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Theorem fldextsubrg 32397
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 32395 . . . . 5 (𝐸/FldExt𝐹𝐸 ∈ Field)
3 fldextfld2 32396 . . . . 5 (𝐸/FldExt𝐹𝐹 ∈ Field)
4 brfldext 32393 . . . . 5 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
52, 3, 4syl2anc 585 . . . 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 497 . 2 (𝐸/FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸))
81, 7eqeltrid 2838 1 (𝐸/FldExt𝐹𝑈 ∈ (SubRing‘𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   class class class wbr 5106  cfv 6497  (class class class)co 7358  Basecbs 17088  s cress 17117  Fieldcfield 20198  SubRingcsubrg 20232  /FldExtcfldext 32384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-iota 6449  df-fv 6505  df-ov 7361  df-fldext 32388
This theorem is referenced by:  fldextsralvec  32401  extdgcl  32402  extdggt0  32403  extdgmul  32407  extdg1id  32409  fldextchr  32411
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