![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > subrngsubg | Structured version Visualization version GIF version |
Description: A subring is a subgroup. (Contributed by AV, 14-Feb-2025.) |
Ref | Expression |
---|---|
subrngsubg | ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) |
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 |
Copyright terms: Public domain | W3C validator |