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| Mirrors > Home > MPE Home > Th. List > subrngringnsg | Structured version Visualization version GIF version | ||
| Description: A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| subrngringnsg | ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | subrngsubg 20553 | . 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) | |
| 2 | subrngrcl 20552 | . . . . . . . . 9 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng) | |
| 3 | rngabl 20153 | . . . . . . . . 9 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel) | |
| 4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Abel) | 
| 5 | 4 | 3anim1i 1152 | . . . . . . 7 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) | 
| 6 | 5 | 3expb 1120 | . . . . . 6 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) | 
| 7 | eqid 2736 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 8 | eqid 2736 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
| 9 | 7, 8 | ablcom 19818 | . . . . . 6 ⊢ ((𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = (𝑦(+g‘𝑅)𝑥)) | 
| 10 | 6, 9 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑦(+g‘𝑅)𝑥)) | 
| 11 | 10 | eleq1d 2825 | . . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(+g‘𝑅)𝑦) ∈ 𝐴 ↔ (𝑦(+g‘𝑅)𝑥) ∈ 𝐴)) | 
| 12 | 11 | biimpd 229 | . . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(+g‘𝑅)𝑦) ∈ 𝐴 → (𝑦(+g‘𝑅)𝑥) ∈ 𝐴)) | 
| 13 | 12 | ralrimivva 3201 | . 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) ∈ 𝐴 → (𝑦(+g‘𝑅)𝑥) ∈ 𝐴)) | 
| 14 | 7, 8 | isnsg2 19175 | . 2 ⊢ (𝐴 ∈ (NrmSGrp‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) ∈ 𝐴 → (𝑦(+g‘𝑅)𝑥) ∈ 𝐴))) | 
| 15 | 1, 13, 14 | sylanbrc 583 | 1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 SubGrpcsubg 19139 NrmSGrpcnsg 19140 Abelcabl 19800 Rngcrng 20150 SubRngcsubrng 20546 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-subg 19142 df-nsg 19143 df-cmn 19801 df-abl 19802 df-rng 20151 df-subrng 20547 | 
| This theorem is referenced by: rng2idlnsg 21277 rng2idlsubgnsg 21280 | 
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