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Theorem stoweidlem9 43076
Description: Lemma for stoweid 43130: here the Stone Weierstrass theorem is proven for the trivial case, T is the empty set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem9.1 (𝜑𝑇 = ∅)
stoweidlem9.2 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
Assertion
Ref Expression
stoweidlem9 (𝜑 → ∃𝑔𝐴𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸)
Distinct variable groups:   𝐴,𝑔   𝑔,𝐸   𝑔,𝐹   𝑡,𝑔,𝑇
Allowed substitution hints:   𝜑(𝑡,𝑔)   𝐴(𝑡)   𝐸(𝑡)   𝐹(𝑡)

Proof of Theorem stoweidlem9
StepHypRef Expression
1 stoweidlem9.1 . . . 4 (𝜑𝑇 = ∅)
2 mpteq1 5115 . . . . 5 (𝑇 = ∅ → (𝑡𝑇 ↦ 1) = (𝑡 ∈ ∅ ↦ 1))
3 mpt0 6473 . . . . 5 (𝑡 ∈ ∅ ↦ 1) = ∅
42, 3eqtrdi 2789 . . . 4 (𝑇 = ∅ → (𝑡𝑇 ↦ 1) = ∅)
51, 4syl 17 . . 3 (𝜑 → (𝑡𝑇 ↦ 1) = ∅)
6 stoweidlem9.2 . . 3 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
75, 6eqeltrrd 2834 . 2 (𝜑 → ∅ ∈ 𝐴)
8 rzal 4392 . . 3 (𝑇 = ∅ → ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸)
91, 8syl 17 . 2 (𝜑 → ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸)
10 fveq1 6667 . . . . . 6 (𝑔 = ∅ → (𝑔𝑡) = (∅‘𝑡))
1110fvoveq1d 7186 . . . . 5 (𝑔 = ∅ → (abs‘((𝑔𝑡) − (𝐹𝑡))) = (abs‘((∅‘𝑡) − (𝐹𝑡))))
1211breq1d 5037 . . . 4 (𝑔 = ∅ → ((abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸 ↔ (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸))
1312ralbidv 3109 . . 3 (𝑔 = ∅ → (∀𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸 ↔ ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸))
1413rspcev 3524 . 2 ((∅ ∈ 𝐴 ∧ ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸) → ∃𝑔𝐴𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸)
157, 9, 14syl2anc 587 1 (𝜑 → ∃𝑔𝐴𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2113  wral 3053  wrex 3054  c0 4209   class class class wbr 5027  cmpt 5107  cfv 6333  (class class class)co 7164  1c1 10609   < clt 10746  cmin 10941  abscabs 14676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6291  df-fun 6335  df-fn 6336  df-fv 6341  df-ov 7167
This theorem is referenced by:  stoweid  43130
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