Theorem sdrgdrng 20792

Description: A sub-division-ring is a division ring. (Contributed by SN, 19-Feb-2025.)

Hypothesis

Ref	Expression
sdrgdrng.1	⊢ 𝑆 = (𝑅 ↾s 𝐴)

Assertion

Ref	Expression
sdrgdrng	⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑆 ∈ DivRing)

Proof of Theorem sdrgdrng

Step	Hyp	Ref	Expression
1		sdrgdrng.1	. 2 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
2		issdrg 20790	. . 3 ⊢ (𝐴 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ DivRing))
3	2	simp3bi 1147	. 2 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ DivRing)
4	1, 3	eqeltrid 2844	1 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑆 ∈ DivRing)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ‘cfv 6560  (class class class)co 7432   ↾_s cress 17275  SubRingcsubrg 20570  DivRingcdr 20730  SubDRingcsdrg 20788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-sdrg 20789

This theorem is referenced by:  sdrgunit  20798  fldextrspunlsplem  33724  fldextrspunlem1  33726  fldextrspunfld  33727  fldextrspundgdvdslem  33731  fldextrspundgdvds  33732  minplymindeg  33752  minplyann  33753  minplyirredlem  33754  minplyirred  33755  irngnminplynz  33756  minplym1p  33757  irredminply  33758  algextdeglem4  33762  algextdeglem8  33766