![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > subrg1cl | Structured version Visualization version GIF version |
Description: A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
subrg1cl.a | β’ 1 = (1rβπ ) |
Ref | Expression |
---|---|
subrg1cl | β’ (π΄ β (SubRingβπ ) β 1 β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
2 | subrg1cl.a | . . . 4 β’ 1 = (1rβπ ) | |
3 | 1, 2 | issubrg 20323 | . . 3 β’ (π΄ β (SubRingβπ ) β ((π β Ring β§ (π βΎs π΄) β Ring) β§ (π΄ β (Baseβπ ) β§ 1 β π΄))) |
4 | 3 | simprbi 497 | . 2 β’ (π΄ β (SubRingβπ ) β (π΄ β (Baseβπ ) β§ 1 β π΄)) |
5 | 4 | simprd 496 | 1 β’ (π΄ β (SubRingβπ ) β 1 β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β wss 3948 βcfv 6543 (class class class)co 7411 Basecbs 17146 βΎs cress 17175 1rcur 20006 Ringcrg 20058 SubRingcsubrg 20319 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7414 df-subrg 20321 |
This theorem is referenced by: subrg1 20333 subrgsubm 20336 issubrg2 20343 subrgint 20346 subsubrg 20349 primefld1cl 20427 zsssubrg 21009 issubassa2 21452 subrgpsr 21545 mplassa 21587 mplbas2 21603 ply1assa 21729 isclmp 24620 taylply2 25887 0ringsubrg 32420 subrgchr 32427 fldgensdrg 32445 primefldgen1 32452 asclply1subcl 32705 drgextlsp 32739 evls1fldgencl 32804 evls1maprhm 32819 ply1annnr 32824 algextdeglem4 32836 evlsmaprhm 41224 cnsrexpcl 41989 rngunsnply 41997 |
Copyright terms: Public domain | W3C validator |