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Theorem sspnv 30707

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 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 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
6		eqid 2735	. . 3 ⊢ (norm_CV‘𝑊) = (norm_CV‘𝑊)
7		sspnv.h	. . 3 ⊢ 𝐻 = (SubSp‘𝑈)
8	1, 2, 3, 4, 5, 6, 7	isssp 30705	. 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 2108 ⊆ wss 3926 ‘cfv 6531 NrmCVeccnv 30565 +_𝑣 cpv 30566 ·_𝑠OLD cns 30568 norm_CVcnmcv 30571 SubSpcss 30702

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 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-fo 6537 df-fv 6539 df-oprab 7409 df-1st 7988 df-2nd 7989 df-vc 30540 df-nv 30573 df-va 30576 df-sm 30578 df-nmcv 30581 df-ssp 30703

This theorem is referenced by: sspg 30709 ssps 30711 sspmlem 30713 sspmval 30714 sspz 30716 sspn 30717 sspimsval 30719 bnsscmcl 30849 minvecolem2 30856 hhshsslem1 31248 hhshsslem2 31249

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