Theorem sdrgss 20708

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 20703	. 2 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))
2		sdrgid.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
3	2	subrgss 20487	. . 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 3922  ‘cfv 6519  (class class class)co 7394  Basecbs 17185   ↾_s cress 17206  SubRingcsubrg 20484  DivRingcdr 20644  SubDRingcsdrg 20701

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 2702  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fv 6527  df-ov 7397  df-subrg 20485  df-sdrg 20702

This theorem is referenced by:  sdrgbas  20709  subsdrg  33256  fldgenidfld  33275  sdrgfldext  33654  fldsdrgfldext  33665  fldsdrgfldext2  33666  fldgenfldext  33671  evls1fldgencl  33673  fldextrspunlsplem  33676  fldextrspunlsp  33677  fldextrspunlem1  33678  fldextrspunfld  33679  fldextrspunlem2  33680  fldextrspundgle  33681  fldextrspundglemul  33682  fldextrspundgdvdslem  33683  fldextrspundgdvds  33684  fldext2rspun  33685  algextdeglem8  33722  rtelextdg2lem  33724  rtelextdg2  33725  constrelextdg2  33745  constrextdg2lem  33746  constrext2chnlem  33748