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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 2735 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
2 | eqid 2735 | . . 3 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
3 | eqid 2735 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
4 | eqid 2735 | . . 3 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
5 | eqid 2735 | . . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
6 | eqid 2735 | . . 3 ⊢ (normCV‘𝑊) = (normCV‘𝑊) | |
7 | sspnv.h | . . 3 ⊢ 𝐻 = (SubSp‘𝑈) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isssp 30753 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈))))) |
9 | 8 | simprbda 498 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 NrmCVeccnv 30613 +𝑣 cpv 30614 ·𝑠OLD cns 30616 normCVcnmcv 30619 SubSpcss 30750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-oprab 7435 df-1st 8013 df-2nd 8014 df-vc 30588 df-nv 30621 df-va 30624 df-sm 30626 df-nmcv 30629 df-ssp 30751 |
This theorem is referenced by: sspg 30757 ssps 30759 sspmlem 30761 sspmval 30762 sspz 30764 sspn 30765 sspimsval 30767 bnsscmcl 30897 minvecolem2 30904 hhshsslem1 31296 hhshsslem2 31297 |
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