesub2dd 11246 . . . . . . . 8 ((𝜑𝑡𝑇) → (1 − 1) ≤ (1 − ((𝑃𝑡)↑𝑁)))
3426, 33eqbrtrrid 5078 . . . . . . 7 ((𝜑𝑡𝑇) → 0 ≤ (1 − ((𝑃𝑡)↑𝑁)))
3522, 25, 34expge0d 13524 . . . . . 6 ((𝜑𝑡𝑇) → 0 ≤ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
363, 8, 15, 23stoweidlem12 42593 . . . . . 6 ((𝜑𝑡𝑇) → (𝑄𝑡) = ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
3735, 36breqtrrd 5070 . . . . 5 ((𝜑𝑡𝑇) → 0 ≤ (𝑄𝑡))
38 0red 10633 . . . . . . . . 9 ((𝜑𝑡𝑇) → 0 ∈ ℝ)
3919, 20, 29expge0d 13524 . . . . . . . . 9 ((𝜑𝑡𝑇) → 0 ≤ ((𝑃𝑡)↑𝑁))
4038, 21, 18, 39lesub2dd 11246 . . . . . . . 8 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ≤ (1 − 0))
41 1m0e1 11746 . . . . . . . 8 (1 − 0) = 1
4240, 41breqtrdi 5083 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ≤ 1)
43 exple1 13536 . . . . . . 7 ((((1 − ((𝑃𝑡)↑𝑁)) ∈ ℝ ∧ 0 ≤ (1 − ((𝑃𝑡)↑𝑁)) ∧ (1 − ((𝑃𝑡)↑𝑁)) ≤ 1) ∧ (𝐾𝑁) ∈ ℕ0) → ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)) ≤ 1)
4422, 34, 42, 25, 43syl31anc 1370 . . . . . 6 ((𝜑𝑡𝑇) → ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)) ≤ 1)
4536, 44eqbrtrd 5064 . . . . 5 ((𝜑𝑡𝑇) → (𝑄𝑡) ≤ 1)
4637, 45jca 515 . . . 4 ((𝜑𝑡𝑇) → (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1))
4746ex 416 . . 3 (𝜑 → (𝑡𝑇 → (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
482, 47ralrimi 3205 . 2 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1))
49 stoweidlem45.3 . . . . 5 𝑉 = {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)}
50 stoweidlem45.7 . . . . 5 (𝜑𝐷 ∈ ℝ+)
51 stoweidlem45.17 . . . . 5 (𝜑𝐸 ∈ ℝ+)
52 stoweidlem45.18 . . . . 5 (𝜑 → (1 − 𝐸) < (1 − (((𝐾 · 𝐷) / 2)↑𝑁)))
5349, 3, 8, 15, 23, 50, 51, 52, 27stoweidlem24 42605 . . . 4 ((𝜑𝑡𝑉) → (1 − 𝐸) < (𝑄𝑡))
5453ex 416 . . 3 (𝜑 → (𝑡𝑉 → (1 − 𝐸) < (𝑄𝑡)))
552, 54ralrimi 3205 . 2 (𝜑 → ∀𝑡𝑉 (1 − 𝐸) < (𝑄𝑡))
56 stoweidlem45.12 . . . . 5 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))
57 stoweidlem45.19 . . . . 5 (𝜑 → (1 / ((𝐾 · 𝐷)↑𝑁)) < 𝐸)
583, 13, 14, 50, 8, 27, 56, 51, 57stoweidlem25 42606 . . . 4 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑄𝑡) < 𝐸)
5958ex 416 . . 3 (𝜑 → (𝑡 ∈ (𝑇𝑈) → (𝑄𝑡) < 𝐸))
602, 59ralrimi 3205 . 2 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)(𝑄𝑡) < 𝐸)
61 nfmpt1 5140 . . . . . . 7 𝑡(𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
623, 61nfcxfr 2977 . . . . . 6 𝑡𝑄
6362nfeq2 2996 . . . . 5 𝑡 𝑦 = 𝑄
64 fveq1 6651 . . . . . . 7 (𝑦 = 𝑄 → (𝑦𝑡) = (𝑄𝑡))
6564breq2d 5054 . . . . . 6 (𝑦 = 𝑄 → (0 ≤ (𝑦𝑡) ↔ 0 ≤ (𝑄𝑡)))
6664breq1d 5052 . . . . . 6 (𝑦 = 𝑄 → ((𝑦𝑡) ≤ 1 ↔ (𝑄𝑡) ≤ 1))
6765, 66anbi12d 633 . . . . 5 (𝑦 = 𝑄 → ((0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
6863, 67ralbid 3220 . . . 4 (𝑦 = 𝑄 → (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
6964breq2d 5054 . . . . 5 (𝑦 = 𝑄 → ((1 − 𝐸) < (𝑦𝑡) ↔ (1 − 𝐸) < (𝑄𝑡)))
7063, 69ralbid 3220 . . . 4 (𝑦 = 𝑄 → (∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡) ↔ ∀𝑡𝑉 (1 − 𝐸) < (𝑄𝑡)))
7164breq1d 5052 . . . . 5 (𝑦 = 𝑄 → ((𝑦𝑡) < 𝐸 ↔ (𝑄𝑡) < 𝐸))
7263, 71ralbid 3220 . . . 4 (𝑦 = 𝑄 → (∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸 ↔ ∀𝑡 ∈ (𝑇𝑈)(𝑄𝑡) < 𝐸))
7368, 70, 723anbi123d 1433 . . 3 (𝑦 = 𝑄 → ((∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸) ↔ (∀𝑡𝑇 (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝐸) < (𝑄𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑄𝑡) < 𝐸)))
7473rspcev 3598 . 2 ((𝑄𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝐸) < (𝑄𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑄𝑡) < 𝐸)) → ∃𝑦𝐴 (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸))
7517, 48, 55, 60, 74syl13anc 1369 1 (𝜑 → ∃𝑦𝐴 (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2960  ∀wral 3130  ∃wrex 3131  {crab 3134   ∖ cdif 3905   class class class wbr 5042   ↦ cmpt 5122  ⟶wf 6330  ‘cfv 6334  (class class class)co 7140  ℝcr 10525  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531   < clt 10664   ≤ cle 10665   − cmin 10859   / cdiv 11286  ℕcn 11625  2c2 11680  ℕ0cn0 11885  ℝ+crp 12377  ↑cexp 13425 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-seq 13365  df-exp 13426 This theorem is referenced by:  stoweidlem49  42630
 Copyright terms: Public domain W3C validator