![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > subrgid | Structured version Visualization version GIF version |
Description: Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
Ref | Expression |
---|---|
subrgss.1 | β’ π΅ = (Baseβπ ) |
Ref | Expression |
---|---|
subrgid | β’ (π β Ring β π΅ β (SubRingβπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 β’ (π β Ring β π β Ring) | |
2 | subrgss.1 | . . . 4 β’ π΅ = (Baseβπ ) | |
3 | 2 | ressid 17193 | . . 3 β’ (π β Ring β (π βΎs π΅) = π ) |
4 | 3, 1 | eqeltrd 2831 | . 2 β’ (π β Ring β (π βΎs π΅) β Ring) |
5 | eqid 2730 | . . . 4 β’ (1rβπ ) = (1rβπ ) | |
6 | 2, 5 | ringidcl 20154 | . . 3 β’ (π β Ring β (1rβπ ) β π΅) |
7 | ssid 4003 | . . 3 β’ π΅ β π΅ | |
8 | 6, 7 | jctil 518 | . 2 β’ (π β Ring β (π΅ β π΅ β§ (1rβπ ) β π΅)) |
9 | 2, 5 | issubrg 20461 | . 2 β’ (π΅ β (SubRingβπ ) β ((π β Ring β§ (π βΎs π΅) β Ring) β§ (π΅ β π΅ β§ (1rβπ ) β π΅))) |
10 | 1, 4, 8, 9 | syl21anbrc 1342 | 1 β’ (π β Ring β π΅ β (SubRingβπ )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 β wss 3947 βcfv 6542 (class class class)co 7411 Basecbs 17148 βΎs cress 17177 1rcur 20075 Ringcrg 20127 SubRingcsubrg 20457 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mgp 20029 df-ur 20076 df-ring 20129 df-subrg 20459 |
This theorem is referenced by: subrgmre 20487 rnrhmsubrg 20495 sdrgid 20551 rlmlmod 20972 ring2idlqus 21068 rlmassa 21644 aspval 21646 evlrhm 21878 evlsscasrng 21879 evlsca 21880 evlsvarsrng 21881 evlvar 21882 mpfsubrg 21885 evl1sca 22073 evl1var 22075 evls1scasrng 22078 evls1varsrng 22079 pf1subrg 22087 pf1ind 22094 evl1gsumadd 22097 evl1varpw 22100 rlmnlm 24425 rlmbn 25109 dvply2 26035 dvnply 26037 taylply 26117 ressply1evl 32921 rgmoddimOLD 32983 fldextid 33026 riccrng1 41400 evlsevl 41445 evlvvval 41447 evlvvvallem 41448 mhphf4 41474 mzpmfp 41787 rgspnval 42212 rgspncl 42213 |
Copyright terms: Public domain | W3C validator |