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Theorem sspnv 30758

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 2740	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
2		eqid 2740	. . 3 ⊢ ( +_𝑣 ‘𝑊) = ( +_𝑣 ‘𝑊)
3		eqid 2740	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
4		eqid 2740	. . 3 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
5		eqid 2740	. . 3 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
6		eqid 2740	. . 3 ⊢ (norm_CV‘𝑊) = (norm_CV‘𝑊)
7		sspnv.h	. . 3 ⊢ 𝐻 = (SubSp‘𝑈)
8	1, 2, 3, 4, 5, 6, 7	isssp 30756	. 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 1087 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 ‘cfv 6573 NrmCVeccnv 30616 +_𝑣 cpv 30617 ·_𝑠OLD cns 30619 norm_CVcnmcv 30622 SubSpcss 30753

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fo 6579 df-fv 6581 df-oprab 7452 df-1st 8030 df-2nd 8031 df-vc 30591 df-nv 30624 df-va 30627 df-sm 30629 df-nmcv 30632 df-ssp 30754

This theorem is referenced by: sspg 30760 ssps 30762 sspmlem 30764 sspmval 30765 sspz 30767 sspn 30768 sspimsval 30770 bnsscmcl 30900 minvecolem2 30907 hhshsslem1 31299 hhshsslem2 31300

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