Theorem fldextfld2 33832

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 5710	. . 3 ⊢ {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} ⊆ (Field × Field)
2		df-br 5073	. . . . 5 ⊢ (𝐸/FldExt𝐹 ↔ ⟨𝐸, 𝐹⟩ ∈ /FldExt)
3	2	biimpi 217	. . . 4 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ /FldExt)
4		df-fldext 33825	. . . 4 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
5	3, 4	eleqtrdi 2849	. . 3 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))})
6	1, 5	sselid 3913	. 2 ⊢ (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ (Field × Field))
7		opelxp2 5661	. 2 ⊢ (⟨𝐸, 𝐹⟩ ∈ (Field × Field) → 𝐹 ∈ Field)
8	6, 7	syl 17	1 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ⟨cop 4561   class class class wbr 5072  {copab 5134   × cxp 5616  ‘cfv 6485  (class class class)co 7356  Basecbs 17170   ↾_s cress 17191  SubRingcsubrg 20541  Fieldcfield 20702  /_FldExtcfldext 33822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-fldext 33825

This theorem is referenced by:  fldextsubrg  33833  fldextress  33835  brfinext  33836  fldextsdrg  33838  fldextsralvec  33839  extdgcl  33840  extdggt0  33841  fldexttr  33842  extdgmul  33847  extdg1id  33850  extdg1b  33851  finextalg  33882