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

Theorem stoweidlem45 46001
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 2735 . . 3 (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁))) = (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁)))
5 eqid 2735 . . 3 (𝑡𝑇 ↦ 1) = (𝑡𝑇 ↦ 1)
6 eqid 2735 . . 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 12585 . . . 4 (𝜑𝑁 ∈ ℕ0)
1614, 15nnexpcld 14281 . . 3 (𝜑 → (𝐾𝑁) ∈ ℕ)
171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16stoweidlem40 45996 . 2 (𝜑𝑄𝐴)
18 1red 11260 . . . . . . . 8 ((𝜑𝑡𝑇) → 1 ∈ ℝ)
198ffvelcdmda 7104 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝑃𝑡) ∈ ℝ)
2015adantr 480 . . . . . . . . 9 ((𝜑𝑡𝑇) → 𝑁 ∈ ℕ0)
2119, 20reexpcld 14200 . . . . . . . 8 ((𝜑𝑡𝑇) → ((𝑃𝑡)↑𝑁) ∈ ℝ)
2218, 21resubcld 11689 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ∈ ℝ)
2314nnnn0d 12585 . . . . . . . . 9 (𝜑𝐾 ∈ ℕ0)
2423, 15nn0expcld 14282 . . . . . . . 8 (𝜑 → (𝐾𝑁) ∈ ℕ0)
2524adantr 480 . . . . . . 7 ((𝜑𝑡𝑇) → (𝐾𝑁) ∈ ℕ0)
26 1m1e0 12336 . . . . . . . 8 (1 − 1) = 0
27 stoweidlem45.11 . . . . . . . . . . . 12 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
2827r19.21bi 3249 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
2928simpld 494 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 0 ≤ (𝑃𝑡))
3028simprd 495 . . . . . . . . . 10 ((𝜑𝑡𝑇) → (𝑃𝑡) ≤ 1)
31 exple1 14213 . . . . . . . . . 10 ((((𝑃𝑡) ∈ ℝ ∧ 0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1) ∧ 𝑁 ∈ ℕ0) → ((𝑃𝑡)↑𝑁) ≤ 1)
3219, 29, 30, 20, 31syl31anc 1372 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝑃𝑡)↑𝑁) ≤ 1)
3321, 18, 18, 32lesub2dd 11878 . . . . . . . 8 ((𝜑𝑡𝑇) → (1 − 1) ≤ (1 − ((𝑃𝑡)↑𝑁)))
3426, 33eqbrtrrid 5184 . . . . . . 7 ((𝜑𝑡𝑇) → 0 ≤ (1 − ((𝑃𝑡)↑𝑁)))
3522, 25, 34expge0d 14201 . . . . . 6 ((𝜑𝑡𝑇) → 0 ≤ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
363, 8, 15, 23stoweidlem12 45968 . . . . . 6 ((𝜑𝑡𝑇) → (𝑄𝑡) = ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
3735, 36breqtrrd 5176 . . . . 5 ((𝜑𝑡𝑇) → 0 ≤ (𝑄𝑡))
38 0red 11262 . . . . . . . . 9 ((𝜑𝑡𝑇) → 0 ∈ ℝ)
3919, 20, 29expge0d 14201 . . . . . . . . 9 ((𝜑𝑡𝑇) → 0 ≤ ((𝑃𝑡)↑𝑁))
4038, 21, 18, 39lesub2dd 11878 . . . . . . . 8 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ≤ (1 − 0))
41 1m0e1 12385 . . . . . . . 8 (1 − 0) = 1
4240, 41breqtrdi 5189 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ≤ 1)
43 exple1 14213 . . . . . . 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 1372 . . . . . 6 ((𝜑𝑡𝑇) → ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)) ≤ 1)
4536, 44eqbrtrd 5170 . . . . 5 ((𝜑𝑡𝑇) → (𝑄𝑡) ≤ 1)
4637, 45jca 511 . . . 4 ((𝜑𝑡𝑇) → (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1))
4746ex 412 . . 3 (𝜑 → (𝑡𝑇 → (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
482, 47ralrimi 3255 . 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 45980 . . . 4 ((𝜑𝑡𝑉) → (1 − 𝐸) < (𝑄𝑡))
5453ex 412 . . 3 (𝜑 → (𝑡𝑉 → (1 − 𝐸) < (𝑄𝑡)))
552, 54ralrimi 3255 . 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 45981 . . . 4 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑄𝑡) < 𝐸)
5958ex 412 . . 3 (𝜑 → (𝑡 ∈ (𝑇𝑈) → (𝑄𝑡) < 𝐸))
602, 59ralrimi 3255 . 2 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)(𝑄𝑡) < 𝐸)
61 nfmpt1 5256 . . . . . . 7 𝑡(𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
623, 61nfcxfr 2901 . . . . . 6 𝑡𝑄
6362nfeq2 2921 . . . . 5 𝑡 𝑦 = 𝑄
64 fveq1 6906 . . . . . . 7 (𝑦 = 𝑄 → (𝑦𝑡) = (𝑄𝑡))
6564breq2d 5160 . . . . . 6 (𝑦 = 𝑄 → (0 ≤ (𝑦𝑡) ↔ 0 ≤ (𝑄𝑡)))
6664breq1d 5158 . . . . . 6 (𝑦 = 𝑄 → ((𝑦𝑡) ≤ 1 ↔ (𝑄𝑡) ≤ 1))
6765, 66anbi12d 632 . . . . 5 (𝑦 = 𝑄 → ((0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
6863, 67ralbid 3271 . . . 4 (𝑦 = 𝑄 → (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
6964breq2d 5160 . . . . 5 (𝑦 = 𝑄 → ((1 − 𝐸) < (𝑦𝑡) ↔ (1 − 𝐸) < (𝑄𝑡)))
7063, 69ralbid 3271 . . . 4 (𝑦 = 𝑄 → (∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡) ↔ ∀𝑡𝑉 (1 − 𝐸) < (𝑄𝑡)))
7164breq1d 5158 . . . . 5 (𝑦 = 𝑄 → ((𝑦𝑡) < 𝐸 ↔ (𝑄𝑡) < 𝐸))
7263, 71ralbid 3271 . . . 4 (𝑦 = 𝑄 → (∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸 ↔ ∀𝑡 ∈ (𝑇𝑈)(𝑄𝑡) < 𝐸))
7368, 70, 723anbi123d 1435 . . 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 3622 . 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 1371 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 395  w3a 1086   = wceq 1537  wnf 1780  wcel 2106  wnfc 2888  wral 3059  wrex 3068  {crab 3433  cdif 3960   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  cn 12264  2c2 12319  0cn0 12524  +crp 13032  cexp 14099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-seq 14040  df-exp 14100
This theorem is referenced by:  stoweidlem49  46005
  Copyright terms: Public domain W3C validator