Theorem subrngrng 20500

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 1138	. 2 ⊢ ((𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)) → (𝑅 ↾s 𝐴) ∈ Rng)
2		eqid 2737	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	2	issubrng 20497	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)))
4		subrngrng.1	. . 3 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
5	4	eleq1i 2828	. 2 ⊢ (𝑆 ∈ Rng ↔ (𝑅 ↾s 𝐴) ∈ Rng)
6	1, 3, 5	3imtr4i 292	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903  ‘cfv 6502  (class class class)co 7370  Basecbs 17150   ↾_s cress 17171  Rngcrng 20104  SubRngcsubrng 20495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fv 6510  df-ov 7373  df-subrng 20496

This theorem is referenced by:  subrngsubg  20502  subrngmcl  20507  subsubrng  20513  pzriprnglem7  21459