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Theorem sdrgss 20818
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 20813 . 2 (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))
2 sdrgid.1 . . . 4 𝐵 = (Base‘𝑅)
32subrgss 20602 . . 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 2108  wss 3976  cfv 6575  (class class class)co 7450  Basecbs 17260  s cress 17289  SubRingcsubrg 20597  DivRingcdr 20753  SubDRingcsdrg 20811
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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447
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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fv 6583  df-ov 7453  df-subrg 20599  df-sdrg 20812
This theorem is referenced by:  sdrgbas  20819  fldgenidfld  33286  fldgenfldext  33680  evls1fldgencl  33682  algextdeglem8  33717  rtelextdg2lem  33719  rtelextdg2  33720  constrelextdg2  33739
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