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Mirrors > Home > MPE Home > Th. List > lssel | Structured version Visualization version GIF version |
Description: A subspace member is a vector. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.) |
Ref | Expression |
---|---|
lssss.v | ⊢ 𝑉 = (Base‘𝑊) |
lssss.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
lssel | ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lssss.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lssss.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | 1, 2 | lssss 19701 | . 2 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
4 | 3 | sselda 3915 | 1 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 Basecbs 16475 LSubSpclss 19696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-lss 19697 |
This theorem is referenced by: lssvsubcl 19708 lssvancl1 19709 lssvancl2 19710 lss0cl 19711 lssvacl 19719 lssvscl 19720 lssvnegcl 19721 lspsnel6 19759 lspsnel5a 19761 lssats2 19765 lsmcl 19848 lsmelval2 19850 lsmcv 19906 ocvin 20363 lsatel 36301 lsmsat 36304 lssatomic 36307 lssats 36308 lsat0cv 36329 lshpkrlem1 36406 lshpkrlem5 36410 lshpkr 36413 dihjat1lem 38724 dochsatshpb 38748 lcfrvalsnN 38837 lcfrlem4 38841 lcfrlem6 38843 lcfrlem16 38854 lcfrlem29 38867 lcfrlem35 38873 mapdval4N 38928 mapdpglem2a 38970 mapdpglem23 38990 |
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