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Mirrors > Home > ILE Home > Th. List > lidlsubg | GIF version |
Description: An ideal is a subgroup of the additive group. (Contributed by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
lidlcl.u | β’ π = (LIdealβπ ) |
Ref | Expression |
---|---|
lidlsubg | β’ ((π β Ring β§ πΌ β π) β πΌ β (SubGrpβπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2187 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
2 | lidlcl.u | . . . 4 β’ π = (LIdealβπ ) | |
3 | 1, 2 | lidlss 13665 | . . 3 β’ (πΌ β π β πΌ β (Baseβπ )) |
4 | 3 | adantl 277 | . 2 β’ ((π β Ring β§ πΌ β π) β πΌ β (Baseβπ )) |
5 | eqid 2187 | . . . 4 β’ (0gβπ ) = (0gβπ ) | |
6 | 2, 5 | lidl0cl 13672 | . . 3 β’ ((π β Ring β§ πΌ β π) β (0gβπ ) β πΌ) |
7 | elex2 2765 | . . 3 β’ ((0gβπ ) β πΌ β βπ π β πΌ) | |
8 | 6, 7 | syl 14 | . 2 β’ ((π β Ring β§ πΌ β π) β βπ π β πΌ) |
9 | eqid 2187 | . . . . . . 7 β’ (+gβπ ) = (+gβπ ) | |
10 | 2, 9 | lidlacl 13673 | . . . . . 6 β’ (((π β Ring β§ πΌ β π) β§ (π₯ β πΌ β§ π¦ β πΌ)) β (π₯(+gβπ )π¦) β πΌ) |
11 | 10 | anassrs 400 | . . . . 5 β’ ((((π β Ring β§ πΌ β π) β§ π₯ β πΌ) β§ π¦ β πΌ) β (π₯(+gβπ )π¦) β πΌ) |
12 | 11 | ralrimiva 2560 | . . . 4 β’ (((π β Ring β§ πΌ β π) β§ π₯ β πΌ) β βπ¦ β πΌ (π₯(+gβπ )π¦) β πΌ) |
13 | eqid 2187 | . . . . . 6 β’ (invgβπ ) = (invgβπ ) | |
14 | 2, 13 | lidlnegcl 13674 | . . . . 5 β’ ((π β Ring β§ πΌ β π β§ π₯ β πΌ) β ((invgβπ )βπ₯) β πΌ) |
15 | 14 | 3expa 1204 | . . . 4 β’ (((π β Ring β§ πΌ β π) β§ π₯ β πΌ) β ((invgβπ )βπ₯) β πΌ) |
16 | 12, 15 | jca 306 | . . 3 β’ (((π β Ring β§ πΌ β π) β§ π₯ β πΌ) β (βπ¦ β πΌ (π₯(+gβπ )π¦) β πΌ β§ ((invgβπ )βπ₯) β πΌ)) |
17 | 16 | ralrimiva 2560 | . 2 β’ ((π β Ring β§ πΌ β π) β βπ₯ β πΌ (βπ¦ β πΌ (π₯(+gβπ )π¦) β πΌ β§ ((invgβπ )βπ₯) β πΌ)) |
18 | ringgrp 13253 | . . . 4 β’ (π β Ring β π β Grp) | |
19 | 18 | adantr 276 | . . 3 β’ ((π β Ring β§ πΌ β π) β π β Grp) |
20 | 1, 9, 13 | issubg2m 13081 | . . 3 β’ (π β Grp β (πΌ β (SubGrpβπ ) β (πΌ β (Baseβπ ) β§ βπ π β πΌ β§ βπ₯ β πΌ (βπ¦ β πΌ (π₯(+gβπ )π¦) β πΌ β§ ((invgβπ )βπ₯) β πΌ)))) |
21 | 19, 20 | syl 14 | . 2 β’ ((π β Ring β§ πΌ β π) β (πΌ β (SubGrpβπ ) β (πΌ β (Baseβπ ) β§ βπ π β πΌ β§ βπ₯ β πΌ (βπ¦ β πΌ (π₯(+gβπ )π¦) β πΌ β§ ((invgβπ )βπ₯) β πΌ)))) |
22 | 4, 8, 17, 21 | mpbir3and 1181 | 1 β’ ((π β Ring β§ πΌ β π) β πΌ β (SubGrpβπ )) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wb 105 β§ w3a 979 = wceq 1363 βwex 1502 β wcel 2158 βwral 2465 β wss 3141 βcfv 5228 (class class class)co 5888 Basecbs 12476 +gcplusg 12551 0gc0g 12723 Grpcgrp 12899 invgcminusg 12900 SubGrpcsubg 13059 Ringcrg 13248 LIdealclidl 13656 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-i2m1 7930 ax-0lt1 7931 ax-0id 7933 ax-rnegex 7934 ax-pre-ltirr 7937 ax-pre-lttrn 7939 ax-pre-ltadd 7941 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-pnf 8008 df-mnf 8009 df-ltxr 8011 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 df-5 8995 df-6 8996 df-7 8997 df-8 8998 df-ndx 12479 df-slot 12480 df-base 12482 df-sets 12483 df-iress 12484 df-plusg 12564 df-mulr 12565 df-sca 12567 df-vsca 12568 df-ip 12569 df-0g 12725 df-mgm 12794 df-sgrp 12827 df-mnd 12840 df-grp 12902 df-minusg 12903 df-sbg 12904 df-subg 13062 df-mgp 13173 df-ur 13212 df-ring 13250 df-subrg 13439 df-lmod 13478 df-lssm 13542 df-sra 13624 df-rgmod 13625 df-lidl 13658 |
This theorem is referenced by: lidlsubcl 13676 dflidl2 13677 df2idl2 13697 2idlcpbl 13712 qus1 13714 qusmul2 13716 quscrng 13720 |
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