Theorem fldextsdrg 33658

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 33651	. . . 4 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐸 ∈ Field)
4	3	flddrngd 20656	. 2 ⊢ (𝜑 → 𝐸 ∈ DivRing)
5		fldextsdrg.1	. . . 4 ⊢ 𝐵 = (Base‘𝐹)
6	5	fldextsubrg 33653	. . 3 ⊢ (𝐸/FldExt𝐹 → 𝐵 ∈ (SubRing‘𝐸))
7	1, 6	syl 17	. 2 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝐸))
8		fldextress 33655	. . . . . 6 ⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
9	1, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
10	5	oveq2i 7405	. . . . 5 ⊢ (𝐸 ↾s 𝐵) = (𝐸 ↾s (Base‘𝐹))
11	9, 10	eqtr4di 2783	. . . 4 ⊢ (𝜑 → 𝐹 = (𝐸 ↾s 𝐵))
12		fldextfld2 33652	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
13	1, 12	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Field)
14	11, 13	eqeltrrd 2830	. . 3 ⊢ (𝜑 → (𝐸 ↾s 𝐵) ∈ Field)
15	14	flddrngd 20656	. 2 ⊢ (𝜑 → (𝐸 ↾s 𝐵) ∈ DivRing)
16		issdrg 20703	. 2 ⊢ (𝐵 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐵 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐵) ∈ DivRing))
17	4, 7, 15, 16	syl3anbrc 1344	1 ⊢ (𝜑 → 𝐵 ∈ (SubDRing‘𝐸))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   class class class wbr 5115  ‘cfv 6519  (class class class)co 7394  Basecbs 17185   ↾_s cress 17206  SubRingcsubrg 20484  DivRingcdr 20644  Fieldcfield 20645  SubDRingcsdrg 20701  /_FldExtcfldext 33642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fv 6527  df-ov 7397  df-field 20647  df-sdrg 20702  df-fldext 33645

This theorem is referenced by:  constrext2chnlem  33748