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Theorem stoweidlem9 45997
Description: Lemma for stoweid 46051: 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 5233 . . . . 5 (𝑇 = ∅ → (𝑡𝑇 ↦ 1) = (𝑡 ∈ ∅ ↦ 1))
3 mpt0 6708 . . . . 5 (𝑡 ∈ ∅ ↦ 1) = ∅
42, 3eqtrdi 2792 . . . 4 (𝑇 = ∅ → (𝑡𝑇 ↦ 1) = ∅)
51, 4syl 17 . . 3 (𝜑 → (𝑡𝑇 ↦ 1) = ∅)
6 stoweidlem9.2 . . 3 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
75, 6eqeltrrd 2841 . 2 (𝜑 → ∅ ∈ 𝐴)
8 rzal 4508 . . 3 (𝑇 = ∅ → ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸)
91, 8syl 17 . 2 (𝜑 → ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸)
10 fveq1 6903 . . . . . 6 (𝑔 = ∅ → (𝑔𝑡) = (∅‘𝑡))
1110fvoveq1d 7451 . . . . 5 (𝑔 = ∅ → (abs‘((𝑔𝑡) − (𝐹𝑡))) = (abs‘((∅‘𝑡) − (𝐹𝑡))))
1211breq1d 5151 . . . 4 (𝑔 = ∅ → ((abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸 ↔ (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸))
1312ralbidv 3177 . . 3 (𝑔 = ∅ → (∀𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸 ↔ ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸))
1413rspcev 3621 . 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 1540  wcel 2108  wral 3060  wrex 3069  c0 4332   class class class wbr 5141  cmpt 5223  cfv 6559  (class class class)co 7429  1c1 11152   < clt 11291  cmin 11488  abscabs 15269
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 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-iota 6512  df-fun 6561  df-fn 6562  df-fv 6567  df-ov 7432
This theorem is referenced by:  stoweid  46051
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