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| Description: A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) | 
| Ref | Expression | 
|---|---|
| subrgring.1 | ⊢ 𝑆 = (𝑅 ↾s 𝐴) | 
| Ref | Expression | 
|---|---|
| subrgring | ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | subrgring.1 | . 2 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
| 2 | eqid 2736 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 3 | eqid 2736 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 4 | 2, 3 | issubrg 20572 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴))) | 
| 5 | 4 | simplbi 497 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring)) | 
| 6 | 5 | simprd 495 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ Ring) | 
| 7 | 1, 6 | eqeltrid 2844 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 ↾s cress 17275 1rcur 20179 Ringcrg 20231 SubRingcsubrg 20570 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-subrg 20571 | 
| This theorem is referenced by: subrgcrng 20576 subrgsubg 20578 subrg1 20583 subrgsubm 20586 subrguss 20588 subrginv 20589 subrgunit 20591 subrgugrp 20592 subrgnzr 20595 subsubrg 20599 resrhm 20602 resrhm2b 20603 issubdrg 20782 imadrhmcl 20799 subdrgint 20805 abvres 20833 sralmod 21195 ring2idlqus 21320 gzrngunitlem 21451 gzrngunit 21452 issubassa3 21887 subrgpsr 21999 mplring 22040 subrgmvrf 22053 subrgascl 22091 subrgasclcl 22092 evlssca 22114 evlsvar 22115 evlsgsumadd 22116 evlsvarpw 22119 mpfconst 22126 mpfproj 22127 mpfsubrg 22128 gsumply1subr 22236 ply1ring 22250 evls1sca 22328 evls1gsumadd 22329 evls1varpw 22332 evls1varpwval 22373 evls1fpws 22374 evls1addd 22376 evls1muld 22377 asclply1subcl 22379 evls1maplmhm 22382 dmatcrng 22509 scmatcrng 22528 scmatsgrp1 22529 scmatsrng1 22530 scmatmhm 22541 scmatrhm 22542 m2cpmrhm 22753 isclmp 25131 reefgim 26495 amgmlem 27034 cntrcrng 33074 ressply10g 33593 evls1subd 33598 evls1fldgencl 33721 0ringirng 33740 ply1annnr 33747 irngnminplynz 33756 algextdeglem6 33764 imacrhmcl 42529 evlsscaval 42579 evlsvarval 42580 evlsbagval 42581 evlsmaprhm 42585 amgmwlem 49376 | 
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