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Theorem stoweidlem9 45294
Description: Lemma for stoweid 45348: 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 5234 . . . . 5 (𝑇 = ∅ → (𝑡𝑇 ↦ 1) = (𝑡 ∈ ∅ ↦ 1))
3 mpt0 6686 . . . . 5 (𝑡 ∈ ∅ ↦ 1) = ∅
42, 3eqtrdi 2782 . . . 4 (𝑇 = ∅ → (𝑡𝑇 ↦ 1) = ∅)
51, 4syl 17 . . 3 (𝜑 → (𝑡𝑇 ↦ 1) = ∅)
6 stoweidlem9.2 . . 3 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
75, 6eqeltrrd 2828 . 2 (𝜑 → ∅ ∈ 𝐴)
8 rzal 4503 . . 3 (𝑇 = ∅ → ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸)
91, 8syl 17 . 2 (𝜑 → ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸)
10 fveq1 6884 . . . . . 6 (𝑔 = ∅ → (𝑔𝑡) = (∅‘𝑡))
1110fvoveq1d 7427 . . . . 5 (𝑔 = ∅ → (abs‘((𝑔𝑡) − (𝐹𝑡))) = (abs‘((∅‘𝑡) − (𝐹𝑡))))
1211breq1d 5151 . . . 4 (𝑔 = ∅ → ((abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸 ↔ (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸))
1312ralbidv 3171 . . 3 (𝑔 = ∅ → (∀𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸 ↔ ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸))
1413rspcev 3606 . 2 ((∅ ∈ 𝐴 ∧ ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸) → ∃𝑔𝐴𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸)
157, 9, 14syl2anc 583 1 (𝜑 → ∃𝑔𝐴𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wral 3055  wrex 3064  c0 4317   class class class wbr 5141  cmpt 5224  cfv 6537  (class class class)co 7405  1c1 11113   < clt 11252  cmin 11448  abscabs 15187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fn 6540  df-fv 6545  df-ov 7408
This theorem is referenced by:  stoweid  45348
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