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Theorem stoweidlem41 46056
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 11261 . . . . . . . 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 5240 . . . 4 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝑦𝑡))) = (𝑡𝑇 ↦ (1 − (𝑦𝑡))))
9 stoweidlem41.2 . . . 4 𝑋 = (𝑡𝑇 ↦ (1 − (𝑦𝑡)))
108, 9eqtr4di 2795 . . 3 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝑦𝑡))) = 𝑋)
11 stoweidlem41.10 . . . . . . 7 ((𝜑𝑤 ∈ ℝ) → (𝑡𝑇𝑤) ∈ 𝐴)
1211stoweidlem4 46019 . . . . . 6 ((𝜑 ∧ 1 ∈ ℝ) → (𝑡𝑇 ↦ 1) ∈ 𝐴)
132, 12mpan2 691 . . . . 5 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
143, 13eqeltrid 2845 . . . 4 (𝜑𝐹𝐴)
15 stoweidlem41.5 . . . 4 (𝜑𝑦𝐴)
16 nfmpt1 5250 . . . . . 6 𝑡(𝑡𝑇 ↦ 1)
173, 16nfcxfr 2903 . . . . 5 𝑡𝐹
18 nfcv 2905 . . . . 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 46048 . . . 4 ((𝜑𝐹𝐴𝑦𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝑦𝑡))) ∈ 𝐴)
2314, 15, 22mpd3an23 1465 . . 3 (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝑦𝑡))) ∈ 𝐴)
2410, 23eqeltrrd 2842 . 2 (𝜑𝑋𝐴)
25 stoweidlem41.6 . . . . . . . 8 (𝜑𝑦:𝑇⟶ℝ)
2625ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑡𝑇) → (𝑦𝑡) ∈ ℝ)
27 1red 11262 . . . . . . 7 ((𝜑𝑡𝑇) → 1 ∈ ℝ)
28 0red 11264 . . . . . . 7 ((𝜑𝑡𝑇) → 0 ∈ ℝ)
29 stoweidlem41.12 . . . . . . . . . 10 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1))
3029r19.21bi 3251 . . . . . . . . 9 ((𝜑𝑡𝑇) → (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1))
3130simprd 495 . . . . . . . 8 ((𝜑𝑡𝑇) → (𝑦𝑡) ≤ 1)
32 1m0e1 12387 . . . . . . . 8 (1 − 0) = 1
3331, 32breqtrrdi 5185 . . . . . . 7 ((𝜑𝑡𝑇) → (𝑦𝑡) ≤ (1 − 0))
3426, 27, 28, 33lesubd 11867 . . . . . 6 ((𝜑𝑡𝑇) → 0 ≤ (1 − (𝑦𝑡)))
35 simpr 484 . . . . . . 7 ((𝜑𝑡𝑇) → 𝑡𝑇)
3627, 26resubcld 11691 . . . . . . 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 5171 . . . . 5 ((𝜑𝑡𝑇) → 0 ≤ (𝑋𝑡))
4030simpld 494 . . . . . . . 8 ((𝜑𝑡𝑇) → 0 ≤ (𝑦𝑡))
4128, 26, 27, 40lesub2dd 11880 . . . . . . 7 ((𝜑𝑡𝑇) → (1 − (𝑦𝑡)) ≤ (1 − 0))
4241, 32breqtrdi 5184 . . . . . 6 ((𝜑𝑡𝑇) → (1 − (𝑦𝑡)) ≤ 1)
4338, 42eqbrtrd 5165 . . . . 5 ((𝜑𝑡𝑇) → (𝑋𝑡) ≤ 1)
4439, 43jca 511 . . . 4 ((𝜑𝑡𝑇) → (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1))
4544ex 412 . . 3 (𝜑 → (𝑡𝑇 → (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
461, 45ralrimi 3257 . 2 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1))
47 stoweidlem41.4 . . . . . . 7 𝑉𝑇
4847sseli 3979 . . . . . 6 (𝑡𝑉𝑡𝑇)
4948, 38sylan2 593 . . . . 5 ((𝜑𝑡𝑉) → (𝑋𝑡) = (1 − (𝑦𝑡)))
50 1red 11262 . . . . . 6 ((𝜑𝑡𝑉) → 1 ∈ ℝ)
51 stoweidlem41.11 . . . . . . . 8 (𝜑𝐸 ∈ ℝ+)
5251rpred 13077 . . . . . . 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 3251 . . . . . 6 ((𝜑𝑡𝑉) → (1 − 𝐸) < (𝑦𝑡))
5750, 53, 54, 56ltsub23d 11868 . . . . 5 ((𝜑𝑡𝑉) → (1 − (𝑦𝑡)) < 𝐸)
5849, 57eqbrtrd 5165 . . . 4 ((𝜑𝑡𝑉) → (𝑋𝑡) < 𝐸)
5958ex 412 . . 3 (𝜑 → (𝑡𝑉 → (𝑋𝑡) < 𝐸))
601, 59ralrimi 3257 . 2 (𝜑 → ∀𝑡𝑉 (𝑋𝑡) < 𝐸)
61 eldifi 4131 . . . . . . 7 (𝑡 ∈ (𝑇𝑈) → 𝑡𝑇)
6261, 26sylan2 593 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑦𝑡) ∈ ℝ)
6352adantr 480 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → 𝐸 ∈ ℝ)
64 1red 11262 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → 1 ∈ ℝ)
65 stoweidlem41.14 . . . . . . 7 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸)
6665r19.21bi 3251 . . . . . 6 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑦𝑡) < 𝐸)
6762, 63, 64, 66ltsub2dd 11876 . . . . 5 ((𝜑𝑡 ∈ (𝑇𝑈)) → (1 − 𝐸) < (1 − (𝑦𝑡)))
6861, 38sylan2 593 . . . . 5 ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑋𝑡) = (1 − (𝑦𝑡)))
6967, 68breqtrrd 5171 . . . 4 ((𝜑𝑡 ∈ (𝑇𝑈)) → (1 − 𝐸) < (𝑋𝑡))
7069ex 412 . . 3 (𝜑 → (𝑡 ∈ (𝑇𝑈) → (1 − 𝐸) < (𝑋𝑡)))
711, 70ralrimi 3257 . 2 (𝜑 → ∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑋𝑡))
72 nfmpt1 5250 . . . . . . 7 𝑡(𝑡𝑇 ↦ (1 − (𝑦𝑡)))
739, 72nfcxfr 2903 . . . . . 6 𝑡𝑋
7473nfeq2 2923 . . . . 5 𝑡 𝑥 = 𝑋
75 fveq1 6905 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑡) = (𝑋𝑡))
7675breq2d 5155 . . . . . 6 (𝑥 = 𝑋 → (0 ≤ (𝑥𝑡) ↔ 0 ≤ (𝑋𝑡)))
7775breq1d 5153 . . . . . 6 (𝑥 = 𝑋 → ((𝑥𝑡) ≤ 1 ↔ (𝑋𝑡) ≤ 1))
7876, 77anbi12d 632 . . . . 5 (𝑥 = 𝑋 → ((0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
7974, 78ralbid 3273 . . . 4 (𝑥 = 𝑋 → (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
8075breq1d 5153 . . . . 5 (𝑥 = 𝑋 → ((𝑥𝑡) < 𝐸 ↔ (𝑋𝑡) < 𝐸))
8174, 80ralbid 3273 . . . 4 (𝑥 = 𝑋 → (∀𝑡𝑉 (𝑥𝑡) < 𝐸 ↔ ∀𝑡𝑉 (𝑋𝑡) < 𝐸))
8275breq2d 5155 . . . . 5 (𝑥 = 𝑋 → ((1 − 𝐸) < (𝑥𝑡) ↔ (1 − 𝐸) < (𝑋𝑡)))
8374, 82ralbid 3273 . . . 4 (𝑥 = 𝑋 → (∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑥𝑡) ↔ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑋𝑡)))
8479, 81, 833anbi123d 1438 . . 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 1374 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 1087   = wceq 1540  wnf 1783  wcel 2108  wral 3061  wrex 3070  cdif 3948  wss 3951   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  +crp 13034
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-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
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-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-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  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-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-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-rp 13035
This theorem is referenced by:  stoweidlem52  46067
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