Theorem fldextress 33654

Description: Field extension implies a structure restriction relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)

Assertion

Ref	Expression
fldextress	⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))

Proof of Theorem fldextress

Step	Hyp	Ref	Expression
1		fldextfld1 33650	. . . 4 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
2		fldextfld2 33651	. . . 4 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
3		brfldext 33648	. . . 4 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
4	1, 2, 3	syl2anc 584	. . 3 ⊢ (𝐸/FldExt𝐹 → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
5	4	ibi 267	. 2 ⊢ (𝐸/FldExt𝐹 → (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))
6	5	simpld 494	1 ⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110   class class class wbr 5089  ‘cfv 6477  (class class class)co 7341  Basecbs 17112   ↾_s cress 17133  SubRingcsubrg 20477  Fieldcfield 20638  /_FldExtcfldext 33641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-xp 5620  df-iota 6433  df-fv 6485  df-ov 7344  df-fldext 33644

This theorem is referenced by:  fldextsdrg  33657  fldextsralvec  33658  extdgcl  33659  extdggt0  33660  extdg1id  33669  fldextchr  33672