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Mirrors > Home > MPE Home > Th. List > Mathboxes > subrgacs | Structured version Visualization version GIF version |
Description: Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Ref | Expression |
---|---|
subrgacs.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
subrgacs | ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | 1 | issubrg3 18808 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubRing‘𝑅) ↔ (𝑥 ∈ (SubGrp‘𝑅) ∧ 𝑥 ∈ (SubMnd‘(mulGrp‘𝑅))))) |
3 | elin 3796 | . . . 4 ⊢ (𝑥 ∈ ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅))) ↔ (𝑥 ∈ (SubGrp‘𝑅) ∧ 𝑥 ∈ (SubMnd‘(mulGrp‘𝑅)))) | |
4 | 2, 3 | syl6bbr 278 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubRing‘𝑅) ↔ 𝑥 ∈ ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅))))) |
5 | 4 | eqrdv 2620 | . 2 ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) = ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅)))) |
6 | subrgacs.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
7 | fvex 6201 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
8 | 6, 7 | eqeltri 2697 | . . . 4 ⊢ 𝐵 ∈ V |
9 | mreacs 16319 | . . . 4 ⊢ (𝐵 ∈ V → (ACS‘𝐵) ∈ (Moore‘𝒫 𝐵)) | |
10 | 8, 9 | mp1i 13 | . . 3 ⊢ (𝑅 ∈ Ring → (ACS‘𝐵) ∈ (Moore‘𝒫 𝐵)) |
11 | ringgrp 18552 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
12 | 6 | subgacs 17629 | . . . 4 ⊢ (𝑅 ∈ Grp → (SubGrp‘𝑅) ∈ (ACS‘𝐵)) |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → (SubGrp‘𝑅) ∈ (ACS‘𝐵)) |
14 | 1 | ringmgp 18553 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
15 | 1, 6 | mgpbas 18495 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
16 | 15 | submacs 17365 | . . . 4 ⊢ ((mulGrp‘𝑅) ∈ Mnd → (SubMnd‘(mulGrp‘𝑅)) ∈ (ACS‘𝐵)) |
17 | 14, 16 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → (SubMnd‘(mulGrp‘𝑅)) ∈ (ACS‘𝐵)) |
18 | mreincl 16259 | . . 3 ⊢ (((ACS‘𝐵) ∈ (Moore‘𝒫 𝐵) ∧ (SubGrp‘𝑅) ∈ (ACS‘𝐵) ∧ (SubMnd‘(mulGrp‘𝑅)) ∈ (ACS‘𝐵)) → ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅))) ∈ (ACS‘𝐵)) | |
19 | 10, 13, 17, 18 | syl3anc 1326 | . 2 ⊢ (𝑅 ∈ Ring → ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅))) ∈ (ACS‘𝐵)) |
20 | 5, 19 | eqeltrd 2701 | 1 ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 𝒫 cpw 4158 ‘cfv 5888 Basecbs 15857 Moorecmre 16242 ACScacs 16245 Mndcmnd 17294 SubMndcsubmnd 17334 Grpcgrp 17422 SubGrpcsubg 17588 mulGrpcmgp 18489 Ringcrg 18547 SubRingcsubrg 18776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-subg 17591 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 |
This theorem is referenced by: sdrgacs 37771 |
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