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Mirrors > Home > MPE Home > Th. List > lsssubg | Structured version Visualization version GIF version |
Description: All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.) |
Ref | Expression |
---|---|
lsssubg.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
lsssubg | ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | lsssubg.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | 1, 2 | lssss 19829 | . . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ (Base‘𝑊)) |
4 | 3 | adantl 485 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ⊆ (Base‘𝑊)) |
5 | 2 | lssn0 19833 | . . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ≠ ∅) |
6 | 5 | adantl 485 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ≠ ∅) |
7 | eqid 2738 | . . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
8 | 7, 2 | lssvacl 19847 | . . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑥(+g‘𝑊)𝑦) ∈ 𝑈) |
9 | 8 | anassrs 471 | . . . . 5 ⊢ ((((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ 𝑈) → (𝑥(+g‘𝑊)𝑦) ∈ 𝑈) |
10 | 9 | ralrimiva 3096 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) → ∀𝑦 ∈ 𝑈 (𝑥(+g‘𝑊)𝑦) ∈ 𝑈) |
11 | eqid 2738 | . . . . . 6 ⊢ (invg‘𝑊) = (invg‘𝑊) | |
12 | 2, 11 | lssvnegcl 19849 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑥 ∈ 𝑈) → ((invg‘𝑊)‘𝑥) ∈ 𝑈) |
13 | 12 | 3expa 1119 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) → ((invg‘𝑊)‘𝑥) ∈ 𝑈) |
14 | 10, 13 | jca 515 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) → (∀𝑦 ∈ 𝑈 (𝑥(+g‘𝑊)𝑦) ∈ 𝑈 ∧ ((invg‘𝑊)‘𝑥) ∈ 𝑈)) |
15 | 14 | ralrimiva 3096 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → ∀𝑥 ∈ 𝑈 (∀𝑦 ∈ 𝑈 (𝑥(+g‘𝑊)𝑦) ∈ 𝑈 ∧ ((invg‘𝑊)‘𝑥) ∈ 𝑈)) |
16 | lmodgrp 19762 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
17 | 16 | adantr 484 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑊 ∈ Grp) |
18 | 1, 7, 11 | issubg2 18414 | . . 3 ⊢ (𝑊 ∈ Grp → (𝑈 ∈ (SubGrp‘𝑊) ↔ (𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∀𝑦 ∈ 𝑈 (𝑥(+g‘𝑊)𝑦) ∈ 𝑈 ∧ ((invg‘𝑊)‘𝑥) ∈ 𝑈)))) |
19 | 17, 18 | syl 17 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑈 ∈ (SubGrp‘𝑊) ↔ (𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∀𝑦 ∈ 𝑈 (𝑥(+g‘𝑊)𝑦) ∈ 𝑈 ∧ ((invg‘𝑊)‘𝑥) ∈ 𝑈)))) |
20 | 4, 6, 15, 19 | mpbir3and 1343 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∀wral 3053 ⊆ wss 3843 ∅c0 4211 ‘cfv 6339 (class class class)co 7172 Basecbs 16588 +gcplusg 16670 Grpcgrp 18221 invgcminusg 18222 SubGrpcsubg 18393 LModclmod 19755 LSubSpclss 19824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-ress 16596 df-plusg 16683 df-0g 16820 df-mgm 17970 df-sgrp 18019 df-mnd 18030 df-grp 18224 df-minusg 18225 df-sbg 18226 df-subg 18396 df-mgp 19361 df-ur 19373 df-ring 19420 df-lmod 19757 df-lss 19825 |
This theorem is referenced by: lsssssubg 19851 islss3 19852 islss4 19855 lspsnsubg 19873 lmhmima 19940 lmhmpreima 19941 reslmhm 19945 reslmhm2 19946 reslmhm2b 19947 lsmcl 19976 lsmelval2 19978 phssip 20476 frlm0 20572 frlmsubgval 20583 frlmgsum 20590 frlmsslsp 20614 lssnlm 23456 cphsscph 24005 cmscsscms 24127 cssbn 24129 eqgvscpbl 31124 qusvscpbl 31125 quslmod 31128 quslmhm 31129 lindsunlem 31279 lbsdiflsp0 31281 dimkerim 31282 qusdimsum 31283 islshpat 36676 lsatcv1 36707 dia2dimlem13 38735 dihvalcqat 38898 dihmeetlem16N 38981 dihmeetlem19N 38984 dochsat 39042 dihjat1lem 39087 dihjat1 39088 dvh3dimatN 39098 dvh2dimatN 39099 dochkrsm 39117 dochexmid 39127 mapdh6dN 39398 hdmap1l6d 39472 pwssplit4 40508 gsumlsscl 45282 |
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