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Mirrors > Home > MPE Home > Th. List > subrgsubg | Structured version Visualization version GIF version |
Description: A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.) |
Ref | Expression |
---|---|
subrgsubg | ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgrcl 19805 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
2 | ringgrp 19567 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Grp) |
4 | eqid 2737 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | 4 | subrgss 19801 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
6 | eqid 2737 | . . . 4 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
7 | 6 | subrgring 19803 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ Ring) |
8 | ringgrp 19567 | . . 3 ⊢ ((𝑅 ↾s 𝐴) ∈ Ring → (𝑅 ↾s 𝐴) ∈ Grp) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ Grp) |
10 | 4 | issubg 18543 | . 2 ⊢ (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ Grp)) |
11 | 3, 5, 9, 10 | syl3anbrc 1345 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ⊆ wss 3866 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 ↾s cress 16784 Grpcgrp 18365 SubGrpcsubg 18537 Ringcrg 19562 SubRingcsubrg 19796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fv 6388 df-ov 7216 df-subg 18540 df-ring 19564 df-subrg 19798 |
This theorem is referenced by: subrg0 19807 subrgbas 19809 subrgacl 19811 issubrg2 19820 subrgint 19822 resrhm 19829 rhmima 19831 subdrgint 19847 primefld0cl 19850 abvres 19875 zsssubrg 20421 gzrngunitlem 20428 zringlpirlem1 20449 zringcyg 20456 zringsubgval 20457 prmirred 20461 zndvds 20514 resubgval 20571 rzgrp 20585 issubassa2 20852 resspsrmul 20942 subrgpsr 20944 mplbas2 20999 gsumply1subr 21155 subrgnrg 23571 sranlm 23582 clmsub 23977 clmneg 23978 clmabs 23980 clmsubcl 23983 isncvsngp 24046 cphsqrtcl3 24084 tcphcph 24134 plypf1 25106 dvply2g 25178 taylply2 25260 circgrp 25441 circsubm 25442 jensenlem2 25870 amgmlem 25872 lgseisenlem4 26259 qrng0 26502 qrngneg 26504 subrgchr 31210 nn0archi 31261 idlinsubrg 31322 drgext0gsca 31393 fedgmullem1 31424 fedgmullem2 31425 rezh 31633 qqhcn 31653 qqhucn 31654 selvval2lem4 39941 fsumcnsrcl 40694 cnsrplycl 40695 rngunsnply 40701 amgmwlem 46177 |
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