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Theorem subrngsubg 20493

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 20492	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2		rnggrp 20102	. . 3 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Grp)
4		eqid 2725	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
5	4	subrngss 20489	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
6		eqid 2725	. . . 4 ⊢ (𝑅 ↾_s 𝐴) = (𝑅 ↾_s 𝐴)
7	6	subrngrng 20491	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑅 ↾_s 𝐴) ∈ Rng)
8		rnggrp 20102	. . 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 19085	. 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 3939 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 ↾_s cress 17208 Grpcgrp 18894 SubGrpcsubg 19079 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 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423

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-sbc 3769 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7419 df-subg 19082 df-abl 19742 df-rng 20097 df-subrng 20487

This theorem is referenced by: subrngringnsg 20494 subrngbas 20495 subrng0 20496 subrngacl 20497 issubrng2 20499 subrngint 20501 rhmimasubrng 20507 rng2idl0 21165 rng2idlsubg0 21168 rngqiprnglinlem2 21186 rngqiprng 21190 rng2idl1cntr 21199

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