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Theorem sdrgss 20681
Description: A division subring is a subset of the base set. (Contributed by Thierry Arnoux, 21-Aug-2023.)
Hypothesis
Ref Expression
sdrgid.1 𝐡 = (Baseβ€˜π‘…)
Assertion
Ref Expression
sdrgss (𝑆 ∈ (SubDRingβ€˜π‘…) β†’ 𝑆 βŠ† 𝐡)

Proof of Theorem sdrgss
StepHypRef Expression
1 issdrg 20676 . 2 (𝑆 ∈ (SubDRingβ€˜π‘…) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRingβ€˜π‘…) ∧ (𝑅 β†Ύs 𝑆) ∈ DivRing))
2 sdrgid.1 . . . 4 𝐡 = (Baseβ€˜π‘…)
32subrgss 20511 . . 3 (𝑆 ∈ (SubRingβ€˜π‘…) β†’ 𝑆 βŠ† 𝐡)
433ad2ant2 1132 . 2 ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRingβ€˜π‘…) ∧ (𝑅 β†Ύs 𝑆) ∈ DivRing) β†’ 𝑆 βŠ† 𝐡)
51, 4sylbi 216 1 (𝑆 ∈ (SubDRingβ€˜π‘…) β†’ 𝑆 βŠ† 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   βŠ† wss 3947  β€˜cfv 6548  (class class class)co 7420  Basecbs 17180   β†Ύs cress 17209  SubRingcsubrg 20506  DivRingcdr 20624  SubDRingcsdrg 20674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-subrg 20508  df-sdrg 20675
This theorem is referenced by:  sdrgbas  20682  fldgenidfld  33017  evls1fldgencl  33358  algextdeglem8  33392
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