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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 2737 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
| 2 | eqid 2737 | . . 3 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
| 3 | eqid 2737 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
| 4 | eqid 2737 | . . 3 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
| 5 | eqid 2737 | . . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
| 6 | eqid 2737 | . . 3 ⊢ (normCV‘𝑊) = (normCV‘𝑊) | |
| 7 | sspnv.h | . . 3 ⊢ 𝐻 = (SubSp‘𝑈) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | isssp 30743 | . 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 1087 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 NrmCVeccnv 30603 +𝑣 cpv 30604 ·𝑠OLD cns 30606 normCVcnmcv 30609 SubSpcss 30740 |
| 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 ax-un 7755 |
| 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-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-oprab 7435 df-1st 8014 df-2nd 8015 df-vc 30578 df-nv 30611 df-va 30614 df-sm 30616 df-nmcv 30619 df-ssp 30741 |
| This theorem is referenced by: sspg 30747 ssps 30749 sspmlem 30751 sspmval 30752 sspz 30754 sspn 30755 sspimsval 30757 bnsscmcl 30887 minvecolem2 30894 hhshsslem1 31286 hhshsslem2 31287 |
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