subrngsubg - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > subrngsubg		Structured version Visualization version GIF version

Theorem subrngsubg 20517

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 20516	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2		rnggrp 20123	. . 3 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Grp)
4		eqid 2736	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
5	4	subrngss 20513	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
6		eqid 2736	. . . 4 ⊢ (𝑅 ↾_s 𝐴) = (𝑅 ↾_s 𝐴)
7	6	subrngrng 20515	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑅 ↾_s 𝐴) ∈ Rng)
8		rnggrp 20123	. . 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 19114	. 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 2109 ⊆ wss 3931 ‘cfv 6536 (class class class)co 7410 Basecbs 17233 ↾_s cress 17256 Grpcgrp 18921 SubGrpcsubg 19108 Rngcrng 20117 SubRngcsubrng 20510

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fv 6544 df-ov 7413 df-subg 19111 df-abl 19769 df-rng 20118 df-subrng 20511

This theorem is referenced by: subrngringnsg 20518 subrngbas 20519 subrng0 20520 subrngacl 20521 issubrng2 20523 subrngint 20525 rhmimasubrng 20531 rng2idl0 21233 rng2idlsubg0 21236 rngqiprnglinlem2 21258 rngqiprng 21262 rng2idl1cntr 21271

W3C validator