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| Mirrors > Home > MPE Home > Th. List > subrgss | Structured version Visualization version GIF version | ||
| Description: A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.) | 
| Ref | Expression | 
|---|---|
| subrgss.1 | ⊢ 𝐵 = (Base‘𝑅) | 
| Ref | Expression | 
|---|---|
| subrgss | ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | subrgss.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
| 2 | eqid 2737 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 3 | 1, 2 | issubrg 20571 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ (1r‘𝑅) ∈ 𝐴))) | 
| 4 | 3 | simprbi 496 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ⊆ 𝐵 ∧ (1r‘𝑅) ∈ 𝐴)) | 
| 5 | 4 | simpld 494 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 1rcur 20178 Ringcrg 20230 SubRingcsubrg 20569 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-subrg 20570 | 
| This theorem is referenced by: subrgsubg 20577 subrg1 20582 subrgsubm 20585 subrgdvds 20586 subrguss 20587 subrginv 20588 subrgdv 20589 subrgmre 20597 subsubrg 20598 issubdrg 20781 sdrgss 20794 sdrgacs 20802 subdrgint 20804 abvres 20832 sralmod 21194 cnsubrg 21445 issubassa3 21886 sraassab 21888 sraassa 21889 sraassaOLD 21890 aspid 21895 issubassa2 21912 resspsrbas 21994 resspsradd 21995 resspsrmul 21996 resspsrvsca 21997 mplassa 22042 ressmplbas2 22045 subrgascl 22090 subrgasclcl 22091 mplind 22094 evlsval2 22111 evlssca 22113 evlsscasrng 22121 mpfconst 22125 mpff 22128 mpfaddcl 22129 mpfmulcl 22130 mpfind 22131 ply1assa 22201 evls1val 22324 evls1rhm 22326 evls1sca 22327 evls1scasrng 22343 pf1f 22354 evls1fpws 22373 evls1vsca 22377 asclply1subcl 22378 evls1maplmhm 22381 sranlm 24705 clmsscn 25112 cphreccllem 25212 cphdivcl 25216 cphabscl 25219 cphsqrtcl2 25220 cphsqrtcl3 25221 cphipcl 25225 4cphipval2 25276 resscdrg 25392 srabn 25394 plypf1 26251 dvply2g 26326 dvply2gOLD 26327 taylply2 26409 taylply2OLD 26410 elrgspn 33250 elrgspnsubrunlem1 33251 elrgspnsubrunlem2 33252 elrgspnsubrun 33253 0ringsubrg 33255 subrdom 33288 fldgenssp 33320 idlinsubrg 33459 ressasclcl 33596 sralvec 33636 lsssra 33639 drgext0g 33640 drgextvsca 33641 drgext0gsca 33642 drgextsubrg 33643 drgextlsp 33644 drgextgsum 33645 fedgmullem1 33680 fedgmullem2 33681 fedgmul 33682 extdggt0 33708 fldexttr 33709 extdg1id 33716 fldextrspunlsp 33724 fldextrspunlem1 33725 fldextrspunfld 33726 elirng 33736 irngss 33737 0ringirng 33739 ply1annnr 33746 imacrhmcl 42524 evlsval3 42569 evlsvvval 42573 evlsbagval 42576 evlsevl 42581 evlsmhpvvval 42605 mhphf 42607 mhphf2 42608 mhphf3 42609 cnsrexpcl 43177 fsumcnsrcl 43178 cnsrplycl 43179 rgspnid 43180 rngunsnply 43181 | 
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