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Theorem fldextsubrg 33679
Description: Field extension implies a subring relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
Hypothesis
Ref Expression
fldextsubrg.1 𝑈 = (Base‘𝐹)
Assertion
Ref Expression
fldextsubrg (𝐸/FldExt𝐹𝑈 ∈ (SubRing‘𝐸))

Proof of Theorem fldextsubrg
StepHypRef Expression
1 fldextsubrg.1 . 2 𝑈 = (Base‘𝐹)
2 fldextfld1 33677 . . . . 5 (𝐸/FldExt𝐹𝐸 ∈ Field)
3 fldextfld2 33678 . . . . 5 (𝐸/FldExt𝐹𝐹 ∈ Field)
4 brfldext 33675 . . . . 5 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
52, 3, 4syl2anc 584 . . . 4 (𝐸/FldExt𝐹 → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
65ibi 267 . . 3 (𝐸/FldExt𝐹 → (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))
76simprd 495 . 2 (𝐸/FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸))
81, 7eqeltrid 2843 1 (𝐸/FldExt𝐹𝑈 ∈ (SubRing‘𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  SubRingcsubrg 20586  Fieldcfield 20747  /FldExtcfldext 33666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-iota 6516  df-fv 6571  df-ov 7434  df-fldext 33670
This theorem is referenced by:  fldextsralvec  33683  extdgcl  33684  extdggt0  33685  extdgmul  33689  extdg1id  33691  fldextchr  33694
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