Theorem fldextress 32731

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 32728	. . . 4 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
2		fldextfld2 32729	. . . 4 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field)
3		brfldext 32726	. . . 4 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
4	1, 2, 3	syl2anc 585	. . 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 496	1 ⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5149  ‘cfv 6544  (class class class)co 7409  Basecbs 17144   ↾_s cress 17173  SubRingcsubrg 20315  Fieldcfield 20358  /_FldExtcfldext 32717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-iota 6496  df-fv 6552  df-ov 7412  df-fldext 32721

This theorem is referenced by:  fldextsralvec  32734  extdgcl  32735  extdggt0  32736  extdg1id  32742  fldextchr  32744