Theorem fldextsdrg 33988

Description: Deduce sub-division-ring from field extension. (Contributed by Thierry Arnoux, 26-Oct-2025.)

Hypotheses

Ref	Expression
fldextsdrg.1	⊢ 𝐵 = (Base‘𝐹)
fldextsdrg.2	⊢ (𝜑 → 𝐸/FldExt𝐹)

Assertion

Ref	Expression
fldextsdrg	⊢ (𝜑 → 𝐵 ∈ (SubDRing‘𝐸))

Proof of Theorem fldextsdrg

Step	Hyp	Ref	Expression
1		fldextsdrg.2	. . . 4 ⊢ (𝜑 → 𝐸/FldExt𝐹)
2		fldextfld1 33981	. . . 4 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → 𝐸 ∈ Field)
4	3	flddrngd 20824	. 2 ⊢ (𝜑 → 𝐸 ∈ DivRing)
5		fldextsdrg.1	. . . 4 ⊢ 𝐵 = (Base‘𝐹)
6	5	fldextsubrg 33983	. . 3 ⊢ (𝐸/FldExt𝐹 → 𝐵 ∈ (SubRing‘𝐸))
7	1, 6	syl 18	. 2 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝐸))
8		fldextress 33985	. . . . . 6 ⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
9	1, 8	syl 18	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
10	5	oveq2i 7422	. . . . 5 ⊢ (𝐸 ↾s 𝐵) = (𝐸 ↾s (Base‘𝐹))
11	9, 10	eqtr4di 2822	. . . 4 ⊢ (𝜑 → 𝐹 = (𝐸 ↾s 𝐵))
12		fldextfld2 33982	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
13	1, 12	syl 18	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Field)
14	11, 13	eqeltrrd 2870	. . 3 ⊢ (𝜑 → (𝐸 ↾s 𝐵) ∈ Field)
15	14	flddrngd 20824	. 2 ⊢ (𝜑 → (𝐸 ↾s 𝐵) ∈ DivRing)
16		issdrg 20868	. 2 ⊢ (𝐵 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐵 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐵) ∈ DivRing))
17	4, 7, 15, 16	syl3anbrc 1360	1 ⊢ (𝜑 → 𝐵 ∈ (SubDRing‘𝐸))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  Basecbs 17268   ↾_s cress 17289  SubRingcsubrg 20653  DivRingcdr 20812  Fieldcfield 20813  SubDRingcsdrg 20866  /_FldExtcfldext 33972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-field 20815  df-sdrg 20867  df-fldext 33975

This theorem is referenced by:  finextalg  34032  constrext2chnlem  34084