Theorem subrngrng 20578

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 2740	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	2	issubrng 20575	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)))
4		subrngrng.1	. . 3 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
5	4	eleq1i 2835	. 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 1537   ∈ wcel 2108   ⊆ wss 3976  ‘cfv 6575  (class class class)co 7450  Basecbs 17260   ↾_s cress 17289  Rngcrng 20181  SubRngcsubrng 20573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fv 6583  df-ov 7453  df-subrng 20574

This theorem is referenced by:  subrngsubg  20580  subrngmcl  20585  subsubrng  20591  pzriprnglem7  21523