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Theorem stoweidlem45 38938
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 2610 . . 3 (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁))) = (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁)))
5 eqid 2610 . . 3 (𝑡𝑇 ↦ 1) = (𝑡𝑇 ↦ 1)
6 eqid 2610 . . 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 11228 . . . 4 (𝜑𝑁 ∈ ℕ0)
1614, 15nnexpcld 12892 . . 3 (𝜑 → (𝐾𝑁) ∈ ℕ)
171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16stoweidlem40 38933 . 2 (𝜑𝑄𝐴)
18 1red 9934 . . . . . . . 8 ((𝜑𝑡𝑇) → 1 ∈ ℝ)
198fnvinran 38196 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝑃𝑡) ∈ ℝ)
2015adantr 480 . . . . . . . . 9 ((𝜑𝑡𝑇) → 𝑁 ∈ ℕ0)
2119, 20reexpcld 12887 . . . . . . . 8 ((𝜑𝑡𝑇) → ((𝑃𝑡)↑𝑁) ∈ ℝ)
2218, 21resubcld 10337 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ∈ ℝ)
2314nnnn0d 11228 . . . . . . . . 9 (𝜑𝐾 ∈ ℕ0)
2423, 15nn0expcld 12893 . . . . . . . 8 (𝜑 → (𝐾𝑁) ∈ ℕ0)
2524adantr 480 . . . . . . 7 ((𝜑𝑡𝑇) → (𝐾𝑁) ∈ ℕ0)
26 1m1e0 10966 . . . . . . . 8 (1 − 1) = 0
27 stoweidlem45.11 . . . . . . . . . . . 12 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
2827r19.21bi 2916 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
2928simpld 474 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 0 ≤ (𝑃𝑡))
3028simprd 478 . . . . . . . . . 10 ((𝜑𝑡𝑇) → (𝑃𝑡) ≤ 1)
31 exple1 12782 . . . . . . . . . 10 ((((𝑃𝑡) ∈ ℝ ∧ 0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1) ∧ 𝑁 ∈ ℕ0) → ((𝑃𝑡)↑𝑁) ≤ 1)
3219, 29, 30, 20, 31syl31anc 1321 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝑃𝑡)↑𝑁) ≤ 1)
3321, 18, 18, 32lesub2dd 10523 . . . . . . . 8 ((𝜑𝑡𝑇) → (1 − 1) ≤ (1 − ((𝑃𝑡)↑𝑁)))
3426, 33syl5eqbrr 4619 . . . . . . 7 ((𝜑𝑡𝑇) → 0 ≤ (1 − ((𝑃𝑡)↑𝑁)))
3522, 25, 34expge0d 12888 . . . . . 6 ((𝜑𝑡𝑇) → 0 ≤ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
363, 8, 15, 23stoweidlem12 38905 . . . . . 6 ((𝜑𝑡𝑇) → (𝑄𝑡) = ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
3735, 36breqtrrd 4611 . . . . 5 ((𝜑𝑡𝑇) → 0 ≤ (𝑄𝑡))
38 0red 9920 . . . . . . . . 9 ((𝜑𝑡𝑇) → 0 ∈ ℝ)
3919, 20, 29expge0d 12888 . . . . . . . . 9 ((𝜑𝑡𝑇) → 0 ≤ ((𝑃𝑡)↑𝑁))
4038, 21, 18, 39lesub2dd 10523 . . . . . . . 8 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ≤ (1 − 0))
41 1m0e1 11008 . . . . . . . 8 (1 − 0) = 1
4240, 41syl6breq 4624 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ≤ 1)
43 exple1 12782 . . . . . . 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 1321 . . . . . 6 ((𝜑𝑡𝑇) → ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)) ≤ 1)
4536, 44eqbrtrd 4605 . . . . 5 ((𝜑𝑡𝑇) → (𝑄𝑡) ≤ 1)
4637, 45jca 553 . . . 4 ((𝜑𝑡𝑇) → (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1))
4746ex 449 . . 3 (𝜑 → (𝑡𝑇 → (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
482, 47ralrimi 2940 . 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 38917 . . . 4 ((𝜑𝑡𝑉) → (1 − 𝐸) < (𝑄𝑡))
5453ex 449 . . 3 (𝜑 → (𝑡𝑉 → (1 − 𝐸) < (𝑄𝑡)))
552, 54ralrimi 2940 . 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 38918 . . . 4 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑄𝑡) < 𝐸)
5958ex 449 . . 3 (𝜑 → (𝑡 ∈ (𝑇𝑈) → (𝑄𝑡) < 𝐸))
602, 59ralrimi 2940 . 2 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)(𝑄𝑡) < 𝐸)
61 nfmpt1 4675 . . . . . . 7 𝑡(𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
623, 61nfcxfr 2749 . . . . . 6 𝑡𝑄
6362nfeq2 2766 . . . . 5 𝑡 𝑦 = 𝑄
64 fveq1 6102 . . . . . . 7 (𝑦 = 𝑄 → (𝑦𝑡) = (𝑄𝑡))
6564breq2d 4595 . . . . . 6 (𝑦 = 𝑄 → (0 ≤ (𝑦𝑡) ↔ 0 ≤ (𝑄𝑡)))
6664breq1d 4593 . . . . . 6 (𝑦 = 𝑄 → ((𝑦𝑡) ≤ 1 ↔ (𝑄𝑡) ≤ 1))
6765, 66anbi12d 743 . . . . 5 (𝑦 = 𝑄 → ((0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
6863, 67ralbid 2966 . . . 4 (𝑦 = 𝑄 → (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
6964breq2d 4595 . . . . 5 (𝑦 = 𝑄 → ((1 − 𝐸) < (𝑦𝑡) ↔ (1 − 𝐸) < (𝑄𝑡)))
7063, 69ralbid 2966 . . . 4 (𝑦 = 𝑄 → (∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡) ↔ ∀𝑡𝑉 (1 − 𝐸) < (𝑄𝑡)))
7164breq1d 4593 . . . . 5 (𝑦 = 𝑄 → ((𝑦𝑡) < 𝐸 ↔ (𝑄𝑡) < 𝐸))
7263, 71ralbid 2966 . . . 4 (𝑦 = 𝑄 → (∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸 ↔ ∀𝑡 ∈ (𝑇𝑈)(𝑄𝑡) < 𝐸))
7368, 70, 723anbi123d 1391 . . 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 3282 . 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 1320 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 383  w3a 1031   = wceq 1475  wnf 1699  wcel 1977  wnfc 2738  wral 2896  wrex 2897  {crab 2900  cdif 3537   class class class wbr 4583  cmpt 4643  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  cn 10897  2c2 10947  0cn0 11169  +crp 11708  cexp 12722
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-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
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-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  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-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-seq 12664  df-exp 12723
This theorem is referenced by:  stoweidlem49  38942
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