Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2821 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | lsssubg.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | 1, 2 | lssss 19708 | . . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ (Base‘𝑊)) |
4 | 3 | adantl 484 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ⊆ (Base‘𝑊)) |
5 | 2 | lssn0 19712 | . . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ≠ ∅) |
6 | 5 | adantl 484 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ≠ ∅) |
7 | eqid 2821 | . . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
8 | 7, 2 | lssvacl 19726 | . . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑥(+g‘𝑊)𝑦) ∈ 𝑈) |
9 | 8 | anassrs 470 | . . . . 5 ⊢ ((((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ 𝑈) → (𝑥(+g‘𝑊)𝑦) ∈ 𝑈) |
10 | 9 | ralrimiva 3182 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) → ∀𝑦 ∈ 𝑈 (𝑥(+g‘𝑊)𝑦) ∈ 𝑈) |
11 | eqid 2821 | . . . . . 6 ⊢ (invg‘𝑊) = (invg‘𝑊) | |
12 | 2, 11 | lssvnegcl 19728 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑥 ∈ 𝑈) → ((invg‘𝑊)‘𝑥) ∈ 𝑈) |
13 | 12 | 3expa 1114 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) → ((invg‘𝑊)‘𝑥) ∈ 𝑈) |
14 | 10, 13 | jca 514 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) → (∀𝑦 ∈ 𝑈 (𝑥(+g‘𝑊)𝑦) ∈ 𝑈 ∧ ((invg‘𝑊)‘𝑥) ∈ 𝑈)) |
15 | 14 | ralrimiva 3182 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → ∀𝑥 ∈ 𝑈 (∀𝑦 ∈ 𝑈 (𝑥(+g‘𝑊)𝑦) ∈ 𝑈 ∧ ((invg‘𝑊)‘𝑥) ∈ 𝑈)) |
16 | lmodgrp 19641 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
17 | 16 | adantr 483 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑊 ∈ Grp) |
18 | 1, 7, 11 | issubg2 18294 | . . 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 1338 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ⊆ wss 3936 ∅c0 4291 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 Grpcgrp 18103 invgcminusg 18104 SubGrpcsubg 18273 LModclmod 19634 LSubSpclss 19703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-mgp 19240 df-ur 19252 df-ring 19299 df-lmod 19636 df-lss 19704 |
This theorem is referenced by: lsssssubg 19730 islss3 19731 islss4 19734 lspsnsubg 19752 lmhmima 19819 lmhmpreima 19820 reslmhm 19824 reslmhm2 19825 reslmhm2b 19826 lsmcl 19855 lsmelval2 19857 phssip 20802 frlm0 20898 frlmsubgval 20909 frlmgsum 20916 frlmsslsp 20940 lssnlm 23310 cphsscph 23854 cmscsscms 23976 cssbn 23978 eqgvscpbl 30919 qusvscpbl 30920 quslmod 30923 quslmhm 30924 lindsunlem 31020 lbsdiflsp0 31022 dimkerim 31023 qusdimsum 31024 islshpat 36168 lsatcv1 36199 dia2dimlem13 38227 dihvalcqat 38390 dihmeetlem16N 38473 dihmeetlem19N 38476 dochsat 38534 dihjat1lem 38579 dihjat1 38580 dvh3dimatN 38590 dvh2dimatN 38591 dochkrsm 38609 dochexmid 38619 mapdh6dN 38890 hdmap1l6d 38964 pwssplit4 39709 gsumlsscl 44451 |
Copyright terms: Public domain | W3C validator |