Theorem fldextfld2 34004

Description: A field extension is only defined if the subfield is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.)

Assertion

Ref	Expression
fldextfld2	⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)

Proof of Theorem fldextfld2

Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opabssxp 5753	. . 3 ⊢ {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} ⊆ (Field × Field)
2		df-br 5109	. . . . 5 ⊢ (𝐸/FldExt𝐹 ↔ ⟨𝐸, 𝐹⟩ ∈ /FldExt)
3	2	biimpi 219	. . . 4 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ /FldExt)
4		df-fldext 33997	. . . 4 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
5	3, 4	eleqtrdi 2871	. . 3 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))})
6	1, 5	sselid 3934	. 2 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ (Field × Field))
7		opelxp2 5704	. 2 ⊢ (⟨𝐸, 𝐹⟩ ∈ (Field × Field) → 𝐹 ∈ Field)
8	6, 7	syl 18	1 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ⟨cop 4594   class class class wbr 5108  {copab 5172   × cxp 5659  ‘cfv 6536  (class class class)co 7410  Basecbs 17268   ↾_s cress 17289  SubRingcsubrg 20653  Fieldcfield 20813  /_FldExtcfldext 33994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5667  df-fldext 33997

This theorem is referenced by:  fldextsubrg  34005  fldextress  34007  brfinext  34008  fldextsdrg  34010  fldextsralvec  34011  extdgcl  34012  extdggt0  34013  fldexttr  34014  extdgmul  34019  extdg1id  34022  extdg1b  34023  finextalg  34054