Theorem fldextsdrg 34329

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 34322	. . . 4 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → 𝐸 ∈ Field)
4	3	flddrngd 20995	. 2 ⊢ (𝜑 → 𝐸 ∈ DivRing)
5		fldextsdrg.1	. . . 4 ⊢ 𝐵 = (Base‘𝐹)
6	5	fldextsubrg 34324	. . 3 ⊢ (𝐸/FldExt𝐹 → 𝐵 ∈ (SubRing‘𝐸))
7	1, 6	syl 18	. 2 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝐸))
8		fldextress 34326	. . . . . 6 ⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
9	1, 8	syl 18	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
10	5	oveq2i 7431	. . . . 5 ⊢ (𝐸 ↾s 𝐵) = (𝐸 ↾s (Base‘𝐹))
11	9, 10	eqtr4di 2813	. . . 4 ⊢ (𝜑 → 𝐹 = (𝐸 ↾s 𝐵))
12		fldextfld2 34323	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
13	1, 12	syl 18	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Field)
14	11, 13	eqeltrrd 2861	. . 3 ⊢ (𝜑 → (𝐸 ↾s 𝐵) ∈ Field)
15	14	flddrngd 20995	. 2 ⊢ (𝜑 → (𝐸 ↾s 𝐵) ∈ DivRing)
16		issdrg 21046	. 2 ⊢ (𝐵 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐵 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐵) ∈ DivRing))
17	4, 7, 15, 16	syl3anbrc 1362	1 ⊢ (𝜑 → 𝐵 ∈ (SubDRing‘𝐸))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Basecbs 17388   ↾_s cress 17409  SubRingcsubrg 20822  DivRingcdr 20981  Fieldcfield 20982  SubDRingcsdrg 21044  /_FldExtcfldext 34313

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-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  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-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  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-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-field 20984  df-sdrg 21045  df-fldext 34316

This theorem is used by:  finextalg  34373  constrext2chnlem  34425