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Theorem stoweidlem41 45997
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 11259 . . . . . . . 8 1 ∈ ℝ
3 stoweidlem41.3 . . . . . . . . 9 𝐹 = (𝑡𝑇 ↦ 1)
43fvmpt2 7027 . . . . . . . 8 ((𝑡𝑇 ∧ 1 ∈ ℝ) → (𝐹𝑡) = 1)
52, 4mpan2 691 . . . . . . 7 (𝑡𝑇 → (𝐹𝑡) = 1)
65adantl 481 . . . . . 6 ((𝜑𝑡𝑇) → (𝐹𝑡) = 1)
76oveq1d 7446 . . . . 5 ((𝜑𝑡𝑇) → ((𝐹𝑡) − (𝑦𝑡)) = (1 − (𝑦𝑡)))
81, 7mpteq2da 5246 . . . 4 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝑦𝑡))) = (𝑡𝑇 ↦ (1 − (𝑦𝑡))))
9 stoweidlem41.2 . . . 4 𝑋 = (𝑡𝑇 ↦ (1 − (𝑦𝑡)))
108, 9eqtr4di 2793 . . 3 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝑦𝑡))) = 𝑋)
11 stoweidlem41.10 . . . . . . 7 ((𝜑𝑤 ∈ ℝ) → (𝑡𝑇𝑤) ∈ 𝐴)
1211stoweidlem4 45960 . . . . . 6 ((𝜑 ∧ 1 ∈ ℝ) → (𝑡𝑇 ↦ 1) ∈ 𝐴)
132, 12mpan2 691 . . . . 5 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
143, 13eqeltrid 2843 . . . 4 (𝜑𝐹𝐴)
15 stoweidlem41.5 . . . 4 (𝜑𝑦𝐴)
16 nfmpt1 5256 . . . . . 6 𝑡(𝑡𝑇 ↦ 1)
173, 16nfcxfr 2901 . . . . 5 𝑡𝐹
18 nfcv 2903 . . . . 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 45989 . . . 4 ((𝜑𝐹𝐴𝑦𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝑦𝑡))) ∈ 𝐴)
2314, 15, 22mpd3an23 1462 . . 3 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝑦𝑡))) ∈ 𝐴)
2410, 23eqeltrrd 2840 . 2 (𝜑𝑋𝐴)
25 stoweidlem41.6 . . . . . . . 8 (𝜑𝑦:𝑇⟶ℝ)
2625ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑡𝑇) → (𝑦𝑡) ∈ ℝ)
27 1red 11260 . . . . . . 7 ((𝜑𝑡𝑇) → 1 ∈ ℝ)
28 0red 11262 . . . . . . 7 ((𝜑𝑡𝑇) → 0 ∈ ℝ)
29 stoweidlem41.12 . . . . . . . . . 10 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1))
3029r19.21bi 3249 . . . . . . . . 9 ((𝜑𝑡𝑇) → (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1))
3130simprd 495 . . . . . . . 8 ((𝜑𝑡𝑇) → (𝑦𝑡) ≤ 1)
32 1m0e1 12385 . . . . . . . 8 (1 − 0) = 1
3331, 32breqtrrdi 5190 . . . . . . 7 ((𝜑𝑡𝑇) → (𝑦𝑡) ≤ (1 − 0))
3426, 27, 28, 33lesubd 11865 . . . . . 6 ((𝜑𝑡𝑇) → 0 ≤ (1 − (𝑦𝑡)))
35 simpr 484 . . . . . . 7 ((𝜑𝑡𝑇) → 𝑡𝑇)
3627, 26resubcld 11689 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − (𝑦𝑡)) ∈ ℝ)
379fvmpt2 7027 . . . . . . 7 ((𝑡𝑇 ∧ (1 − (𝑦𝑡)) ∈ ℝ) → (𝑋𝑡) = (1 − (𝑦𝑡)))
3835, 36, 37syl2anc 584 . . . . . 6 ((𝜑𝑡𝑇) → (𝑋𝑡) = (1 − (𝑦𝑡)))
3934, 38breqtrrd 5176 . . . . 5 ((𝜑𝑡𝑇) → 0 ≤ (𝑋𝑡))
4030simpld 494 . . . . . . . 8 ((𝜑𝑡𝑇) → 0 ≤ (𝑦𝑡))
4128, 26, 27, 40lesub2dd 11878 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − (𝑦𝑡)) ≤ (1 − 0))
4241, 32breqtrdi 5189 . . . . . 6 ((𝜑𝑡𝑇) → (1 − (𝑦𝑡)) ≤ 1)
4338, 42eqbrtrd 5170 . . . . 5 ((𝜑𝑡𝑇) → (𝑋𝑡) ≤ 1)
4439, 43jca 511 . . . 4 ((𝜑𝑡𝑇) → (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1))
4544ex 412 . . 3 (𝜑 → (𝑡𝑇 → (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
461, 45ralrimi 3255 . 2 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1))
47 stoweidlem41.4 . . . . . . 7 𝑉𝑇
4847sseli 3991 . . . . . 6 (𝑡𝑉𝑡𝑇)
4948, 38sylan2 593 . . . . 5 ((𝜑𝑡𝑉) → (𝑋𝑡) = (1 − (𝑦𝑡)))
50 1red 11260 . . . . . 6 ((𝜑𝑡𝑉) → 1 ∈ ℝ)
51 stoweidlem41.11 . . . . . . . 8 (𝜑𝐸 ∈ ℝ+)
5251rpred 13075 . . . . . . 7 (𝜑𝐸 ∈ ℝ)
5352adantr 480 . . . . . 6 ((𝜑𝑡𝑉) → 𝐸 ∈ ℝ)
5448, 26sylan2 593 . . . . . 6 ((𝜑𝑡𝑉) → (𝑦𝑡) ∈ ℝ)
55 stoweidlem41.13 . . . . . . 7 (𝜑 → ∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡))
5655r19.21bi 3249 . . . . . 6 ((𝜑𝑡𝑉) → (1 − 𝐸) < (𝑦𝑡))
5750, 53, 54, 56ltsub23d 11866 . . . . 5 ((𝜑𝑡𝑉) → (1 − (𝑦𝑡)) < 𝐸)
5849, 57eqbrtrd 5170 . . . 4 ((𝜑𝑡𝑉) → (𝑋𝑡) < 𝐸)
5958ex 412 . . 3 (𝜑 → (𝑡𝑉 → (𝑋𝑡) < 𝐸))
601, 59ralrimi 3255 . 2 (𝜑 → ∀𝑡𝑉 (𝑋𝑡) < 𝐸)
61 eldifi 4141 . . . . . . 7 (𝑡 ∈ (𝑇𝑈) → 𝑡𝑇)
6261, 26sylan2 593 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑦𝑡) ∈ ℝ)
6352adantr 480 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → 𝐸 ∈ ℝ)
64 1red 11260 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → 1 ∈ ℝ)
65 stoweidlem41.14 . . . . . . 7 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸)
6665r19.21bi 3249 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑦𝑡) < 𝐸)
6762, 63, 64, 66ltsub2dd 11874 . . . . 5 ((𝜑𝑡 ∈ (𝑇𝑈)) → (1 − 𝐸) < (1 − (𝑦𝑡)))
6861, 38sylan2 593 . . . . 5 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑋𝑡) = (1 − (𝑦𝑡)))
6967, 68breqtrrd 5176 . . . 4 ((𝜑𝑡 ∈ (𝑇𝑈)) → (1 − 𝐸) < (𝑋𝑡))
7069ex 412 . . 3 (𝜑 → (𝑡 ∈ (𝑇𝑈) → (1 − 𝐸) < (𝑋𝑡)))
711, 70ralrimi 3255 . 2 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑋𝑡))
72 nfmpt1 5256 . . . . . . 7 𝑡(𝑡𝑇 ↦ (1 − (𝑦𝑡)))
739, 72nfcxfr 2901 . . . . . 6 𝑡𝑋
7473nfeq2 2921 . . . . 5 𝑡 𝑥 = 𝑋
75 fveq1 6906 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑡) = (𝑋𝑡))
7675breq2d 5160 . . . . . 6 (𝑥 = 𝑋 → (0 ≤ (𝑥𝑡) ↔ 0 ≤ (𝑋𝑡)))
7775breq1d 5158 . . . . . 6 (𝑥 = 𝑋 → ((𝑥𝑡) ≤ 1 ↔ (𝑋𝑡) ≤ 1))
7876, 77anbi12d 632 . . . . 5 (𝑥 = 𝑋 → ((0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
7974, 78ralbid 3271 . . . 4 (𝑥 = 𝑋 → (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
8075breq1d 5158 . . . . 5 (𝑥 = 𝑋 → ((𝑥𝑡) < 𝐸 ↔ (𝑋𝑡) < 𝐸))
8174, 80ralbid 3271 . . . 4 (𝑥 = 𝑋 → (∀𝑡𝑉 (𝑥𝑡) < 𝐸 ↔ ∀𝑡𝑉 (𝑋𝑡) < 𝐸))
8275breq2d 5160 . . . . 5 (𝑥 = 𝑋 → ((1 − 𝐸) < (𝑥𝑡) ↔ (1 − 𝐸) < (𝑋𝑡)))
8374, 82ralbid 3271 . . . 4 (𝑥 = 𝑋 → (∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑥𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑋𝑡)))
8479, 81, 833anbi123d 1435 . . 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 3622 . 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 1371 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 395  w3a 1086   = wceq 1537  wnf 1780  wcel 2106  wral 3059  wrex 3068  cdif 3960  wss 3963   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  +crp 13032
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-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
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-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-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  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-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-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-rp 13033
This theorem is referenced by:  stoweidlem52  46008
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