Theorem sdrgss 20678

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 20673	. 2 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))
2		sdrgid.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
3	2	subrgss 20457	. . 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 1540   ∈ wcel 2109   ⊆ wss 3903  ‘cfv 6482  (class class class)co 7349  Basecbs 17120   ↾_s cress 17141  SubRingcsubrg 20454  DivRingcdr 20614  SubDRingcsdrg 20671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-subrg 20455  df-sdrg 20672

This theorem is referenced by:  sdrgbas  20679  subsdrg  33238  fldgenidfld  33257  sdrgfldext  33623  fldsdrgfldext  33634  fldsdrgfldext2  33635  fldgenfldext  33641  evls1fldgencl  33643  fldextrspunlsplem  33646  fldextrspunlsp  33647  fldextrspunlem1  33648  fldextrspunfld  33649  fldextrspunlem2  33650  fldextrspundgle  33651  fldextrspundglemul  33652  fldextrspundgdvdslem  33653  fldextrspundgdvds  33654  fldext2rspun  33655  extdgfialglem1  33665  extdgfialglem2  33666  algextdeglem8  33697  rtelextdg2lem  33699  rtelextdg2  33700  constrelextdg2  33720  constrextdg2lem  33721  constrext2chnlem  33723