Theorem sdrgss 21077

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 21072	. 2 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))
2		sdrgid.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
3	2	subrgss 20851	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵)
4	3	3ad2ant2 1152	. 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing) → 𝑆 ⊆ 𝐵)
5	1, 4	sylbi 220	1 ⊢ (𝑆 ∈ (SubDRing‘𝑅) → 𝑆 ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  ‘cfv 6555  (class class class)co 7440  Basecbs 17413   ↾_s cress 17434  SubRingcsubrg 20848  DivRingcdr 21007  SubDRingcsdrg 21070

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fv 6563  df-ov 7443  df-subrg 20849  df-sdrg 21071

This theorem is used by:  sdrgbas  21078  subsdrg  33927  fldgenidfld  33946  sdrgfldext  34349  fldsdrgfldext  34360  fldsdrgfldext2  34361  fldgenfldext  34367  evls1fldgencl  34369  fldextrspunlsplem  34372  fldextrspunlsp  34373  fldextrspunlem1  34374  fldextrspunfld  34375  fldextrspunlem2  34376  fldextrspundgle  34377  fldextrspundglemul  34378  fldextrspundgdvdslem  34379  fldextrspundgdvds  34380  fldext2rspun  34381  extdgfialglem1  34391  extdgfialglem2  34392  algextdeglem8  34423  rtelextdg2lem  34425  rtelextdg2  34426  constrelextdg2  34446  constrextdg2lem  34447  constrext2chnlem  34449