Theorem fldextsubrg 34063

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 34061	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
3		fldextfld2 34062	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
4		brfldext 34059	. . . . 5 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
5	2, 3, 4	syl2anc 596	. . . 4 ⊢ (𝐸/FldExt𝐹 → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
6	5	ibi 270	. . 3 ⊢ (𝐸/FldExt𝐹 → (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))
7	6	simprd 501	. 2 ⊢ (𝐸/FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸))
8	1, 7	eqeltrid 2870	1 ⊢ (𝐸/FldExt𝐹 → 𝑈 ∈ (SubRing‘𝐸))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  Basecbs 17294   ↾_s cress 17315  SubRingcsubrg 20698  Fieldcfield 20858  /_FldExtcfldext 34052

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-xp 5672  df-iota 6499  df-fv 6551  df-ov 7426  df-fldext 34055

This theorem is used by:  fldextsdrg  34068  fldextsralvec  34069  extdgcl  34070  extdggt0  34071  extdgmul  34077  extdg1id  34080  fldextchr  34083