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Mirrors > Home > ILE Home > Th. List > subrgid | 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 19 | . 2 β’ (π β Ring β π β Ring) | |
2 | subrgss.1 | . . 3 β’ π΅ = (Baseβπ ) | |
3 | 2 | ringressid 13374 | . 2 β’ (π β Ring β (π βΎs π΅) β Ring) |
4 | eqid 2189 | . . . 4 β’ (1rβπ ) = (1rβπ ) | |
5 | 2, 4 | ringidcl 13335 | . . 3 β’ (π β Ring β (1rβπ ) β π΅) |
6 | ssid 3190 | . . 3 β’ π΅ β π΅ | |
7 | 5, 6 | jctil 312 | . 2 β’ (π β Ring β (π΅ β π΅ β§ (1rβπ ) β π΅)) |
8 | 2, 4 | issubrg 13529 | . 2 β’ (π΅ β (SubRingβπ ) β ((π β Ring β§ (π βΎs π΅) β Ring) β§ (π΅ β π΅ β§ (1rβπ ) β π΅))) |
9 | 1, 3, 7, 8 | syl21anbrc 1184 | 1 β’ (π β Ring β π΅ β (SubRingβπ )) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1364 β wcel 2160 β wss 3144 βcfv 5231 (class class class)co 5891 Basecbs 12480 βΎs cress 12481 1rcur 13274 Ringcrg 13311 SubRingcsubrg 13525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-addcom 7929 ax-addass 7931 ax-i2m1 7934 ax-0lt1 7935 ax-0id 7937 ax-rnegex 7938 ax-pre-ltirr 7941 ax-pre-lttrn 7943 ax-pre-ltadd 7945 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8012 df-mnf 8013 df-ltxr 8015 df-inn 8938 df-2 8996 df-3 8997 df-ndx 12483 df-slot 12484 df-base 12486 df-sets 12487 df-iress 12488 df-plusg 12568 df-mulr 12569 df-0g 12729 df-mgm 12798 df-sgrp 12831 df-mnd 12844 df-grp 12914 df-minusg 12915 df-mgp 13236 df-ur 13275 df-ring 13313 df-subrg 13527 |
This theorem is referenced by: rlmlmod 13741 |
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