Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem41 Structured version   Visualization version   GIF version

Theorem stoweidlem41 40576
 Description: This lemma is used to prove that there exists x as in Lemma 1 of [BrosowskiDeutsh] p. 90: 0 <= x(t) <= 1 for all t in T, x(t) < epsilon for all t in V, x(t) > 1 - epsilon for all t in T \ U. Here we prove the very last step of the proof of Lemma 1: "The result follows from taking x = 1 - qn";. Here 𝐸 is used to represent ε in the paper, and 𝑦 to represent qn in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem41.1 𝑡𝜑
stoweidlem41.2 𝑋 = (𝑡𝑇 ↦ (1 − (𝑦𝑡)))
stoweidlem41.3 𝐹 = (𝑡𝑇 ↦ 1)
stoweidlem41.4 𝑉𝑇
stoweidlem41.5 (𝜑𝑦𝐴)
stoweidlem41.6 (𝜑𝑦:𝑇⟶ℝ)
stoweidlem41.7 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem41.8 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
stoweidlem41.9 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stoweidlem41.10 ((𝜑𝑤 ∈ ℝ) → (𝑡𝑇𝑤) ∈ 𝐴)
stoweidlem41.11 (𝜑𝐸 ∈ ℝ+)
stoweidlem41.12 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1))
stoweidlem41.13 (𝜑 → ∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡))
stoweidlem41.14 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸)
Assertion
Ref Expression
stoweidlem41 (𝜑 → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝐸 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑥𝑡)))
Distinct variable groups:   𝑓,𝑔,𝑡,𝑦   𝐴,𝑓,𝑔,𝑡   𝑓,𝐹,𝑔   𝑇,𝑓,𝑔,𝑡   𝜑,𝑓,𝑔   𝑤,𝑡,𝐴   𝑥,𝑡,𝐴   𝑤,𝑇   𝜑,𝑤   𝑥,𝐸   𝑥,𝑇   𝑥,𝑈   𝑥,𝑉   𝑥,𝑋
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑡)   𝐴(𝑦)   𝑇(𝑦)   𝑈(𝑦,𝑤,𝑡,𝑓,𝑔)   𝐸(𝑦,𝑤,𝑡,𝑓,𝑔)   𝐹(𝑥,𝑦,𝑤,𝑡)   𝑉(𝑦,𝑤,𝑡,𝑓,𝑔)   𝑋(𝑦,𝑤,𝑡,𝑓,𝑔)

Proof of Theorem stoweidlem41
StepHypRef Expression
1 stoweidlem41.1 . . . . 5 𝑡𝜑
2 1re 10077 . . . . . . . 8 1 ∈ ℝ
3 stoweidlem41.3 . . . . . . . . 9 𝐹 = (𝑡𝑇 ↦ 1)
43fvmpt2 6330 . . . . . . . 8 ((𝑡𝑇 ∧ 1 ∈ ℝ) → (𝐹𝑡) = 1)
52, 4mpan2 707 . . . . . . 7 (𝑡𝑇 → (𝐹𝑡) = 1)
65adantl 481 . . . . . 6 ((𝜑𝑡𝑇) → (𝐹𝑡) = 1)
76oveq1d 6705 . . . . 5 ((𝜑𝑡𝑇) → ((𝐹𝑡) − (𝑦𝑡)) = (1 − (𝑦𝑡)))
81, 7mpteq2da 4776 . . . 4 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝑦𝑡))) = (𝑡𝑇 ↦ (1 − (𝑦𝑡))))
9 stoweidlem41.2 . . . 4 𝑋 = (𝑡𝑇 ↦ (1 − (𝑦𝑡)))
108, 9syl6eqr 2703 . . 3 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝑦𝑡))) = 𝑋)
11 stoweidlem41.10 . . . . . . 7 ((𝜑𝑤 ∈ ℝ) → (𝑡𝑇𝑤) ∈ 𝐴)
1211stoweidlem4 40539 . . . . . 6 ((𝜑 ∧ 1 ∈ ℝ) → (𝑡𝑇 ↦ 1) ∈ 𝐴)
132, 12mpan2 707 . . . . 5 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
143, 13syl5eqel 2734 . . . 4 (𝜑𝐹𝐴)
15 stoweidlem41.5 . . . 4 (𝜑𝑦𝐴)
16 nfmpt1 4780 . . . . . 6 𝑡(𝑡𝑇 ↦ 1)
173, 16nfcxfr 2791 . . . . 5 𝑡𝐹
18 nfcv 2793 . . . . 5 𝑡𝑦
19 stoweidlem41.7 . . . . 5 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
20 stoweidlem41.8 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
21 stoweidlem41.9 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
2217, 18, 1, 19, 20, 21, 11stoweidlem33 40568 . . . 4 ((𝜑𝐹𝐴𝑦𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝑦𝑡))) ∈ 𝐴)
2314, 15, 22mpd3an23 1466 . . 3 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝑦𝑡))) ∈ 𝐴)
2410, 23eqeltrrd 2731 . 2 (𝜑𝑋𝐴)
25 stoweidlem41.6 . . . . . . . 8 (𝜑𝑦:𝑇⟶ℝ)
2625ffvelrnda 6399 . . . . . . 7 ((𝜑𝑡𝑇) → (𝑦𝑡) ∈ ℝ)
27 1red 10093 . . . . . . 7 ((𝜑𝑡𝑇) → 1 ∈ ℝ)
28 0red 10079 . . . . . . 7 ((𝜑𝑡𝑇) → 0 ∈ ℝ)
29 stoweidlem41.12 . . . . . . . . . 10 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1))
3029r19.21bi 2961 . . . . . . . . 9 ((𝜑𝑡𝑇) → (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1))
3130simprd 478 . . . . . . . 8 ((𝜑𝑡𝑇) → (𝑦𝑡) ≤ 1)
32 1m0e1 11169 . . . . . . . 8 (1 − 0) = 1
3331, 32syl6breqr 4727 . . . . . . 7 ((𝜑𝑡𝑇) → (𝑦𝑡) ≤ (1 − 0))
3426, 27, 28, 33lesubd 10669 . . . . . 6 ((𝜑𝑡𝑇) → 0 ≤ (1 − (𝑦𝑡)))
35 simpr 476 . . . . . . 7 ((𝜑𝑡𝑇) → 𝑡𝑇)
3627, 26resubcld 10496 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − (𝑦𝑡)) ∈ ℝ)
379fvmpt2 6330 . . . . . . 7 ((𝑡𝑇 ∧ (1 − (𝑦𝑡)) ∈ ℝ) → (𝑋𝑡) = (1 − (𝑦𝑡)))
3835, 36, 37syl2anc 694 . . . . . 6 ((𝜑𝑡𝑇) → (𝑋𝑡) = (1 − (𝑦𝑡)))
3934, 38breqtrrd 4713 . . . . 5 ((𝜑𝑡𝑇) → 0 ≤ (𝑋𝑡))
4030simpld 474 . . . . . . . 8 ((𝜑𝑡𝑇) → 0 ≤ (𝑦𝑡))
4128, 26, 27, 40lesub2dd 10682 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − (𝑦𝑡)) ≤ (1 − 0))
4241, 32syl6breq 4726 . . . . . 6 ((𝜑𝑡𝑇) → (1 − (𝑦𝑡)) ≤ 1)
4338, 42eqbrtrd 4707 . . . . 5 ((𝜑𝑡𝑇) → (𝑋𝑡) ≤ 1)
4439, 43jca 553 . . . 4 ((𝜑𝑡𝑇) → (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1))
4544ex 449 . . 3 (𝜑 → (𝑡𝑇 → (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
461, 45ralrimi 2986 . 2 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1))
47 stoweidlem41.4 . . . . . . 7 𝑉𝑇
4847sseli 3632 . . . . . 6 (𝑡𝑉𝑡𝑇)
4948, 38sylan2 490 . . . . 5 ((𝜑𝑡𝑉) → (𝑋𝑡) = (1 − (𝑦𝑡)))
50 1red 10093 . . . . . 6 ((𝜑𝑡𝑉) → 1 ∈ ℝ)
51 stoweidlem41.11 . . . . . . . 8 (𝜑𝐸 ∈ ℝ+)
5251rpred 11910 . . . . . . 7 (𝜑𝐸 ∈ ℝ)
5352adantr 480 . . . . . 6 ((𝜑𝑡𝑉) → 𝐸 ∈ ℝ)
5448, 26sylan2 490 . . . . . 6 ((𝜑𝑡𝑉) → (𝑦𝑡) ∈ ℝ)
55 stoweidlem41.13 . . . . . . 7 (𝜑 → ∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡))
5655r19.21bi 2961 . . . . . 6 ((𝜑𝑡𝑉) → (1 − 𝐸) < (𝑦𝑡))
5750, 53, 54, 56ltsub23d 10670 . . . . 5 ((𝜑𝑡𝑉) → (1 − (𝑦𝑡)) < 𝐸)
5849, 57eqbrtrd 4707 . . . 4 ((𝜑𝑡𝑉) → (𝑋𝑡) < 𝐸)
5958ex 449 . . 3 (𝜑 → (𝑡𝑉 → (𝑋𝑡) < 𝐸))
601, 59ralrimi 2986 . 2 (𝜑 → ∀𝑡𝑉 (𝑋𝑡) < 𝐸)
61 eldifi 3765 . . . . . . 7 (𝑡 ∈ (𝑇𝑈) → 𝑡𝑇)
6261, 26sylan2 490 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑦𝑡) ∈ ℝ)
6352adantr 480 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → 𝐸 ∈ ℝ)
64 1red 10093 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → 1 ∈ ℝ)
65 stoweidlem41.14 . . . . . . 7 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸)
6665r19.21bi 2961 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑦𝑡) < 𝐸)
6762, 63, 64, 66ltsub2dd 10678 . . . . 5 ((𝜑𝑡 ∈ (𝑇𝑈)) → (1 − 𝐸) < (1 − (𝑦𝑡)))
6861, 38sylan2 490 . . . . 5 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑋𝑡) = (1 − (𝑦𝑡)))
6967, 68breqtrrd 4713 . . . 4 ((𝜑𝑡 ∈ (𝑇𝑈)) → (1 − 𝐸) < (𝑋𝑡))
7069ex 449 . . 3 (𝜑 → (𝑡 ∈ (𝑇𝑈) → (1 − 𝐸) < (𝑋𝑡)))
711, 70ralrimi 2986 . 2 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑋𝑡))
72 nfmpt1 4780 . . . . . . 7 𝑡(𝑡𝑇 ↦ (1 − (𝑦𝑡)))
739, 72nfcxfr 2791 . . . . . 6 𝑡𝑋
7473nfeq2 2809 . . . . 5 𝑡 𝑥 = 𝑋
75 fveq1 6228 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑡) = (𝑋𝑡))
7675breq2d 4697 . . . . . 6 (𝑥 = 𝑋 → (0 ≤ (𝑥𝑡) ↔ 0 ≤ (𝑋𝑡)))
7775breq1d 4695 . . . . . 6 (𝑥 = 𝑋 → ((𝑥𝑡) ≤ 1 ↔ (𝑋𝑡) ≤ 1))
7876, 77anbi12d 747 . . . . 5 (𝑥 = 𝑋 → ((0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
7974, 78ralbid 3012 . . . 4 (𝑥 = 𝑋 → (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
8075breq1d 4695 . . . . 5 (𝑥 = 𝑋 → ((𝑥𝑡) < 𝐸 ↔ (𝑋𝑡) < 𝐸))
8174, 80ralbid 3012 . . . 4 (𝑥 = 𝑋 → (∀𝑡𝑉 (𝑥𝑡) < 𝐸 ↔ ∀𝑡𝑉 (𝑋𝑡) < 𝐸))
8275breq2d 4697 . . . . 5 (𝑥 = 𝑋 → ((1 − 𝐸) < (𝑥𝑡) ↔ (1 − 𝐸) < (𝑋𝑡)))
8374, 82ralbid 3012 . . . 4 (𝑥 = 𝑋 → (∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑥𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑋𝑡)))
8479, 81, 833anbi123d 1439 . . 3 (𝑥 = 𝑋 → ((∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝐸 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑥𝑡)) ↔ (∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑋𝑡) < 𝐸 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑋𝑡))))
8584rspcev 3340 . 2 ((𝑋𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑋𝑡) < 𝐸 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑋𝑡))) → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝐸 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑥𝑡)))
8624, 46, 60, 71, 85syl13anc 1368 1 (𝜑 → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝐸 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑥𝑡)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523  Ⅎwnf 1748   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   ∖ cdif 3604   ⊆ wss 3607   class class class wbr 4685   ↦ cmpt 4762  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112   ≤ cle 10113   − cmin 10304  ℝ+crp 11870 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-rp 11871 This theorem is referenced by:  stoweidlem52  40587
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