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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 20934 | . 2 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
| 4 | 3 | sselda 3983 | 1 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 Basecbs 17247 LSubSpclss 20929 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-lss 20930 |
| This theorem is referenced by: lssvacl 20941 lssvsubcl 20942 lssvancl1 20943 lssvancl2 20944 lss0cl 20945 lssvscl 20953 lssvnegcl 20954 ellspsn6 20992 ellspsn5 20994 lssats2 20998 lsmcl 21082 lsmelval2 21084 lsmcv 21143 ocvin 21692 lsatel 39006 lsmsat 39009 lssatomic 39012 lssats 39013 lsat0cv 39034 lshpkrlem1 39111 lshpkrlem5 39115 lshpkr 39118 dihjat1lem 41430 dochsatshpb 41454 lcfrvalsnN 41543 lcfrlem4 41547 lcfrlem6 41549 lcfrlem16 41560 lcfrlem29 41573 lcfrlem35 41579 mapdval4N 41634 mapdpglem2a 41676 mapdpglem23 41696 |
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