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Mirrors > Home > MPE Home > Th. List > subrgring | Structured version Visualization version GIF version |
Description: A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
subrgring.1 | β’ π = (π βΎs π΄) |
Ref | Expression |
---|---|
subrgring | β’ (π΄ β (SubRingβπ ) β π β Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgring.1 | . 2 β’ π = (π βΎs π΄) | |
2 | eqid 2731 | . . . . 5 β’ (Baseβπ ) = (Baseβπ ) | |
3 | eqid 2731 | . . . . 5 β’ (1rβπ ) = (1rβπ ) | |
4 | 2, 3 | issubrg 20462 | . . . 4 β’ (π΄ β (SubRingβπ ) β ((π β Ring β§ (π βΎs π΄) β Ring) β§ (π΄ β (Baseβπ ) β§ (1rβπ ) β π΄))) |
5 | 4 | simplbi 497 | . . 3 β’ (π΄ β (SubRingβπ ) β (π β Ring β§ (π βΎs π΄) β Ring)) |
6 | 5 | simprd 495 | . 2 β’ (π΄ β (SubRingβπ ) β (π βΎs π΄) β Ring) |
7 | 1, 6 | eqeltrid 2836 | 1 β’ (π΄ β (SubRingβπ ) β π β Ring) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 β wss 3948 βcfv 6543 (class class class)co 7412 Basecbs 17149 βΎs cress 17178 1rcur 20076 Ringcrg 20128 SubRingcsubrg 20458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 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 7415 df-subrg 20460 |
This theorem is referenced by: subrgcrng 20466 subrgsubg 20468 subrg1 20473 subrgsubm 20476 subrguss 20478 subrginv 20479 subrgunit 20481 subrgugrp 20482 subrgnzr 20485 subsubrg 20489 resrhm 20492 resrhm2b 20493 issubdrg 20545 imadrhmcl 20557 subdrgint 20563 abvres 20591 sralmod 20955 ring2idlqus 21069 gzrngunitlem 21211 gzrngunit 21212 issubassa3 21640 subrgpsr 21759 mplring 21798 subrgmvrf 21809 subrgascl 21847 subrgasclcl 21848 evlssca 21872 evlsvar 21873 evlsgsumadd 21874 evlsvarpw 21877 mpfconst 21884 mpfproj 21885 mpfsubrg 21886 gsumply1subr 21977 ply1ring 21991 evls1sca 22063 evls1gsumadd 22064 evls1varpw 22067 dmatcrng 22225 scmatcrng 22244 scmatsgrp1 22245 scmatsrng1 22246 scmatmhm 22257 scmatrhm 22258 m2cpmrhm 22469 isclmp 24845 reefgim 26199 amgmlem 26731 cntrcrng 32485 evls1varpwval 32920 evls1fpws 32921 evls1addd 32923 evls1muld 32924 ressply10g 32931 asclply1subcl 32935 evls1fldgencl 33034 0ringirng 33043 evls1maplmhm 33050 ply1annnr 33054 irngnminplynz 33061 algextdeglem6 33068 imacrhmcl 41394 evlsscaval 41439 evlsvarval 41440 evlsbagval 41441 evlsmaprhm 41445 amgmwlem 47937 |
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