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Mathbox for Stefan O'Rear |
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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 2825 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | 1 | issubrg3 19171 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubRing‘𝑅) ↔ (𝑥 ∈ (SubGrp‘𝑅) ∧ 𝑥 ∈ (SubMnd‘(mulGrp‘𝑅))))) |
3 | elin 4025 | . . . 4 ⊢ (𝑥 ∈ ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅))) ↔ (𝑥 ∈ (SubGrp‘𝑅) ∧ 𝑥 ∈ (SubMnd‘(mulGrp‘𝑅)))) | |
4 | 2, 3 | syl6bbr 281 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubRing‘𝑅) ↔ 𝑥 ∈ ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅))))) |
5 | 4 | eqrdv 2823 | . 2 ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) = ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅)))) |
6 | subrgacs.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
7 | 6 | fvexi 6451 | . . . 4 ⊢ 𝐵 ∈ V |
8 | mreacs 16678 | . . . 4 ⊢ (𝐵 ∈ V → (ACS‘𝐵) ∈ (Moore‘𝒫 𝐵)) | |
9 | 7, 8 | mp1i 13 | . . 3 ⊢ (𝑅 ∈ Ring → (ACS‘𝐵) ∈ (Moore‘𝒫 𝐵)) |
10 | ringgrp 18913 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
11 | 6 | subgacs 17987 | . . . 4 ⊢ (𝑅 ∈ Grp → (SubGrp‘𝑅) ∈ (ACS‘𝐵)) |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → (SubGrp‘𝑅) ∈ (ACS‘𝐵)) |
13 | 1 | ringmgp 18914 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
14 | 1, 6 | mgpbas 18856 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
15 | 14 | submacs 17725 | . . . 4 ⊢ ((mulGrp‘𝑅) ∈ Mnd → (SubMnd‘(mulGrp‘𝑅)) ∈ (ACS‘𝐵)) |
16 | 13, 15 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → (SubMnd‘(mulGrp‘𝑅)) ∈ (ACS‘𝐵)) |
17 | mreincl 16619 | . . 3 ⊢ (((ACS‘𝐵) ∈ (Moore‘𝒫 𝐵) ∧ (SubGrp‘𝑅) ∈ (ACS‘𝐵) ∧ (SubMnd‘(mulGrp‘𝑅)) ∈ (ACS‘𝐵)) → ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅))) ∈ (ACS‘𝐵)) | |
18 | 9, 12, 16, 17 | syl3anc 1494 | . 2 ⊢ (𝑅 ∈ Ring → ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅))) ∈ (ACS‘𝐵)) |
19 | 5, 18 | eqeltrd 2906 | 1 ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∩ cin 3797 𝒫 cpw 4380 ‘cfv 6127 Basecbs 16229 Moorecmre 16602 ACScacs 16605 Mndcmnd 17654 SubMndcsubmnd 17694 Grpcgrp 17783 SubGrpcsubg 17946 mulGrpcmgp 18850 Ringcrg 18908 SubRingcsubrg 19139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-0g 16462 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-grp 17786 df-minusg 17787 df-subg 17949 df-mgp 18851 df-ur 18863 df-ring 18910 df-subrg 19141 |
This theorem is referenced by: sdrgacs 38613 |
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