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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 2821 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | 1 | issubrg3 19563 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubRing‘𝑅) ↔ (𝑥 ∈ (SubGrp‘𝑅) ∧ 𝑥 ∈ (SubMnd‘(mulGrp‘𝑅))))) |
3 | elin 4169 | . . . 4 ⊢ (𝑥 ∈ ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅))) ↔ (𝑥 ∈ (SubGrp‘𝑅) ∧ 𝑥 ∈ (SubMnd‘(mulGrp‘𝑅)))) | |
4 | 2, 3 | syl6bbr 291 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑥 ∈ (SubRing‘𝑅) ↔ 𝑥 ∈ ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅))))) |
5 | 4 | eqrdv 2819 | . 2 ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) = ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅)))) |
6 | subrgacs.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
7 | 6 | fvexi 6684 | . . . 4 ⊢ 𝐵 ∈ V |
8 | mreacs 16929 | . . . 4 ⊢ (𝐵 ∈ V → (ACS‘𝐵) ∈ (Moore‘𝒫 𝐵)) | |
9 | 7, 8 | mp1i 13 | . . 3 ⊢ (𝑅 ∈ Ring → (ACS‘𝐵) ∈ (Moore‘𝒫 𝐵)) |
10 | ringgrp 19302 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
11 | 6 | subgacs 18313 | . . . 4 ⊢ (𝑅 ∈ Grp → (SubGrp‘𝑅) ∈ (ACS‘𝐵)) |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → (SubGrp‘𝑅) ∈ (ACS‘𝐵)) |
13 | 1 | ringmgp 19303 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
14 | 1, 6 | mgpbas 19245 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
15 | 14 | submacs 17991 | . . . 4 ⊢ ((mulGrp‘𝑅) ∈ Mnd → (SubMnd‘(mulGrp‘𝑅)) ∈ (ACS‘𝐵)) |
16 | 13, 15 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → (SubMnd‘(mulGrp‘𝑅)) ∈ (ACS‘𝐵)) |
17 | mreincl 16870 | . . 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 1367 | . 2 ⊢ (𝑅 ∈ Ring → ((SubGrp‘𝑅) ∩ (SubMnd‘(mulGrp‘𝑅))) ∈ (ACS‘𝐵)) |
19 | 5, 18 | eqeltrd 2913 | 1 ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 𝒫 cpw 4539 ‘cfv 6355 Basecbs 16483 Moorecmre 16853 ACScacs 16856 Mndcmnd 17911 SubMndcsubmnd 17955 Grpcgrp 18103 SubGrpcsubg 18273 mulGrpcmgp 19239 Ringcrg 19297 SubRingcsubrg 19531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-subg 18276 df-mgp 19240 df-ur 19252 df-ring 19299 df-subrg 19533 |
This theorem is referenced by: sdrgacs 19580 |
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