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Theorem subrngsubg 20460

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 20459	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2		rnggrp 20069	. . 3 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Grp)
4		eqid 2730	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
5	4	subrngss 20456	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
6		eqid 2730	. . . 4 ⊢ (𝑅 ↾_s 𝐴) = (𝑅 ↾_s 𝐴)
7	6	subrngrng 20458	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑅 ↾_s 𝐴) ∈ Rng)
8		rnggrp 20069	. . 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 19031	. 2 ⊢ (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝑅) ∧ (𝑅 ↾_s 𝐴) ∈ Grp))
11	3, 5, 9, 10	syl3anbrc 1344	1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2110 ⊆ wss 3900 ‘cfv 6477 (class class class)co 7341 Basecbs 17112 ↾_s cress 17133 Grpcgrp 18838 SubGrpcsubg 19025 Rngcrng 20063 SubRngcsubrng 20453

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3394 df-v 3436 df-sbc 3740 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6433 df-fun 6479 df-fv 6485 df-ov 7344 df-subg 19028 df-abl 19688 df-rng 20064 df-subrng 20454

This theorem is referenced by: subrngringnsg 20461 subrngbas 20462 subrng0 20463 subrngacl 20464 issubrng2 20466 subrngint 20468 rhmimasubrng 20474 rng2idl0 21197 rng2idlsubg0 21200 rngqiprnglinlem2 21222 rngqiprng 21226 rng2idl1cntr 21235

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