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Mirrors > Home > MPE Home > Th. List > 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 2728 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | 1 | issubrg3 20538 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubRing‘𝑅) ↔ (𝑥 ∈ (SubGrp‘𝑅) ∧ 𝑥 ∈ (SubMnd‘(mulGrp‘𝑅))))) |
3 | elin 3963 | . . . 4 ⊢ (𝑥 ∈ ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅))) ↔ (𝑥 ∈ (SubGrp‘𝑅) ∧ 𝑥 ∈ (SubMnd‘(mulGrp‘𝑅)))) | |
4 | 2, 3 | bitr4di 289 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubRing‘𝑅) ↔ 𝑥 ∈ ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅))))) |
5 | 4 | eqrdv 2726 | . 2 ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) = ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅)))) |
6 | subrgacs.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
7 | 6 | fvexi 6911 | . . . 4 ⊢ 𝐵 ∈ V |
8 | mreacs 17637 | . . . 4 ⊢ (𝐵 ∈ V → (ACS‘𝐵) ∈ (Moore‘𝒫 𝐵)) | |
9 | 7, 8 | mp1i 13 | . . 3 ⊢ (𝑅 ∈ Ring → (ACS‘𝐵) ∈ (Moore‘𝒫 𝐵)) |
10 | ringgrp 20177 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
11 | 6 | subgacs 19115 | . . . 4 ⊢ (𝑅 ∈ Grp → (SubGrp‘𝑅) ∈ (ACS‘𝐵)) |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → (SubGrp‘𝑅) ∈ (ACS‘𝐵)) |
13 | 1 | ringmgp 20178 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
14 | 1, 6 | mgpbas 20079 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
15 | 14 | submacs 18778 | . . . 4 ⊢ ((mulGrp‘𝑅) ∈ Mnd → (SubMnd‘(mulGrp‘𝑅)) ∈ (ACS‘𝐵)) |
16 | 13, 15 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → (SubMnd‘(mulGrp‘𝑅)) ∈ (ACS‘𝐵)) |
17 | mreincl 17578 | . . 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 1369 | . 2 ⊢ (𝑅 ∈ Ring → ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅))) ∈ (ACS‘𝐵)) |
19 | 5, 18 | eqeltrd 2829 | 1 ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∩ cin 3946 𝒫 cpw 4603 ‘cfv 6548 Basecbs 17179 Moorecmre 17561 ACScacs 17564 Mndcmnd 18693 SubMndcsubmnd 18738 Grpcgrp 18889 SubGrpcsubg 19074 mulGrpcmgp 20073 Ringcrg 20172 SubRingcsubrg 20505 |
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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-0g 17422 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-grp 18892 df-minusg 18893 df-subg 19077 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-subrng 20482 df-subrg 20507 |
This theorem is referenced by: sdrgacs 20688 |
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