Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem9 Structured version   Visualization version   GIF version

Theorem stoweidlem9 43550
Description: Lemma for stoweid 43604: 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 5167 . . . . 5 (𝑇 = ∅ → (𝑡𝑇 ↦ 1) = (𝑡 ∈ ∅ ↦ 1))
3 mpt0 6575 . . . . 5 (𝑡 ∈ ∅ ↦ 1) = ∅
42, 3eqtrdi 2794 . . . 4 (𝑇 = ∅ → (𝑡𝑇 ↦ 1) = ∅)
51, 4syl 17 . . 3 (𝜑 → (𝑡𝑇 ↦ 1) = ∅)
6 stoweidlem9.2 . . 3 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
75, 6eqeltrrd 2840 . 2 (𝜑 → ∅ ∈ 𝐴)
8 rzal 4439 . . 3 (𝑇 = ∅ → ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸)
91, 8syl 17 . 2 (𝜑 → ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸)
10 fveq1 6773 . . . . . 6 (𝑔 = ∅ → (𝑔𝑡) = (∅‘𝑡))
1110fvoveq1d 7297 . . . . 5 (𝑔 = ∅ → (abs‘((𝑔𝑡) − (𝐹𝑡))) = (abs‘((∅‘𝑡) − (𝐹𝑡))))
1211breq1d 5084 . . . 4 (𝑔 = ∅ → ((abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸 ↔ (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸))
1312ralbidv 3112 . . 3 (𝑔 = ∅ → (∀𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸 ↔ ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸))
1413rspcev 3561 . 2 ((∅ ∈ 𝐴 ∧ ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸) → ∃𝑔𝐴𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸)
157, 9, 14syl2anc 584 1 (𝜑 → ∃𝑔𝐴𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  wral 3064  wrex 3065  c0 4256   class class class wbr 5074  cmpt 5157  cfv 6433  (class class class)co 7275  1c1 10872   < clt 11009  cmin 11205  abscabs 14945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-ov 7278
This theorem is referenced by:  stoweid  43604
  Copyright terms: Public domain W3C validator