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Mirrors > Home > MPE Home > Th. List > subrgbas | Structured version Visualization version GIF version |
Description: Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
subrgbas.b | ⊢ 𝑆 = (𝑅 ↾s 𝐴) |
Ref | Expression |
---|---|
subrgbas | ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgsubg 19534 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) | |
2 | subrgbas.b | . . 3 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
3 | 2 | subgbas 18275 | . 2 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 = (Base‘𝑆)) |
4 | 1, 3 | syl 17 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 ↾s cress 16476 SubGrpcsubg 18265 SubRingcsubrg 19524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-subg 18268 df-ring 19292 df-subrg 19526 |
This theorem is referenced by: subrg1 19538 subrgmcl 19540 subrgdvds 19542 subrguss 19543 subrginv 19544 subrgdv 19545 subrgunit 19546 issubdrg 19553 subsubrg 19554 abvres 19603 qsssubdrg 20150 gzrngunitlem 20156 gzrngunit 20157 issubassa3 20554 sraassa 20556 resspsrbas 20653 resspsradd 20654 resspsrmul 20655 resspsrvsca 20656 subrgpsr 20657 subrgascl 20737 subrgasclcl 20738 dmatcrng 21107 scmatcrng 21126 scmatstrbas 21131 sranlm 23290 isclmi 23682 plypf1 24809 idlinsubrg 31016 |
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