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Theorem fldextfld2 33247
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 β†’ 𝐹 ∈ Field)

Proof of Theorem fldextfld2
Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opabssxp 5761 . . 3 {βŸ¨π‘’, π‘“βŸ© ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 β†Ύs (Baseβ€˜π‘“)) ∧ (Baseβ€˜π‘“) ∈ (SubRingβ€˜π‘’)))} βŠ† (Field Γ— Field)
2 df-br 5142 . . . . 5 (𝐸/FldExt𝐹 ↔ ⟨𝐸, 𝐹⟩ ∈ /FldExt)
32biimpi 215 . . . 4 (𝐸/FldExt𝐹 β†’ ⟨𝐸, 𝐹⟩ ∈ /FldExt)
4 df-fldext 33239 . . . 4 /FldExt = {βŸ¨π‘’, π‘“βŸ© ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 β†Ύs (Baseβ€˜π‘“)) ∧ (Baseβ€˜π‘“) ∈ (SubRingβ€˜π‘’)))}
53, 4eleqtrdi 2837 . . 3 (𝐸/FldExt𝐹 β†’ ⟨𝐸, 𝐹⟩ ∈ {βŸ¨π‘’, π‘“βŸ© ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 β†Ύs (Baseβ€˜π‘“)) ∧ (Baseβ€˜π‘“) ∈ (SubRingβ€˜π‘’)))})
61, 5sselid 3975 . 2 (𝐸/FldExt𝐹 β†’ ⟨𝐸, 𝐹⟩ ∈ (Field Γ— Field))
7 opelxp2 5712 . 2 (⟨𝐸, 𝐹⟩ ∈ (Field Γ— Field) β†’ 𝐹 ∈ Field)
86, 7syl 17 1 (𝐸/FldExt𝐹 β†’ 𝐹 ∈ Field)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4629   class class class wbr 5141  {copab 5203   Γ— cxp 5667  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153   β†Ύs cress 17182  SubRingcsubrg 20469  Fieldcfield 20588  /FldExtcfldext 33235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-fldext 33239
This theorem is referenced by:  fldextsubrg  33248  fldextress  33249  brfinext  33250  fldextsralvec  33252  extdgcl  33253  extdggt0  33254  fldexttr  33255  extdgmul  33258  extdg1id  33260  extdg1b  33261
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