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 20477

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 20476	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2		rnggrp 20086	. . 3 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Grp)
4		eqid 2733	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
5	4	subrngss 20473	. 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
6		eqid 2733	. . . 4 ⊢ (𝑅 ↾_s 𝐴) = (𝑅 ↾_s 𝐴)
7	6	subrngrng 20475	. . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑅 ↾_s 𝐴) ∈ Rng)
8		rnggrp 20086	. . 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 19049	. 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 2113 ⊆ wss 3899 ‘cfv 6489 (class class class)co 7355 Basecbs 17130 ↾_s cress 17151 Grpcgrp 18856 SubGrpcsubg 19043 Rngcrng 20080 SubRngcsubrng 20470

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374

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 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-sbc 3739 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fv 6497 df-ov 7358 df-subg 19046 df-abl 19705 df-rng 20081 df-subrng 20471

This theorem is referenced by: subrngringnsg 20478 subrngbas 20479 subrng0 20480 subrngacl 20481 issubrng2 20483 subrngint 20485 rhmimasubrng 20491 rng2idl0 21214 rng2idlsubg0 21217 rngqiprnglinlem2 21239 rngqiprng 21243 rng2idl1cntr 21252

W3C validator