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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 20447 | . . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng) | |
2 | rnggrp 20059 | . . 3 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Grp) |
4 | eqid 2731 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | 4 | subrngss 20444 | . 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
6 | eqid 2731 | . . . 4 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
7 | 6 | subrngrng 20446 | . . 3 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑅 ↾s 𝐴) ∈ Rng) |
8 | rnggrp 20059 | . . 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 1342 | 1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3948 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 ↾s cress 17180 Grpcgrp 18861 SubGrpcsubg 19043 Rngcrng 20053 SubRngcsubrng 20441 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-subg 19046 df-abl 19699 df-rng 20054 df-subrng 20442 |
This theorem is referenced by: subrngringnsg 20449 subrngbas 20450 subrng0 20451 subrngacl 20452 issubrng2 20454 subrngint 20456 rhmimasubrng 20462 rng2idl0 21120 rng2idlsubg0 21123 rngqiprnglinlem2 21141 rngqiprng 21145 rng2idl1cntr 21154 |
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