sspnv - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > sspnv		Structured version Visualization version GIF version

Theorem sspnv 30655

Description: A subspace is a normed complex vector space. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
sspnv.h	⊢ 𝐻 = (SubSp‘𝑈)

**Assertion**
Ref	Expression
sspnv	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec)

**Proof of Theorem sspnv**
Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
2		eqid 2729	. . 3 ⊢ ( +_𝑣 ‘𝑊) = ( +_𝑣 ‘𝑊)
3		eqid 2729	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
4		eqid 2729	. . 3 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
5		eqid 2729	. . 3 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
6		eqid 2729	. . 3 ⊢ (norm_CV‘𝑊) = (norm_CV‘𝑊)
7		sspnv.h	. . 3 ⊢ 𝐻 = (SubSp‘𝑈)
8	1, 2, 3, 4, 5, 6, 7	isssp 30653	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +_𝑣 ‘𝑊) ⊆ ( +_𝑣 ‘𝑈) ∧ ( ·_𝑠OLD ‘𝑊) ⊆ ( ·_𝑠OLD ‘𝑈) ∧ (norm_CV‘𝑊) ⊆ (norm_CV‘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 1540 ∈ wcel 2109 ⊆ wss 3914 ‘cfv 6511 NrmCVeccnv 30513 +_𝑣 cpv 30514 ·_𝑠OLD cns 30516 norm_CVcnmcv 30519 SubSpcss 30650

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-fo 6517 df-fv 6519 df-oprab 7391 df-1st 7968 df-2nd 7969 df-vc 30488 df-nv 30521 df-va 30524 df-sm 30526 df-nmcv 30529 df-ssp 30651

This theorem is referenced by: sspg 30657 ssps 30659 sspmlem 30661 sspmval 30662 sspz 30664 sspn 30665 sspimsval 30667 bnsscmcl 30797 minvecolem2 30804 hhshsslem1 31196 hhshsslem2 31197

W3C validator