Theorem sdrgss 20761

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) → 𝑆 ⊆ 𝐵)

Proof of Theorem sdrgss

Step	Hyp	Ref	Expression
1		issdrg 20756	. 2 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))
2		sdrgid.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
3	2	subrgss 20539	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵)
4	3	3ad2ant2 1134	. 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing) → 𝑆 ⊆ 𝐵)
5	1, 4	sylbi 217	1 ⊢ (𝑆 ∈ (SubDRing‘𝑅) → 𝑆 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ⊆ wss 3931  ‘cfv 6540  (class class class)co 7412  Basecbs 17228   ↾_s cress 17251  SubRingcsubrg 20536  DivRingcdr 20696  SubDRingcsdrg 20754

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 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6493  df-fun 6542  df-fv 6548  df-ov 7415  df-subrg 20537  df-sdrg 20755

This theorem is referenced by:  sdrgbas  20762  subsdrg  33231  fldgenidfld  33250  sdrgfldext  33629  fldsdrgfldext  33640  fldsdrgfldext2  33641  fldgenfldext  33646  evls1fldgencl  33648  fldextrspunlsplem  33651  fldextrspunlsp  33652  fldextrspunlem1  33653  fldextrspunfld  33654  fldextrspunlem2  33655  fldextrspundgle  33656  fldextrspundglemul  33657  fldextrspundgdvdslem  33658  fldextrspundgdvds  33659  fldext2rspun  33660  algextdeglem8  33695  rtelextdg2lem  33697  rtelextdg2  33698  constrelextdg2  33718  constrextdg2lem  33719  constrext2chnlem  33721