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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 18995 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
2 | ringgrp 18760 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Grp) |
4 | eqid 2771 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | 4 | subrgss 18991 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
6 | eqid 2771 | . . . 4 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
7 | 6 | subrgring 18993 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ Ring) |
8 | ringgrp 18760 | . . 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 17802 | . 2 ⊢ (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ Grp)) |
11 | 3, 5, 9, 10 | syl3anbrc 1428 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2145 ⊆ wss 3723 ‘cfv 6031 (class class class)co 6793 Basecbs 16064 ↾s cress 16065 Grpcgrp 17630 SubGrpcsubg 17796 Ringcrg 18755 SubRingcsubrg 18986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fv 6039 df-ov 6796 df-subg 17799 df-ring 18757 df-subrg 18988 |
This theorem is referenced by: subrg0 18997 subrgbas 18999 subrgacl 19001 issubrg2 19010 subrgint 19012 resrhm 19019 rhmima 19021 abvres 19049 issubassa2 19560 resspsrmul 19632 subrgpsr 19634 mplbas2 19685 gsumply1subr 19819 zsssubrg 20019 gzrngunitlem 20026 zringlpirlem1 20047 zringcyg 20054 prmirred 20058 zndvds 20113 resubgval 20172 subrgnrg 22697 sranlm 22708 clmsub 23099 clmneg 23100 clmabs 23102 clmsubcl 23105 isncvsngp 23168 cphsqrtcl3 23206 tchcph 23255 plypf1 24188 dvply2g 24260 taylply2 24342 circgrp 24519 circsubm 24520 rzgrp 24521 jensenlem2 24935 amgmlem 24937 lgseisenlem4 25324 qrng0 25531 qrngneg 25533 subrgchr 30134 nn0archi 30183 rezh 30355 qqhcn 30375 qqhucn 30376 fsumcnsrcl 38262 cnsrplycl 38263 rngunsnply 38269 zringsubgval 42711 amgmwlem 43079 |
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