Theorem subrngrng 20435

Description: A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.)

Hypothesis

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

Assertion

Ref	Expression
subrngrng	⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng)

Proof of Theorem subrngrng

Step	Hyp	Ref	Expression
1		simp2 1137	. 2 ⊢ ((𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)) → (𝑅 ↾s 𝐴) ∈ Rng)
2		eqid 2729	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	2	issubrng 20432	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)))
4		subrngrng.1	. . 3 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
5	4	eleq1i 2819	. 2 ⊢ (𝑆 ∈ Rng ↔ (𝑅 ↾s 𝐴) ∈ Rng)
6	1, 3, 5	3imtr4i 292	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3903  ‘cfv 6482  (class class class)co 7349  Basecbs 17120   ↾_s cress 17141  Rngcrng 20037  SubRngcsubrng 20430

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 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  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 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-subrng 20431

This theorem is referenced by:  subrngsubg  20437  subrngmcl  20442  subsubrng  20448  pzriprnglem7  21394