Theorem sdrgdrng 20874

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 20872	. . 3 ⊢ (𝐴 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ DivRing))
3	2	simp3bi 1165	. 2 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ DivRing)
4	1, 3	eqeltrid 2867	1 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑆 ∈ DivRing)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ‘cfv 6538  (class class class)co 7412   ↾_s cress 17291  SubRingcsubrg 20655  DivRingcdr 20814  SubDRingcsdrg 20870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-sdrg 20871

This theorem is referenced by:  sdrgunit  20880  subsdrg  33600  fldextrspunlsplem  34044  fldextrspunlem1  34046  fldextrspunfld  34047  fldextrspundgdvdslem  34051  fldextrspundgdvds  34052  extdgfialglem1  34063  minplymindeg  34079  minplyann  34080  minplyirredlem  34081  minplyirred  34082  irngnminplynz  34083  minplym1p  34084  minplynzm1p  34085  irredminply  34087  algextdeglem4  34091  algextdeglem8  34095