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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 19997 | . 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 1543 ∈ wcel 2111 ‘cfv 6397 Basecbs 16784 LSubSpclss 19992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-br 5068 df-opab 5130 df-mpt 5150 df-id 5469 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-iota 6355 df-fun 6399 df-fv 6405 df-ov 7234 df-lss 19993 |
This theorem is referenced by: lssvsubcl 20004 lssvancl1 20005 lssvancl2 20006 lss0cl 20007 lssvacl 20015 lssvscl 20016 lssvnegcl 20017 lspsnel6 20055 lspsnel5a 20057 lssats2 20061 lsmcl 20144 lsmelval2 20146 lsmcv 20202 ocvin 20660 lsatel 36782 lsmsat 36785 lssatomic 36788 lssats 36789 lsat0cv 36810 lshpkrlem1 36887 lshpkrlem5 36891 lshpkr 36894 dihjat1lem 39205 dochsatshpb 39229 lcfrvalsnN 39318 lcfrlem4 39322 lcfrlem6 39324 lcfrlem16 39335 lcfrlem29 39348 lcfrlem35 39354 mapdval4N 39409 mapdpglem2a 39451 mapdpglem23 39471 |
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