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Theorem sdrgss 20820
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 20815 . 2 (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))
2 sdrgid.1 . . . 4 𝐵 = (Base‘𝑅)
32subrgss 20598 . . 3 (𝑆 ∈ (SubRing‘𝑅) → 𝑆𝐵)
433ad2ant2 1135 . 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 1539  wcel 2108  wss 3966  cfv 6569  (class class class)co 7438  Basecbs 17254  s cress 17283  SubRingcsubrg 20595  DivRingcdr 20755  SubDRingcsdrg 20813
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fv 6577  df-ov 7441  df-subrg 20596  df-sdrg 20814
This theorem is referenced by:  sdrgbas  20821  fldgenidfld  33331  fldgenfldext  33725  evls1fldgencl  33727  algextdeglem8  33762  rtelextdg2lem  33764  rtelextdg2  33765  constrelextdg2  33784
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