Theorem subrngsubg 20612

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

Assertion

Ref	Expression
subrngsubg	⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))

Proof of Theorem subrngsubg

Step	Hyp	Ref	Expression
1		subrngrcl 20611	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2		rnggrp 20214	. . 3 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Grp)
4		eqid 2763	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
5	4	subrngss 20608	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
6		eqid 2763	. . . 4 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴)
7	6	subrngrng 20610	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑅 ↾s 𝐴) ∈ Rng)
8		rnggrp 20214	. . 3 ⊢ ((𝑅 ↾s 𝐴) ∈ Rng → (𝑅 ↾s 𝐴) ∈ Grp)
9	7, 8	syl 17	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑅 ↾s 𝐴) ∈ Grp)
10	4	issubg 19178	. 2 ⊢ (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ Grp))
11	3, 5, 9, 10	syl3anbrc 1358	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))

Syntax hints:   → wi 4   ∈ wcel 2143   ⊆ wss 3905  ‘cfv 6521  (class class class)co 7396  Basecbs 17255   ↾_s cress 17276  Grpcgrp 18985  SubGrpcsubg 19172  Rngcrng 20208  SubRngcsubrng 20605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-subg 19175  df-abl 19833  df-rng 20209  df-subrng 20606

This theorem is referenced by:  subrngringnsg  20613  subrngbas  20614  subrng0  20615  subrngacl  20616  issubrng2  20618  subrngint  20620  rhmimasubrng  20626  rng2idl0  21344  rng2idlsubg0  21347  rngqiprnglinlem2  21369  rngqiprng  21373  rng2idl1cntr  21382