Theorem subrngrng 20491

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 1134	. 2 ⊢ ((𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)) → (𝑅 ↾s 𝐴) ∈ Rng)
2		eqid 2725	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	2	issubrng 20488	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)))
4		subrngrng.1	. . 3 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
5	4	eleq1i 2816	. 2 ⊢ (𝑆 ∈ Rng ↔ (𝑅 ↾s 𝐴) ∈ Rng)
6	1, 3, 5	3imtr4i 291	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng)

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ⊆ wss 3945  ‘cfv 6547  (class class class)co 7417  Basecbs 17179   ↾_s cress 17208  Rngcrng 20096  SubRngcsubrng 20486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6499  df-fun 6549  df-fv 6555  df-ov 7420  df-subrng 20487

This theorem is referenced by:  subrngsubg  20493  subrngmcl  20498  subsubrng  20504  pzriprnglem7  21417