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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 19469 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
2 | ringgrp 19231 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Grp) |
4 | eqid 2818 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | 4 | subrgss 19465 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅)) |
6 | eqid 2818 | . . . 4 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
7 | 6 | subrgring 19467 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ Ring) |
8 | ringgrp 19231 | . . 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 18217 | . 2 ⊢ (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝑅) ∧ (𝑅 ↾s 𝐴) ∈ Grp)) |
11 | 3, 5, 9, 10 | syl3anbrc 1335 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 ↾s cress 16472 Grpcgrp 18041 SubGrpcsubg 18211 Ringcrg 19226 SubRingcsubrg 19460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-subg 18214 df-ring 19228 df-subrg 19462 |
This theorem is referenced by: subrg0 19471 subrgbas 19473 subrgacl 19475 issubrg2 19484 subrgint 19486 resrhm 19493 rhmima 19495 subdrgint 19511 primefld0cl 19514 abvres 19539 issubassa2 20049 resspsrmul 20125 subrgpsr 20127 mplbas2 20179 gsumply1subr 20330 zsssubrg 20531 gzrngunitlem 20538 zringlpirlem1 20559 zringcyg 20566 prmirred 20570 zndvds 20624 resubgval 20681 rzgrp 20695 subrgnrg 23209 sranlm 23220 clmsub 23611 clmneg 23612 clmabs 23614 clmsubcl 23617 isncvsngp 23680 cphsqrtcl3 23718 tcphcph 23767 plypf1 24729 dvply2g 24801 taylply2 24883 circgrp 25063 circsubm 25064 jensenlem2 25492 amgmlem 25494 lgseisenlem4 25881 qrng0 26124 qrngneg 26126 subrgchr 30792 nn0archi 30843 drgext0gsca 30893 fedgmullem1 30924 fedgmullem2 30925 rezh 31111 qqhcn 31131 qqhucn 31132 selvval2lem4 39014 fsumcnsrcl 39644 cnsrplycl 39645 rngunsnply 39651 zringsubgval 44377 amgmwlem 44831 |
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