Theorem fldextsdrg 33792

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 33785	. . . 4 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐸 ∈ Field)
4	3	flddrngd 20678	. 2 ⊢ (𝜑 → 𝐸 ∈ DivRing)
5		fldextsdrg.1	. . . 4 ⊢ 𝐵 = (Base‘𝐹)
6	5	fldextsubrg 33787	. . 3 ⊢ (𝐸/FldExt𝐹 → 𝐵 ∈ (SubRing‘𝐸))
7	1, 6	syl 17	. 2 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝐸))
8		fldextress 33789	. . . . . 6 ⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
9	1, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
10	5	oveq2i 7371	. . . . 5 ⊢ (𝐸 ↾s 𝐵) = (𝐸 ↾s (Base‘𝐹))
11	9, 10	eqtr4di 2790	. . . 4 ⊢ (𝜑 → 𝐹 = (𝐸 ↾s 𝐵))
12		fldextfld2 33786	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
13	1, 12	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Field)
14	11, 13	eqeltrrd 2838	. . 3 ⊢ (𝜑 → (𝐸 ↾s 𝐵) ∈ Field)
15	14	flddrngd 20678	. 2 ⊢ (𝜑 → (𝐸 ↾s 𝐵) ∈ DivRing)
16		issdrg 20725	. 2 ⊢ (𝐵 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐵 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐵) ∈ DivRing))
17	4, 7, 15, 16	syl3anbrc 1345	1 ⊢ (𝜑 → 𝐵 ∈ (SubDRing‘𝐸))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   class class class wbr 5099  ‘cfv 6493  (class class class)co 7360  Basecbs 17140   ↾_s cress 17161  SubRingcsubrg 20506  DivRingcdr 20666  Fieldcfield 20667  SubDRingcsdrg 20723  /_FldExtcfldext 33776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7363  df-field 20669  df-sdrg 20724  df-fldext 33779

This theorem is referenced by:  finextalg  33836  constrext2chnlem  33888