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	⊢ (𝜑 → 𝑇 = ∅)
stoweidlem9.2	⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)

Assertion

Ref	Expression
stoweidlem9	⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 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

Step	Hyp	Ref	Expression
1		stoweidlem9.1	. . . 4 ⊢ (𝜑 → 𝑇 = ∅)
2		mpteq1 5167	. . . . 5 ⊢ (𝑇 = ∅ → (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ ∅ ↦ 1))
3		mpt0 6575	. . . . 5 ⊢ (𝑡 ∈ ∅ ↦ 1) = ∅
4	2, 3	eqtrdi 2794	. . . 4 ⊢ (𝑇 = ∅ → (𝑡 ∈ 𝑇 ↦ 1) = ∅)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) = ∅)
6		stoweidlem9.2	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
7	5, 6	eqeltrrd 2840	. 2 ⊢ (𝜑 → ∅ ∈ 𝐴)
8		rzal 4439	. . 3 ⊢ (𝑇 = ∅ → ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸)
9	1, 8	syl 17	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸)
10		fveq1 6773	. . . . . 6 ⊢ (𝑔 = ∅ → (𝑔‘𝑡) = (∅‘𝑡))
11	10	fvoveq1d 7297	. . . . 5 ⊢ (𝑔 = ∅ → (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) = (abs‘((∅‘𝑡) − (𝐹‘𝑡))))
12	11	breq1d 5084	. . . 4 ⊢ (𝑔 = ∅ → ((abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸))
13	12	ralbidv 3112	. . 3 ⊢ (𝑔 = ∅ → (∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸))
14	13	rspcev 3561	. 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸)
15	7, 9, 14	syl2anc 584	1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸)

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