Theorem fldextsubrg 34328

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 34326	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
3		fldextfld2 34327	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
4		brfldext 34324	. . . . 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 2864	1 ⊢ (𝐸/FldExt𝐹 → 𝑈 ∈ (SubRing‘𝐸))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6539  (class class class)co 7421  Basecbs 17393   ↾_s cress 17414  SubRingcsubrg 20827  Fieldcfield 20987  /_FldExtcfldext 34317

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-iota 6494  df-fv 6547  df-ov 7424  df-fldext 34320

This theorem is used by:  fldextsdrg  34333  fldextsralvec  34334  extdgcl  34335  extdggt0  34336  extdgmul  34342  extdg1id  34345  fldextchr  34348