Theorem fldextsdrg 34166

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 34159	. . . 4 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → 𝐸 ∈ Field)
4	3	flddrngd 20908	. 2 ⊢ (𝜑 → 𝐸 ∈ DivRing)
5		fldextsdrg.1	. . . 4 ⊢ 𝐵 = (Base‘𝐹)
6	5	fldextsubrg 34161	. . 3 ⊢ (𝐸/FldExt𝐹 → 𝐵 ∈ (SubRing‘𝐸))
7	1, 6	syl 18	. 2 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝐸))
8		fldextress 34163	. . . . . 6 ⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
9	1, 8	syl 18	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
10	5	oveq2i 7427	. . . . 5 ⊢ (𝐸 ↾s 𝐵) = (𝐸 ↾s (Base‘𝐹))
11	9, 10	eqtr4di 2815	. . . 4 ⊢ (𝜑 → 𝐹 = (𝐸 ↾s 𝐵))
12		fldextfld2 34160	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
13	1, 12	syl 18	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Field)
14	11, 13	eqeltrrd 2863	. . 3 ⊢ (𝜑 → (𝐸 ↾s 𝐵) ∈ Field)
15	14	flddrngd 20908	. 2 ⊢ (𝜑 → (𝐸 ↾s 𝐵) ∈ DivRing)
16		issdrg 20958	. 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 1570   ∈ wcel 2145   class class class wbr 5107  ‘cfv 6537  (class class class)co 7416  Basecbs 17305   ↾_s cress 17326  SubRingcsubrg 20735  DivRingcdr 20894  Fieldcfield 20895  SubDRingcsdrg 20956  /_FldExtcfldext 34150

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

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-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7419  df-field 20897  df-sdrg 20957  df-fldext 34153

This theorem is used by:  finextalg  34210  constrext2chnlem  34262