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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 18985 | . 2 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
4 | 3 | sselda 3636 | 1 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 Basecbs 15904 LSubSpclss 18980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 df-ov 6693 df-lss 18981 |
This theorem is referenced by: lssvsubcl 18992 lssvancl1 18993 lssvancl2 18994 lss0cl 18995 lssvacl 19002 lssvscl 19003 lssvnegcl 19004 lspsnel6 19042 lspsnel5a 19044 lssats2 19048 lsmcl 19131 lsmelval2 19133 lsmcv 19189 ocvin 20066 lsatel 34610 lsmsat 34613 lssatomic 34616 lssats 34617 lsat0cv 34638 lshpkrlem1 34715 lshpkrlem5 34719 lshpkr 34722 dihjat1lem 37034 dochsatshpb 37058 lcfrvalsnN 37147 lcfrlem4 37151 lcfrlem6 37153 lcfrlem16 37164 lcfrlem29 37177 lcfrlem35 37183 mapdval4N 37238 mapdpglem2a 37280 mapdpglem23 37300 |
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