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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 20951 | . 2 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
4 | 3 | sselda 3994 | 1 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 Basecbs 17244 LSubSpclss 20946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-lss 20947 |
This theorem is referenced by: lssvacl 20958 lssvsubcl 20959 lssvancl1 20960 lssvancl2 20961 lss0cl 20962 lssvscl 20970 lssvnegcl 20971 ellspsn6 21009 ellspsn5 21011 lssats2 21015 lsmcl 21099 lsmelval2 21101 lsmcv 21160 ocvin 21709 lsatel 38986 lsmsat 38989 lssatomic 38992 lssats 38993 lsat0cv 39014 lshpkrlem1 39091 lshpkrlem5 39095 lshpkr 39098 dihjat1lem 41410 dochsatshpb 41434 lcfrvalsnN 41523 lcfrlem4 41527 lcfrlem6 41529 lcfrlem16 41540 lcfrlem29 41553 lcfrlem35 41559 mapdval4N 41614 mapdpglem2a 41656 mapdpglem23 41676 |
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