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Theorem fldextsubrg 33702
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 33700 . . . . 5 (𝐸/FldExt𝐹𝐸 ∈ Field)
3 fldextfld2 33701 . . . . 5 (𝐸/FldExt𝐹𝐹 ∈ Field)
4 brfldext 33698 . . . . 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 2845 1 (𝐸/FldExt𝐹𝑈 ∈ (SubRing‘𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  SubRingcsubrg 20569  Fieldcfield 20730  /FldExtcfldext 33689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-iota 6514  df-fv 6569  df-ov 7434  df-fldext 33693
This theorem is referenced by:  fldextsralvec  33706  extdgcl  33707  extdggt0  33708  extdgmul  33714  extdg1id  33716  fldextchr  33719
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