Theorem sdrgdrng 21080

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 21078	. . 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 2868	1 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑆 ∈ DivRing)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ‘cfv 6561  (class class class)co 7446   ↾_s cress 17440  SubRingcsubrg 20854  DivRingcdr 21013  SubDRingcsdrg 21076

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 5253  ax-nul 5264  ax-pr 5396

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 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-sdrg 21077

This theorem is used by:  sdrgunit  21086  subsdrg  33944  fldextrspunlsplem  34388  fldextrspunlem1  34390  fldextrspunfld  34391  fldextrspundgdvdslem  34395  fldextrspundgdvds  34396  extdgfialglem1  34407  minplymindeg  34423  minplyann  34424  minplyirredlem  34425  minplyirred  34426  irngnminplynz  34427  minplym1p  34428  minplynzm1p  34429  irredminply  34431  algextdeglem4  34435  algextdeglem8  34439