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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 | ⊢ 𝑈 = (LIdeal‘𝑅) |
Ref | Expression |
---|---|
lidlsubg | ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 𝐼 ∈ (SubGrp‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2189 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | lidlcl.u | . . . 4 ⊢ 𝑈 = (LIdeal‘𝑅) | |
3 | 1, 2 | lidlss 13753 | . . 3 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑅)) |
4 | 3 | adantl 277 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 𝐼 ⊆ (Base‘𝑅)) |
5 | eqid 2189 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | 2, 5 | lidl0cl 13760 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0g‘𝑅) ∈ 𝐼) |
7 | elex2 2768 | . . 3 ⊢ ((0g‘𝑅) ∈ 𝐼 → ∃𝑗 𝑗 ∈ 𝐼) | |
8 | 6, 7 | syl 14 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ∃𝑗 𝑗 ∈ 𝐼) |
9 | eqid 2189 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
10 | 2, 9 | lidlacl 13761 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼)) → (𝑥(+g‘𝑅)𝑦) ∈ 𝐼) |
11 | 10 | anassrs 400 | . . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (𝑥(+g‘𝑅)𝑦) ∈ 𝐼) |
12 | 11 | ralrimiva 2563 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ 𝐼) → ∀𝑦 ∈ 𝐼 (𝑥(+g‘𝑅)𝑦) ∈ 𝐼) |
13 | eqid 2189 | . . . . . 6 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
14 | 2, 13 | lidlnegcl 13762 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑥 ∈ 𝐼) → ((invg‘𝑅)‘𝑥) ∈ 𝐼) |
15 | 14 | 3expa 1205 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ 𝐼) → ((invg‘𝑅)‘𝑥) ∈ 𝐼) |
16 | 12, 15 | jca 306 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ 𝐼) → (∀𝑦 ∈ 𝐼 (𝑥(+g‘𝑅)𝑦) ∈ 𝐼 ∧ ((invg‘𝑅)‘𝑥) ∈ 𝐼)) |
17 | 16 | ralrimiva 2563 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥(+g‘𝑅)𝑦) ∈ 𝐼 ∧ ((invg‘𝑅)‘𝑥) ∈ 𝐼)) |
18 | ringgrp 13316 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
19 | 18 | adantr 276 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 𝑅 ∈ Grp) |
20 | 1, 9, 13 | issubg2m 13094 | . . 3 ⊢ (𝑅 ∈ Grp → (𝐼 ∈ (SubGrp‘𝑅) ↔ (𝐼 ⊆ (Base‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥(+g‘𝑅)𝑦) ∈ 𝐼 ∧ ((invg‘𝑅)‘𝑥) ∈ 𝐼)))) |
21 | 19, 20 | syl 14 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝐼 ∈ (SubGrp‘𝑅) ↔ (𝐼 ⊆ (Base‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥(+g‘𝑅)𝑦) ∈ 𝐼 ∧ ((invg‘𝑅)‘𝑥) ∈ 𝐼)))) |
22 | 4, 8, 17, 21 | mpbir3and 1182 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 𝐼 ∈ (SubGrp‘𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∃wex 1503 ∈ wcel 2160 ∀wral 2468 ⊆ wss 3144 ‘cfv 5231 (class class class)co 5891 Basecbs 12480 +gcplusg 12555 0gc0g 12727 Grpcgrp 12911 invgcminusg 12912 SubGrpcsubg 13072 Ringcrg 13311 LIdealclidl 13744 |
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-1st 6159 df-2nd 6160 df-pnf 8012 df-mnf 8013 df-ltxr 8015 df-inn 8938 df-2 8996 df-3 8997 df-4 8998 df-5 8999 df-6 9000 df-7 9001 df-8 9002 df-ndx 12483 df-slot 12484 df-base 12486 df-sets 12487 df-iress 12488 df-plusg 12568 df-mulr 12569 df-sca 12571 df-vsca 12572 df-ip 12573 df-0g 12729 df-mgm 12798 df-sgrp 12831 df-mnd 12844 df-grp 12914 df-minusg 12915 df-sbg 12916 df-subg 13075 df-mgp 13236 df-ur 13275 df-ring 13313 df-subrg 13527 df-lmod 13566 df-lssm 13630 df-sra 13712 df-rgmod 13713 df-lidl 13746 |
This theorem is referenced by: lidlsubcl 13764 dflidl2 13765 df2idl2 13785 2idlcpbl 13800 qus1 13802 qusmul2 13804 quscrng 13808 |
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