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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 18758 | . 2 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
4 | 3 | sselda 3568 | 1 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 Basecbs 15695 LSubSpclss 18753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-lss 18754 |
This theorem is referenced by: lssvsubcl 18765 lssvancl1 18766 lssvancl2 18767 lss0cl 18768 lssvacl 18775 lssvscl 18776 lssvnegcl 18777 lspsnel6 18815 lspsnel5a 18817 lssats2 18821 lsmcl 18904 lsmelval2 18906 lsmcv 18962 ocvin 19837 lsatel 33310 lsmsat 33313 lssatomic 33316 lssats 33317 lsat0cv 33338 lshpkrlem1 33415 lshpkrlem5 33419 lshpkr 33422 dihjat1lem 35735 dochsatshpb 35759 lcfrvalsnN 35848 lcfrlem4 35852 lcfrlem6 35854 lcfrlem16 35865 lcfrlem29 35878 lcfrlem35 35884 mapdval4N 35939 mapdpglem2a 35981 mapdpglem23 36001 |
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