Theorem stoweidlem9 46531

Description: Lemma for stoweid 46585: 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 5183	. . . . 5 ⊢ (𝑇 = ∅ → (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ ∅ ↦ 1))
3		mpt0 6652	. . . . 5 ⊢ (𝑡 ∈ ∅ ↦ 1) = ∅
4	2, 3	eqtrdi 2807	. . . 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 2857	. 2 ⊢ (𝜑 → ∅ ∈ 𝐴)
8		rzal 4442	. . 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 6855	. . . . . 6 ⊢ (𝑔 = ∅ → (𝑔‘𝑡) = (∅‘𝑡))
11	10	fvoveq1d 7407	. . . . 5 ⊢ (𝑔 = ∅ → (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) = (abs‘((∅‘𝑡) − (𝐹‘𝑡))))
12	11	breq1d 5104	. . . 4 ⊢ (𝑔 = ∅ → ((abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸))
13	12	ralbidv 3179	. . 3 ⊢ (𝑔 = ∅ → (∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸 ↔ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸))
14	13	rspcev 3576	. 2 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((∅‘𝑡) − (𝐹‘𝑡))) < 𝐸) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸)
15	7, 9, 14	syl2anc 592	1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸)

Syntax hints:   → wi 4   = wceq 1554   ∈ wcel 2136  ∀wral 3070  ∃wrex 3080  ∅c0 4280   class class class wbr 5094   ↦ cmpt 5175  ‘cfv 6510  (class class class)co 7385  1c1 11064   < clt 11206   − cmin 11404  abscabs 15237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6512  df-fn 6513  df-fv 6518  df-ov 7388

This theorem is referenced by:  stoweid  46585