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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  wnf 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  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112  cle 10113  cmin 10304  +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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