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Mirrors > Home > ILE Home > Th. List > subrgcrng | GIF version |
Description: A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
subrgring.1 | β’ π = (π βΎs π΄) |
Ref | Expression |
---|---|
subrgcrng | β’ ((π β CRing β§ π΄ β (SubRingβπ )) β π β CRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgring.1 | . . . 4 β’ π = (π βΎs π΄) | |
2 | 1 | subrgring 13283 | . . 3 β’ (π΄ β (SubRingβπ ) β π β Ring) |
3 | 2 | adantl 277 | . 2 β’ ((π β CRing β§ π΄ β (SubRingβπ )) β π β Ring) |
4 | eqid 2177 | . . . 4 β’ (mulGrpβπ ) = (mulGrpβπ ) | |
5 | 1, 4 | mgpress 13072 | . . 3 β’ ((π β CRing β§ π΄ β (SubRingβπ )) β ((mulGrpβπ ) βΎs π΄) = (mulGrpβπ)) |
6 | eqidd 2178 | . . . 4 β’ ((π β CRing β§ π΄ β (SubRingβπ )) β ((mulGrpβπ ) βΎs π΄) = ((mulGrpβπ ) βΎs π΄)) | |
7 | 4 | crngmgp 13118 | . . . . 5 β’ (π β CRing β (mulGrpβπ ) β CMnd) |
8 | 7 | adantr 276 | . . . 4 β’ ((π β CRing β§ π΄ β (SubRingβπ )) β (mulGrpβπ ) β CMnd) |
9 | eqid 2177 | . . . . . . 7 β’ (mulGrpβπ) = (mulGrpβπ) | |
10 | 9 | ringmgp 13116 | . . . . . 6 β’ (π β Ring β (mulGrpβπ) β Mnd) |
11 | 3, 10 | syl 14 | . . . . 5 β’ ((π β CRing β§ π΄ β (SubRingβπ )) β (mulGrpβπ) β Mnd) |
12 | 5, 11 | eqeltrd 2254 | . . . 4 β’ ((π β CRing β§ π΄ β (SubRingβπ )) β ((mulGrpβπ ) βΎs π΄) β Mnd) |
13 | simpr 110 | . . . 4 β’ ((π β CRing β§ π΄ β (SubRingβπ )) β π΄ β (SubRingβπ )) | |
14 | 6, 8, 12, 13 | subcmnd 13060 | . . 3 β’ ((π β CRing β§ π΄ β (SubRingβπ )) β ((mulGrpβπ ) βΎs π΄) β CMnd) |
15 | 5, 14 | eqeltrrd 2255 | . 2 β’ ((π β CRing β§ π΄ β (SubRingβπ )) β (mulGrpβπ) β CMnd) |
16 | 9 | iscrng 13117 | . 2 β’ (π β CRing β (π β Ring β§ (mulGrpβπ) β CMnd)) |
17 | 3, 15, 16 | sylanbrc 417 | 1 β’ ((π β CRing β§ π΄ β (SubRingβπ )) β π β CRing) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 βcfv 5215 (class class class)co 5872 βΎs cress 12455 Mndcmnd 12749 CMndccmn 13019 mulGrpcmgp 13061 Ringcrg 13110 CRingccrg 13111 SubRingcsubrg 13276 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-pre-ltirr 7920 ax-pre-lttrn 7922 ax-pre-ltadd 7924 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-fv 5223 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-ltxr 7993 df-inn 8916 df-2 8974 df-3 8975 df-ndx 12457 df-slot 12458 df-base 12460 df-sets 12461 df-iress 12462 df-plusg 12541 df-mulr 12542 df-cmn 13021 df-mgp 13062 df-ring 13112 df-cring 13113 df-subrg 13278 |
This theorem is referenced by: zringcrng 13351 |
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