![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 20357 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) | |
2 | subrgbas.b | . . 3 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
3 | 2 | subgbas 19004 | . 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 1542 ∈ wcel 2107 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 ↾s cress 17169 SubGrpcsubg 18994 SubRingcsubrg 20347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-1cn 11164 ax-addcl 11166 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-nn 12209 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-subg 18997 df-ring 20049 df-subrg 20349 |
This theorem is referenced by: subrg1 20361 subrgmcl 20363 subrgdvds 20365 subrguss 20366 subrginv 20367 subrgdv 20368 subrgunit 20369 issubdrg 20377 subsubrg 20378 abvres 20435 qsssubdrg 20989 gzrngunitlem 20995 gzrngunit 20996 issubassa3 21404 sraassa 21406 resspsrbas 21517 resspsradd 21518 resspsrmul 21519 resspsrvsca 21520 subrgpsr 21521 subrgascl 21609 subrgasclcl 21610 dmatcrng 21986 scmatcrng 22005 scmatstrbas 22010 sranlm 24183 isclmi 24575 plypf1 25708 sdrgdvcl 32366 sdrginvcl 32367 idlinsubrg 32507 evlsvvval 41085 evlsscaval 41086 evlsbagval 41088 evlsevl 41093 evlsmhpvvval 41117 mhphf 41119 |
Copyright terms: Public domain | W3C validator |