MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sdrgss Structured version   Visualization version   GIF version

Theorem sdrgss 20811
Description: A division subring is a subset of the base set. (Contributed by Thierry Arnoux, 21-Aug-2023.)
Hypothesis
Ref Expression
sdrgid.1 𝐵 = (Base‘𝑅)
Assertion
Ref Expression
sdrgss (𝑆 ∈ (SubDRing‘𝑅) → 𝑆𝐵)

Proof of Theorem sdrgss
StepHypRef Expression
1 issdrg 20806 . 2 (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))
2 sdrgid.1 . . . 4 𝐵 = (Base‘𝑅)
32subrgss 20595 . . 3 (𝑆 ∈ (SubRing‘𝑅) → 𝑆𝐵)
433ad2ant2 1134 . 2 ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing) → 𝑆𝐵)
51, 4sylbi 217 1 (𝑆 ∈ (SubDRing‘𝑅) → 𝑆𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1537  wcel 2103  wss 3970  cfv 6572  (class class class)co 7445  Basecbs 17253  s cress 17282  SubRingcsubrg 20590  DivRingcdr 20746  SubDRingcsdrg 20804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fv 6580  df-ov 7448  df-subrg 20592  df-sdrg 20805
This theorem is referenced by:  sdrgbas  20812  fldgenidfld  33276  fldgenfldext  33670  evls1fldgencl  33672  algextdeglem8  33707  rtelextdg2lem  33709  rtelextdg2  33710  constrelextdg2  33729
  Copyright terms: Public domain W3C validator