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| 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 20551 | . . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng) | |
| 2 | rnggrp 20155 | . . 3 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Grp) |
| 4 | eqid 2737 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 5 | 4 | subrngss 20548 | . 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
| 6 | eqid 2737 | . . . 4 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
| 7 | 6 | subrngrng 20550 | . . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑅 ↾s 𝐴) ∈ Rng) |
| 8 | rnggrp 20155 | . . 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 19144 | . 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 2108 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 Grpcgrp 18951 SubGrpcsubg 19138 Rngcrng 20149 SubRngcsubrng 20545 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-subg 19141 df-abl 19801 df-rng 20150 df-subrng 20546 |
| This theorem is referenced by: subrngringnsg 20553 subrngbas 20554 subrng0 20555 subrngacl 20556 issubrng2 20558 subrngint 20560 rhmimasubrng 20566 rng2idl0 21277 rng2idlsubg0 21280 rngqiprnglinlem2 21302 rngqiprng 21306 rng2idl1cntr 21315 |
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