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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 20560 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
2 | ringgrp 20221 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Grp) |
4 | eqid 2726 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | 4 | subrgss 20556 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
6 | eqid 2726 | . . . 4 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
7 | 6 | subrgring 20558 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ Ring) |
8 | ringgrp 20221 | . . 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 19120 | . 2 ⊢ (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ Grp)) |
11 | 3, 5, 9, 10 | syl3anbrc 1340 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ⊆ wss 3947 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 ↾s cress 17242 Grpcgrp 18928 SubGrpcsubg 19114 Ringcrg 20216 SubRingcsubrg 20551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fv 6562 df-ov 7427 df-subg 19117 df-ring 20218 df-subrg 20553 |
This theorem is referenced by: subrg0 20563 subrgbas 20565 subrgacl 20567 issubrg2 20576 subrgint 20579 resrhm 20585 resrhm2b 20586 rhmima 20588 subdrgint 20782 primefld0cl 20785 abvres 20810 zsssubrg 21422 gzrngunitlem 21429 zringlpirlem1 21452 zringcyg 21459 zringsubgval 21460 prmirred 21464 zndvds 21547 resubgval 21605 rzgrp 21619 issubassa2 21889 resspsrmul 21985 subrgpsr 21987 mplbas2 22049 gsumply1subr 22223 subrgnrg 24681 sranlm 24692 clmsub 25098 clmneg 25099 clmabs 25101 clmsubcl 25104 isncvsngp 25168 cphsqrtcl3 25206 tcphcph 25256 plypf1 26239 dvply2g 26312 dvply2gOLD 26313 taylply2 26395 taylply2OLD 26396 circgrp 26579 circsubm 26580 jensenlem2 27016 amgmlem 27018 lgseisenlem4 27407 qrng0 27650 qrngneg 27652 subrgchr 33102 subrdom 33137 1fldgenq 33172 nn0archi 33222 idlinsubrg 33306 ressply10g 33439 ressply1invg 33441 ressply1sub 33442 evls1subd 33444 drgext0gsca 33488 fedgmullem1 33524 fedgmullem2 33525 evls1fldgencl 33556 irngss 33563 algextdeglem1 33584 algextdeglem2 33585 algextdeglem3 33586 algextdeglem4 33587 algextdeglem5 33588 rtelextdg2lem 33604 2sqr3minply 33607 rezh 33786 qqhcn 33806 qqhucn 33807 fsumcnsrcl 42827 cnsrplycl 42828 rngunsnply 42834 amgmwlem 48550 |
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