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Theorem stoweidlem45 40765
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 2760 . . 3 (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁))) = (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁)))
5 eqid 2760 . . 3 (𝑡𝑇 ↦ 1) = (𝑡𝑇 ↦ 1)
6 eqid 2760 . . 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 11543 . . . 4 (𝜑𝑁 ∈ ℕ0)
1614, 15nnexpcld 13224 . . 3 (𝜑 → (𝐾𝑁) ∈ ℕ)
171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16stoweidlem40 40760 . 2 (𝜑𝑄𝐴)
18 1red 10247 . . . . . . . 8 ((𝜑𝑡𝑇) → 1 ∈ ℝ)
198ffvelrnda 6522 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝑃𝑡) ∈ ℝ)
2015adantr 472 . . . . . . . . 9 ((𝜑𝑡𝑇) → 𝑁 ∈ ℕ0)
2119, 20reexpcld 13219 . . . . . . . 8 ((𝜑𝑡𝑇) → ((𝑃𝑡)↑𝑁) ∈ ℝ)
2218, 21resubcld 10650 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ∈ ℝ)
2314nnnn0d 11543 . . . . . . . . 9 (𝜑𝐾 ∈ ℕ0)
2423, 15nn0expcld 13225 . . . . . . . 8 (𝜑 → (𝐾𝑁) ∈ ℕ0)
2524adantr 472 . . . . . . 7 ((𝜑𝑡𝑇) → (𝐾𝑁) ∈ ℕ0)
26 1m1e0 11281 . . . . . . . 8 (1 − 1) = 0
27 stoweidlem45.11 . . . . . . . . . . . 12 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
2827r19.21bi 3070 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
2928simpld 477 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 0 ≤ (𝑃𝑡))
3028simprd 482 . . . . . . . . . 10 ((𝜑𝑡𝑇) → (𝑃𝑡) ≤ 1)
31 exple1 13114 . . . . . . . . . 10 ((((𝑃𝑡) ∈ ℝ ∧ 0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1) ∧ 𝑁 ∈ ℕ0) → ((𝑃𝑡)↑𝑁) ≤ 1)
3219, 29, 30, 20, 31syl31anc 1480 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝑃𝑡)↑𝑁) ≤ 1)
3321, 18, 18, 32lesub2dd 10836 . . . . . . . 8 ((𝜑𝑡𝑇) → (1 − 1) ≤ (1 − ((𝑃𝑡)↑𝑁)))
3426, 33syl5eqbrr 4840 . . . . . . 7 ((𝜑𝑡𝑇) → 0 ≤ (1 − ((𝑃𝑡)↑𝑁)))
3522, 25, 34expge0d 13220 . . . . . 6 ((𝜑𝑡𝑇) → 0 ≤ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
363, 8, 15, 23stoweidlem12 40732 . . . . . 6 ((𝜑𝑡𝑇) → (𝑄𝑡) = ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
3735, 36breqtrrd 4832 . . . . 5 ((𝜑𝑡𝑇) → 0 ≤ (𝑄𝑡))
38 0red 10233 . . . . . . . . 9 ((𝜑𝑡𝑇) → 0 ∈ ℝ)
3919, 20, 29expge0d 13220 . . . . . . . . 9 ((𝜑𝑡𝑇) → 0 ≤ ((𝑃𝑡)↑𝑁))
4038, 21, 18, 39lesub2dd 10836 . . . . . . . 8 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ≤ (1 − 0))
41 1m0e1 11323 . . . . . . . 8 (1 − 0) = 1
4240, 41syl6breq 4845 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ≤ 1)
43 exple1 13114 . . . . . . 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 1480 . . . . . 6 ((𝜑𝑡𝑇) → ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)) ≤ 1)
4536, 44eqbrtrd 4826 . . . . 5 ((𝜑𝑡𝑇) → (𝑄𝑡) ≤ 1)
4637, 45jca 555 . . . 4 ((𝜑𝑡𝑇) → (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1))
4746ex 449 . . 3 (𝜑 → (𝑡𝑇 → (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
482, 47ralrimi 3095 . 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 40744 . . . 4 ((𝜑𝑡𝑉) → (1 − 𝐸) < (𝑄𝑡))
5453ex 449 . . 3 (𝜑 → (𝑡𝑉 → (1 − 𝐸) < (𝑄𝑡)))
552, 54ralrimi 3095 . 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 40745 . . . 4 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑄𝑡) < 𝐸)
5958ex 449 . . 3 (𝜑 → (𝑡 ∈ (𝑇𝑈) → (𝑄𝑡) < 𝐸))
602, 59ralrimi 3095 . 2 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)(𝑄𝑡) < 𝐸)
61 nfmpt1 4899 . . . . . . 7 𝑡(𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
623, 61nfcxfr 2900 . . . . . 6 𝑡𝑄
6362nfeq2 2918 . . . . 5 𝑡 𝑦 = 𝑄
64 fveq1 6351 . . . . . . 7 (𝑦 = 𝑄 → (𝑦𝑡) = (𝑄𝑡))
6564breq2d 4816 . . . . . 6 (𝑦 = 𝑄 → (0 ≤ (𝑦𝑡) ↔ 0 ≤ (𝑄𝑡)))
6664breq1d 4814 . . . . . 6 (𝑦 = 𝑄 → ((𝑦𝑡) ≤ 1 ↔ (𝑄𝑡) ≤ 1))
6765, 66anbi12d 749 . . . . 5 (𝑦 = 𝑄 → ((0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
6863, 67ralbid 3121 . . . 4 (𝑦 = 𝑄 → (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
6964breq2d 4816 . . . . 5 (𝑦 = 𝑄 → ((1 − 𝐸) < (𝑦𝑡) ↔ (1 − 𝐸) < (𝑄𝑡)))
7063, 69ralbid 3121 . . . 4 (𝑦 = 𝑄 → (∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡) ↔ ∀𝑡𝑉 (1 − 𝐸) < (𝑄𝑡)))
7164breq1d 4814 . . . . 5 (𝑦 = 𝑄 → ((𝑦𝑡) < 𝐸 ↔ (𝑄𝑡) < 𝐸))
7263, 71ralbid 3121 . . . 4 (𝑦 = 𝑄 → (∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸 ↔ ∀𝑡 ∈ (𝑇𝑈)(𝑄𝑡) < 𝐸))
7368, 70, 723anbi123d 1548 . . 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 3449 . 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 1479 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 1072   = wceq 1632  wnf 1857  wcel 2139  wnfc 2889  wral 3050  wrex 3051  {crab 3054  cdif 3712   class class class wbr 4804  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6813  cr 10127  0cc0 10128  1c1 10129   + caddc 10131   · cmul 10133   < clt 10266  cle 10267  cmin 10458   / cdiv 10876  cn 11212  2c2 11262  0cn0 11484  +crp 12025  cexp 13054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-n0 11485  df-z 11570  df-uz 11880  df-rp 12026  df-seq 12996  df-exp 13055
This theorem is referenced by:  stoweidlem49  40769
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