Theorem fldextsubrg 33645

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/FldExt𝐹 → 𝑈 ∈ (SubRing‘𝐸))

Proof of Theorem fldextsubrg

Step	Hyp	Ref	Expression
1		fldextsubrg.1	. 2 ⊢ 𝑈 = (Base‘𝐹)
2		fldextfld1 33643	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
3		fldextfld2 33644	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
4		brfldext 33641	. . . . 5 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
5	2, 3, 4	syl2anc 584	. . . 4 ⊢ (𝐸/FldExt𝐹 → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
6	5	ibi 267	. . 3 ⊢ (𝐸/FldExt𝐹 → (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))
7	6	simprd 495	. 2 ⊢ (𝐸/FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸))
8	1, 7	eqeltrid 2832	1 ⊢ (𝐸/FldExt𝐹 → 𝑈 ∈ (SubRing‘𝐸))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  Basecbs 17179   ↾_s cress 17200  SubRingcsubrg 20478  Fieldcfield 20639  /_FldExtcfldext 33634

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-iota 6464  df-fv 6519  df-ov 7390  df-fldext 33637

This theorem is referenced by:  fldextsdrg  33650  fldextsralvec  33651  extdgcl  33652  extdggt0  33653  extdgmul  33659  extdg1id  33661  fldextchr  33664