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Theorem fldextfld2 33543
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/FldExt𝐹𝐹 ∈ Field)

Proof of Theorem fldextfld2
Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opabssxp 5765 . . 3 {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} ⊆ (Field × Field)
2 df-br 5145 . . . . 5 (𝐸/FldExt𝐹 ↔ ⟨𝐸, 𝐹⟩ ∈ /FldExt)
32biimpi 215 . . . 4 (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ /FldExt)
4 df-fldext 33535 . . . 4 /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
53, 4eleqtrdi 2836 . . 3 (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))})
61, 5sselid 3977 . 2 (𝐸/FldExt𝐹 → ⟨𝐸, 𝐹⟩ ∈ (Field × Field))
7 opelxp2 5716 . 2 (⟨𝐸, 𝐹⟩ ∈ (Field × Field) → 𝐹 ∈ Field)
86, 7syl 17 1 (𝐸/FldExt𝐹𝐹 ∈ Field)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  cop 4630   class class class wbr 5144  {copab 5206   × cxp 5671  cfv 6544  (class class class)co 7414  Basecbs 17206  s cress 17235  SubRingcsubrg 20545  Fieldcfield 20702  /FldExtcfldext 33531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-xp 5679  df-fldext 33535
This theorem is referenced by:  fldextsubrg  33544  fldextress  33545  brfinext  33546  fldextsralvec  33548  extdgcl  33549  extdggt0  33550  fldexttr  33551  extdgmul  33554  extdg1id  33556  extdg1b  33557
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