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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 2726 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | lsssubg.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | 1, 2 | lssss 20780 | . . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ (Base‘𝑊)) |
4 | 3 | adantl 481 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ⊆ (Base‘𝑊)) |
5 | 2 | lssn0 20784 | . . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ≠ ∅) |
6 | 5 | adantl 481 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ≠ ∅) |
7 | eqid 2726 | . . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
8 | 7, 2 | lssvacl 20787 | . . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑥(+g‘𝑊)𝑦) ∈ 𝑈) |
9 | 8 | anassrs 467 | . . . . 5 ⊢ ((((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ 𝑈) → (𝑥(+g‘𝑊)𝑦) ∈ 𝑈) |
10 | 9 | ralrimiva 3140 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) → ∀𝑦 ∈ 𝑈 (𝑥(+g‘𝑊)𝑦) ∈ 𝑈) |
11 | eqid 2726 | . . . . . 6 ⊢ (invg‘𝑊) = (invg‘𝑊) | |
12 | 2, 11 | lssvnegcl 20800 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑥 ∈ 𝑈) → ((invg‘𝑊)‘𝑥) ∈ 𝑈) |
13 | 12 | 3expa 1115 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) → ((invg‘𝑊)‘𝑥) ∈ 𝑈) |
14 | 10, 13 | jca 511 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) → (∀𝑦 ∈ 𝑈 (𝑥(+g‘𝑊)𝑦) ∈ 𝑈 ∧ ((invg‘𝑊)‘𝑥) ∈ 𝑈)) |
15 | 14 | ralrimiva 3140 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → ∀𝑥 ∈ 𝑈 (∀𝑦 ∈ 𝑈 (𝑥(+g‘𝑊)𝑦) ∈ 𝑈 ∧ ((invg‘𝑊)‘𝑥) ∈ 𝑈)) |
16 | lmodgrp 20710 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
17 | 16 | adantr 480 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑊 ∈ Grp) |
18 | 1, 7, 11 | issubg2 19065 | . . 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 1339 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 ⊆ wss 3943 ∅c0 4317 ‘cfv 6536 (class class class)co 7404 Basecbs 17150 +gcplusg 17203 Grpcgrp 18860 invgcminusg 18861 SubGrpcsubg 19044 LModclmod 20703 LSubSpclss 20775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18863 df-minusg 18864 df-sbg 18865 df-subg 19047 df-mgp 20037 df-ur 20084 df-ring 20137 df-lmod 20705 df-lss 20776 |
This theorem is referenced by: lsssssubg 20802 islss3 20803 islss4 20806 lspsnsubg 20824 lmhmima 20892 lmhmpreima 20893 reslmhm 20897 reslmhm2 20898 reslmhm2b 20899 lsmcl 20928 lsmelval2 20930 phssip 21546 frlm0 21644 frlmsubgval 21655 frlmgsum 21662 frlmsslsp 21686 lssnlm 24568 cphsscph 25129 cmscsscms 25251 cssbn 25253 eqgvscpbl 32967 qusvscpbl 32968 quslmod 32975 quslmhm 32976 ply1degltdimlem 33224 lindsunlem 33226 lbsdiflsp0 33228 dimkerim 33229 qusdimsum 33230 islshpat 38399 lsatcv1 38430 dia2dimlem13 40459 dihvalcqat 40622 dihmeetlem16N 40705 dihmeetlem19N 40708 dochsat 40766 dihjat1lem 40811 dihjat1 40812 dvh3dimatN 40822 dvh2dimatN 40823 dochkrsm 40841 dochexmid 40851 mapdh6dN 41122 hdmap1l6d 41196 pwssplit4 42391 gsumlsscl 47317 |
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