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Mirrors > Home > MPE Home > Th. List > subrgring | Structured version Visualization version Unicode version |
Description: A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
subrgring.1 | ↾s |
Ref | Expression |
---|---|
subrgring | SubRing |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgring.1 | . 2 ↾s | |
2 | eqid 2622 | . . . . 5 | |
3 | eqid 2622 | . . . . 5 | |
4 | 2, 3 | issubrg 18780 | . . . 4 SubRing ↾s |
5 | 4 | simplbi 476 | . . 3 SubRing ↾s |
6 | 5 | simprd 479 | . 2 SubRing ↾s |
7 | 1, 6 | syl5eqel 2705 | 1 SubRing |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wss 3574 cfv 5888 (class class class)co 6650 cbs 15857 ↾s cress 15858 cur 18501 crg 18547 SubRingcsubrg 18776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-subrg 18778 |
This theorem is referenced by: subrgcrng 18784 subrgsubg 18786 subrg1 18790 subrgmcl 18792 subrgsubm 18793 subrguss 18795 subrginv 18796 subrgunit 18798 subrgugrp 18799 issubdrg 18805 subsubrg 18806 resrhm 18809 abvres 18839 sralmod 19187 subrgnzr 19268 issubassa 19324 subrgpsr 19419 mplring 19452 subrgmvrf 19462 subrgascl 19498 subrgasclcl 19499 evlssca 19522 evlsvar 19523 mpfconst 19530 mpfproj 19531 mpfsubrg 19532 gsumply1subr 19604 ply1ring 19618 evls1sca 19688 evls1gsumadd 19689 evls1varpw 19691 gzrngunitlem 19811 gzrngunit 19812 dmatcrng 20308 scmatcrng 20327 scmatsgrp1 20328 scmatsrng1 20329 scmatmhm 20340 scmatrhm 20341 scmatrngiso 20342 m2cpmrhm 20551 isclmp 22897 reefgim 24204 amgmlem 24716 amgmwlem 42548 |
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