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Theorem subrngsubg 20452

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 20451	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2		rnggrp 20063	. . 3 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Grp)
4		eqid 2726	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
5	4	subrngss 20448	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
6		eqid 2726	. . . 4 ⊢ (𝑅 ↾_s 𝐴) = (𝑅 ↾_s 𝐴)
7	6	subrngrng 20450	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑅 ↾_s 𝐴) ∈ Rng)
8		rnggrp 20063	. . 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 19053	. 2 ⊢ (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝑅) ∧ (𝑅 ↾_s 𝐴) ∈ Grp))
11	3, 5, 9, 10	syl3anbrc 1340	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2098 ⊆ wss 3943 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 ↾_s cress 17182 Grpcgrp 18863 SubGrpcsubg 19047 Rngcrng 20057 SubRngcsubrng 20445

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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fv 6545 df-ov 7408 df-subg 19050 df-abl 19703 df-rng 20058 df-subrng 20446

This theorem is referenced by: subrngringnsg 20453 subrngbas 20454 subrng0 20455 subrngacl 20456 issubrng2 20458 subrngint 20460 rhmimasubrng 20466 rng2idl0 21124 rng2idlsubg0 21127 rngqiprnglinlem2 21145 rngqiprng 21149 rng2idl1cntr 21158

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