Theorem sdrgss 20743

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 20738	. 2 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))
2		sdrgid.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
3	2	subrgss 20522	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵)
4	3	3ad2ant2 1135	. 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 1087   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903  ‘cfv 6502  (class class class)co 7370  Basecbs 17150   ↾_s cress 17171  SubRingcsubrg 20519  DivRingcdr 20679  SubDRingcsdrg 20736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fv 6510  df-ov 7373  df-subrg 20520  df-sdrg 20737

This theorem is referenced by:  sdrgbas  20744  subsdrg  33398  fldgenidfld  33417  sdrgfldext  33834  fldsdrgfldext  33845  fldsdrgfldext2  33846  fldgenfldext  33852  evls1fldgencl  33854  fldextrspunlsplem  33857  fldextrspunlsp  33858  fldextrspunlem1  33859  fldextrspunfld  33860  fldextrspunlem2  33861  fldextrspundgle  33862  fldextrspundglemul  33863  fldextrspundgdvdslem  33864  fldextrspundgdvds  33865  fldext2rspun  33866  extdgfialglem1  33876  extdgfialglem2  33877  algextdeglem8  33908  rtelextdg2lem  33910  rtelextdg2  33911  constrelextdg2  33931  constrextdg2lem  33932  constrext2chnlem  33934