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Mirrors > Home > MPE Home > Th. List > sspnv | Structured version Visualization version GIF version |
Description: A subspace is a normed complex vector space. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspnv.h | ⊢ 𝐻 = (SubSp‘𝑈) |
Ref | Expression |
---|---|
sspnv | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
2 | eqid 2738 | . . 3 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
3 | eqid 2738 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
4 | eqid 2738 | . . 3 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
5 | eqid 2738 | . . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
6 | eqid 2738 | . . 3 ⊢ (normCV‘𝑊) = (normCV‘𝑊) | |
7 | sspnv.h | . . 3 ⊢ 𝐻 = (SubSp‘𝑈) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isssp 29094 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈))))) |
9 | 8 | simprbda 499 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ⊆ wss 3886 ‘cfv 6426 NrmCVeccnv 28954 +𝑣 cpv 28955 ·𝑠OLD cns 28957 normCVcnmcv 28960 SubSpcss 29091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-fo 6432 df-fv 6434 df-oprab 7271 df-1st 7820 df-2nd 7821 df-vc 28929 df-nv 28962 df-va 28965 df-sm 28967 df-nmcv 28970 df-ssp 29092 |
This theorem is referenced by: sspg 29098 ssps 29100 sspmlem 29102 sspmval 29103 sspz 29105 sspn 29106 sspimsval 29108 bnsscmcl 29238 minvecolem2 29245 hhshsslem1 29637 hhshsslem2 29638 |
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