Theorem brfldext 32785

Description: The field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.)

Assertion

Ref	Expression
brfldext	⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))

Proof of Theorem brfldext

Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 483	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑒 = 𝐸)
2	1	eleq1d 2818	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ∈ Field ↔ 𝐸 ∈ Field))
3		simpr 485	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
4	3	eleq1d 2818	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑓 ∈ Field ↔ 𝐹 ∈ Field))
5	2, 4	anbi12d 631	. . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ↔ (𝐸 ∈ Field ∧ 𝐹 ∈ Field)))
6	3	fveq2d 6895	. . . . . . 7 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
7	1, 6	oveq12d 7429	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ↾s (Base‘𝑓)) = (𝐸 ↾s (Base‘𝐹)))
8	3, 7	eqeq12d 2748	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑓 = (𝑒 ↾s (Base‘𝑓)) ↔ 𝐹 = (𝐸 ↾s (Base‘𝐹))))
9	1	fveq2d 6895	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (SubRing‘𝑒) = (SubRing‘𝐸))
10	6, 9	eleq12d 2827	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((Base‘𝑓) ∈ (SubRing‘𝑒) ↔ (Base‘𝐹) ∈ (SubRing‘𝐸)))
11	8, 10	anbi12d 631	. . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)) ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
12	5, 11	anbi12d 631	. . 3 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒))) ↔ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) ∧ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))))
13		df-fldext 32780	. . 3 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
14	12, 13	brabga 5534	. 2 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) ∧ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))))
15	14	bianabs 542	1 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  ‘cfv 6543  (class class class)co 7411  Basecbs 17146   ↾_s cress 17175  SubRingcsubrg 20319  Fieldcfield 20362  /_FldExtcfldext 32776

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-iota 6495  df-fv 6551  df-ov 7414  df-fldext 32780

This theorem is referenced by:  ccfldextrr  32786  fldextsubrg  32789  fldextress  32790  fldexttr  32796  fldextid  32797  extdgmul  32799  algextdeglem4  32836