![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > subrngsubg | 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 13699 | . . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng) | |
2 | rnggrp 13434 | . . 3 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Grp) |
4 | eqid 2193 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | 4 | subrngss 13696 | . 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
6 | eqid 2193 | . . . 4 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
7 | 6 | subrngrng 13698 | . . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑅 ↾s 𝐴) ∈ Rng) |
8 | rnggrp 13434 | . . 3 ⊢ ((𝑅 ↾s 𝐴) ∈ Rng → (𝑅 ↾s 𝐴) ∈ Grp) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑅 ↾s 𝐴) ∈ Grp) |
10 | 4 | issubg 13243 | . 2 ⊢ (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ Grp)) |
11 | 3, 5, 9, 10 | syl3anbrc 1183 | 1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2164 ⊆ wss 3153 ‘cfv 5254 (class class class)co 5918 Basecbs 12618 ↾s cress 12619 Grpcgrp 13072 SubGrpcsubg 13237 Rngcrng 13428 SubRngcsubrng 13693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-cnex 7963 ax-resscn 7964 ax-1re 7966 ax-addrcl 7969 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-fv 5262 df-ov 5921 df-inn 8983 df-2 9041 df-3 9042 df-ndx 12621 df-slot 12622 df-base 12624 df-plusg 12708 df-mulr 12709 df-subg 13240 df-abl 13357 df-rng 13429 df-subrng 13694 |
This theorem is referenced by: subrngringnsg 13701 subrngbas 13702 subrng0 13703 subrngacl 13704 issubrng2 13706 subrngintm 13708 rng2idl0 14015 rng2idlsubg0 14018 |
Copyright terms: Public domain | W3C validator |