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Theorem sspnv 30797

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 2736	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
2		eqid 2736	. . 3 ⊢ ( +_𝑣 ‘𝑊) = ( +_𝑣 ‘𝑊)
3		eqid 2736	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
4		eqid 2736	. . 3 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
5		eqid 2736	. . 3 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
6		eqid 2736	. . 3 ⊢ (norm_CV‘𝑊) = (norm_CV‘𝑊)
7		sspnv.h	. . 3 ⊢ 𝐻 = (SubSp‘𝑈)
8	1, 2, 3, 4, 5, 6, 7	isssp 30795	. 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 1542 ∈ wcel 2114 ⊆ wss 3889 ‘cfv 6498 NrmCVeccnv 30655 +_𝑣 cpv 30656 ·_𝑠OLD cns 30658 norm_CVcnmcv 30661 SubSpcss 30792

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-fo 6504 df-fv 6506 df-oprab 7371 df-1st 7942 df-2nd 7943 df-vc 30630 df-nv 30663 df-va 30666 df-sm 30668 df-nmcv 30671 df-ssp 30793

This theorem is referenced by: sspg 30799 ssps 30801 sspmlem 30803 sspmval 30804 sspz 30806 sspn 30807 sspimsval 30809 bnsscmcl 30939 minvecolem2 30946 hhshsslem1 31338 hhshsslem2 31339

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