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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 13280 | . . 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 5215 (class class class)co 5872 Basecbs 12454 βΎs cress 12455 1rcur 13073 Ringcrg 13110 SubRingcsubrg 13276 |
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 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-cnex 7899 ax-resscn 7900 ax-1re 7902 ax-addrcl 7905 |
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 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-fv 5223 df-ov 5875 df-inn 8916 df-ndx 12457 df-slot 12458 df-base 12460 df-subrg 13278 |
This theorem is referenced by: subrgsubg 13286 subrg1 13290 subrgmcl 13292 subrgsubm 13293 subrgdvds 13294 subrguss 13295 subrginv 13296 subrgdv 13297 subrgunit 13298 subrgugrp 13299 subrgintm 13302 subsubrg 13304 subrgpropd 13307 |
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