Theorem sdrgdrng 20707

Description: A sub-division-ring is a division ring. (Contributed by SN, 19-Feb-2025.)

Hypothesis

Ref	Expression
sdrgdrng.1	⊢ 𝑆 = (𝑅 ↾s 𝐴)

Assertion

Ref	Expression
sdrgdrng	⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑆 ∈ DivRing)

Proof of Theorem sdrgdrng

Step	Hyp	Ref	Expression
1		sdrgdrng.1	. 2 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
2		issdrg 20705	. . 3 ⊢ (𝐴 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ DivRing))
3	2	simp3bi 1147	. 2 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ DivRing)
4	1, 3	eqeltrid 2837	1 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑆 ∈ DivRing)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ‘cfv 6486  (class class class)co 7352   ↾_s cress 17143  SubRingcsubrg 20486  DivRingcdr 20646  SubDRingcsdrg 20703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  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 6442  df-fun 6488  df-fv 6494  df-ov 7355  df-sdrg 20704

This theorem is referenced by:  sdrgunit  20713  subsdrg  33271  fldextrspunlsplem  33707  fldextrspunlem1  33709  fldextrspunfld  33710  fldextrspundgdvdslem  33714  fldextrspundgdvds  33715  extdgfialglem1  33726  minplymindeg  33742  minplyann  33743  minplyirredlem  33744  minplyirred  33745  irngnminplynz  33746  minplym1p  33747  minplynzm1p  33748  irredminply  33750  algextdeglem4  33754  algextdeglem8  33758