Theorem fldextsubrg 33399

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 33397	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
3		fldextfld2 33398	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
4		brfldext 33395	. . . . 5 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
5	2, 3, 4	syl2anc 582	. . . 4 ⊢ (𝐸/FldExt𝐹 → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
6	5	ibi 266	. . 3 ⊢ (𝐸/FldExt𝐹 → (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))
7	6	simprd 494	. 2 ⊢ (𝐸/FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸))
8	1, 7	eqeltrid 2829	1 ⊢ (𝐸/FldExt𝐹 → 𝑈 ∈ (SubRing‘𝐸))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   class class class wbr 5143  ‘cfv 6542  (class class class)co 7415  Basecbs 17177   ↾_s cress 17206  SubRingcsubrg 20508  Fieldcfield 20627  /_FldExtcfldext 33386

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 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-xp 5678  df-iota 6494  df-fv 6550  df-ov 7418  df-fldext 33390

This theorem is referenced by:  fldextsralvec  33403  extdgcl  33404  extdggt0  33405  extdgmul  33409  extdg1id  33411  fldextchr  33413