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Theorem stoweidlem45 46060
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 2737 . . 3 (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁))) = (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁)))
5 eqid 2737 . . 3 (𝑡𝑇 ↦ 1) = (𝑡𝑇 ↦ 1)
6 eqid 2737 . . 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 12587 . . . 4 (𝜑𝑁 ∈ ℕ0)
1614, 15nnexpcld 14284 . . 3 (𝜑 → (𝐾𝑁) ∈ ℕ)
171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16stoweidlem40 46055 . 2 (𝜑𝑄𝐴)
18 1red 11262 . . . . . . . 8 ((𝜑𝑡𝑇) → 1 ∈ ℝ)
198ffvelcdmda 7104 . . . . . . . . 9 ((𝜑𝑡𝑇) → (𝑃𝑡) ∈ ℝ)
2015adantr 480 . . . . . . . . 9 ((𝜑𝑡𝑇) → 𝑁 ∈ ℕ0)
2119, 20reexpcld 14203 . . . . . . . 8 ((𝜑𝑡𝑇) → ((𝑃𝑡)↑𝑁) ∈ ℝ)
2218, 21resubcld 11691 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ∈ ℝ)
2314nnnn0d 12587 . . . . . . . . 9 (𝜑𝐾 ∈ ℕ0)
2423, 15nn0expcld 14285 . . . . . . . 8 (𝜑 → (𝐾𝑁) ∈ ℕ0)
2524adantr 480 . . . . . . 7 ((𝜑𝑡𝑇) → (𝐾𝑁) ∈ ℕ0)
26 1m1e0 12338 . . . . . . . 8 (1 − 1) = 0
27 stoweidlem45.11 . . . . . . . . . . . 12 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
2827r19.21bi 3251 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))
2928simpld 494 . . . . . . . . . 10 ((𝜑𝑡𝑇) → 0 ≤ (𝑃𝑡))
3028simprd 495 . . . . . . . . . 10 ((𝜑𝑡𝑇) → (𝑃𝑡) ≤ 1)
31 exple1 14216 . . . . . . . . . 10 ((((𝑃𝑡) ∈ ℝ ∧ 0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1) ∧ 𝑁 ∈ ℕ0) → ((𝑃𝑡)↑𝑁) ≤ 1)
3219, 29, 30, 20, 31syl31anc 1375 . . . . . . . . 9 ((𝜑𝑡𝑇) → ((𝑃𝑡)↑𝑁) ≤ 1)
3321, 18, 18, 32lesub2dd 11880 . . . . . . . 8 ((𝜑𝑡𝑇) → (1 − 1) ≤ (1 − ((𝑃𝑡)↑𝑁)))
3426, 33eqbrtrrid 5179 . . . . . . 7 ((𝜑𝑡𝑇) → 0 ≤ (1 − ((𝑃𝑡)↑𝑁)))
3522, 25, 34expge0d 14204 . . . . . 6 ((𝜑𝑡𝑇) → 0 ≤ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
363, 8, 15, 23stoweidlem12 46027 . . . . . 6 ((𝜑𝑡𝑇) → (𝑄𝑡) = ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
3735, 36breqtrrd 5171 . . . . 5 ((𝜑𝑡𝑇) → 0 ≤ (𝑄𝑡))
38 0red 11264 . . . . . . . . 9 ((𝜑𝑡𝑇) → 0 ∈ ℝ)
3919, 20, 29expge0d 14204 . . . . . . . . 9 ((𝜑𝑡𝑇) → 0 ≤ ((𝑃𝑡)↑𝑁))
4038, 21, 18, 39lesub2dd 11880 . . . . . . . 8 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ≤ (1 − 0))
41 1m0e1 12387 . . . . . . . 8 (1 − 0) = 1
4240, 41breqtrdi 5184 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − ((𝑃𝑡)↑𝑁)) ≤ 1)
43 exple1 14216 . . . . . . 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 1375 . . . . . 6 ((𝜑𝑡𝑇) → ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)) ≤ 1)
4536, 44eqbrtrd 5165 . . . . 5 ((𝜑𝑡𝑇) → (𝑄𝑡) ≤ 1)
4637, 45jca 511 . . . 4 ((𝜑𝑡𝑇) → (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1))
4746ex 412 . . 3 (𝜑 → (𝑡𝑇 → (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
482, 47ralrimi 3257 . 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 46039 . . . 4 ((𝜑𝑡𝑉) → (1 − 𝐸) < (𝑄𝑡))
5453ex 412 . . 3 (𝜑 → (𝑡𝑉 → (1 − 𝐸) < (𝑄𝑡)))
552, 54ralrimi 3257 . 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 46040 . . . 4 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑄𝑡) < 𝐸)
5958ex 412 . . 3 (𝜑 → (𝑡 ∈ (𝑇𝑈) → (𝑄𝑡) < 𝐸))
602, 59ralrimi 3257 . 2 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)(𝑄𝑡) < 𝐸)
61 nfmpt1 5250 . . . . . . 7 𝑡(𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
623, 61nfcxfr 2903 . . . . . 6 𝑡𝑄
6362nfeq2 2923 . . . . 5 𝑡 𝑦 = 𝑄
64 fveq1 6905 . . . . . . 7 (𝑦 = 𝑄 → (𝑦𝑡) = (𝑄𝑡))
6564breq2d 5155 . . . . . 6 (𝑦 = 𝑄 → (0 ≤ (𝑦𝑡) ↔ 0 ≤ (𝑄𝑡)))
6664breq1d 5153 . . . . . 6 (𝑦 = 𝑄 → ((𝑦𝑡) ≤ 1 ↔ (𝑄𝑡) ≤ 1))
6765, 66anbi12d 632 . . . . 5 (𝑦 = 𝑄 → ((0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
6863, 67ralbid 3273 . . . 4 (𝑦 = 𝑄 → (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑄𝑡) ∧ (𝑄𝑡) ≤ 1)))
6964breq2d 5155 . . . . 5 (𝑦 = 𝑄 → ((1 − 𝐸) < (𝑦𝑡) ↔ (1 − 𝐸) < (𝑄𝑡)))
7063, 69ralbid 3273 . . . 4 (𝑦 = 𝑄 → (∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡) ↔ ∀𝑡𝑉 (1 − 𝐸) < (𝑄𝑡)))
7164breq1d 5153 . . . . 5 (𝑦 = 𝑄 → ((𝑦𝑡) < 𝐸 ↔ (𝑄𝑡) < 𝐸))
7263, 71ralbid 3273 . . . 4 (𝑦 = 𝑄 → (∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸 ↔ ∀𝑡 ∈ (𝑇𝑈)(𝑄𝑡) < 𝐸))
7368, 70, 723anbi123d 1438 . . 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 1374 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 1087   = wceq 1540  wnf 1783  wcel 2108  wnfc 2890  wral 3061  wrex 3070  {crab 3436  cdif 3948   class class class wbr 5143  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295  cle 11296  cmin 11492   / cdiv 11920  cn 12266  2c2 12321  0cn0 12526  +crp 13034  cexp 14102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-seq 14043  df-exp 14103
This theorem is referenced by:  stoweidlem49  46064
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