![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > subrgrcl | GIF version |
Description: Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.) |
Ref | Expression |
---|---|
subrgrcl | β’ (π΄ β (SubRingβπ ) β π β Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
2 | eqid 2177 | . . . 4 β’ (1rβπ ) = (1rβπ ) | |
3 | 1, 2 | issubrg 13302 | . . 3 β’ (π΄ β (SubRingβπ ) β ((π β Ring β§ (π βΎs π΄) β Ring) β§ (π΄ β (Baseβπ ) β§ (1rβπ ) β π΄))) |
4 | 3 | simplbi 274 | . 2 β’ (π΄ β (SubRingβπ ) β (π β Ring β§ (π βΎs π΄) β Ring)) |
5 | 4 | simpld 112 | 1 β’ (π΄ β (SubRingβπ ) β π β Ring) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wcel 2148 β wss 3129 βcfv 5216 (class class class)co 5874 Basecbs 12456 βΎs cress 12457 1rcur 13095 Ringcrg 13132 SubRingcsubrg 13298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-fv 5224 df-ov 5877 df-inn 8918 df-ndx 12459 df-slot 12460 df-base 12462 df-subrg 13300 |
This theorem is referenced by: subrgsubg 13308 subrg1 13312 subrgmcl 13314 subrgsubm 13315 subrgdvds 13316 subrguss 13317 subrginv 13318 subrgdv 13319 subrgunit 13320 subrgugrp 13321 subrgintm 13324 subsubrg 13326 subrgpropd 13329 |
Copyright terms: Public domain | W3C validator |