Theorem brfldext 33675

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 482	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑒 = 𝐸)
2	1	eleq1d 2824	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ∈ Field ↔ 𝐸 ∈ Field))
3		simpr 484	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
4	3	eleq1d 2824	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑓 ∈ Field ↔ 𝐹 ∈ Field))
5	2, 4	anbi12d 632	. . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ↔ (𝐸 ∈ Field ∧ 𝐹 ∈ Field)))
6	3	fveq2d 6911	. . . . . . 7 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
7	1, 6	oveq12d 7449	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ↾s (Base‘𝑓)) = (𝐸 ↾s (Base‘𝐹)))
8	3, 7	eqeq12d 2751	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑓 = (𝑒 ↾s (Base‘𝑓)) ↔ 𝐹 = (𝐸 ↾s (Base‘𝐹))))
9	1	fveq2d 6911	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (SubRing‘𝑒) = (SubRing‘𝐸))
10	6, 9	eleq12d 2833	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((Base‘𝑓) ∈ (SubRing‘𝑒) ↔ (Base‘𝐹) ∈ (SubRing‘𝐸)))
11	8, 10	anbi12d 632	. . . 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 632	. . 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 33670	. . 3 ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))}
14	12, 13	brabga 5544	. 2 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) ∧ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))))
15	14	bianabs 541	1 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  Basecbs 17245   ↾_s cress 17274  SubRingcsubrg 20586  Fieldcfield 20747  /_FldExtcfldext 33666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-iota 6516  df-fv 6571  df-ov 7434  df-fldext 33670

This theorem is referenced by:  ccfldextrr  33676  fldextsubrg  33679  fldextress  33680  fldexttr  33686  fldextid  33687  extdgmul  33689  fldgenfldext  33693  algextdeglem4  33726