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Theorem sdrgsubrg 20683
Description: A sub-division-ring is a subring. (Contributed by SN, 19-Feb-2025.)
Assertion
Ref Expression
sdrgsubrg (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 ∈ (SubRing‘𝑅))

Proof of Theorem sdrgsubrg
StepHypRef Expression
1 issdrg 20680 . 2 (𝐴 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝐴) ∈ DivRing))
21simp2bi 1143 1 (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 ∈ (SubRing‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  cfv 6543  (class class class)co 7416  s cress 17208  SubRingcsubrg 20510  DivRingcdr 20628  SubDRingcsdrg 20678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7419  df-sdrg 20679
This theorem is referenced by:  sdrgunit  20688  imadrhmcl  20689  evls1fldgencl  33415  minplyann  33436  minplyirredlem  33437  minplyirred  33438  irngnminplynz  33439  minplym1p  33440  irredminply  33441  algextdeglem4  33445  algextdeglem5  33446  algextdeglem6  33447  algextdeglem7  33448  algextdeglem8  33449
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