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Theorem stoweidlem9 40743
Description: Lemma for stoweid 40797: 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 4871 . . . . 5 (𝑇 = ∅ → (𝑡𝑇 ↦ 1) = (𝑡 ∈ ∅ ↦ 1))
3 mpt0 6161 . . . . 5 (𝑡 ∈ ∅ ↦ 1) = ∅
42, 3syl6eq 2821 . . . 4 (𝑇 = ∅ → (𝑡𝑇 ↦ 1) = ∅)
51, 4syl 17 . . 3 (𝜑 → (𝑡𝑇 ↦ 1) = ∅)
6 stoweidlem9.2 . . 3 (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)
75, 6eqeltrrd 2851 . 2 (𝜑 → ∅ ∈ 𝐴)
8 rzal 4214 . . 3 (𝑇 = ∅ → ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸)
91, 8syl 17 . 2 (𝜑 → ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸)
10 fveq1 6331 . . . . . . 7 (𝑔 = ∅ → (𝑔𝑡) = (∅‘𝑡))
1110oveq1d 6808 . . . . . 6 (𝑔 = ∅ → ((𝑔𝑡) − (𝐹𝑡)) = ((∅‘𝑡) − (𝐹𝑡)))
1211fveq2d 6336 . . . . 5 (𝑔 = ∅ → (abs‘((𝑔𝑡) − (𝐹𝑡))) = (abs‘((∅‘𝑡) − (𝐹𝑡))))
1312breq1d 4796 . . . 4 (𝑔 = ∅ → ((abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸 ↔ (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸))
1413ralbidv 3135 . . 3 (𝑔 = ∅ → (∀𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸 ↔ ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸))
1514rspcev 3460 . 2 ((∅ ∈ 𝐴 ∧ ∀𝑡𝑇 (abs‘((∅‘𝑡) − (𝐹𝑡))) < 𝐸) → ∃𝑔𝐴𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸)
167, 9, 15syl2anc 573 1 (𝜑 → ∃𝑔𝐴𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wral 3061  wrex 3062  c0 4063   class class class wbr 4786  cmpt 4863  cfv 6031  (class class class)co 6793  1c1 10139   < clt 10276  cmin 10468  abscabs 14182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-ov 6796
This theorem is referenced by:  stoweid  40797
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