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Theorem sspid 30754

Description: A normed complex vector space is a subspace of itself. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)

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

**Assertion**
Ref	Expression
sspid	⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ 𝐻)

**Proof of Theorem sspid**
Step	Hyp	Ref	Expression
1		ssid 4018	. . . 4 ⊢ ( +_𝑣 ‘𝑈) ⊆ ( +_𝑣 ‘𝑈)
2		ssid 4018	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑈) ⊆ ( ·_𝑠OLD ‘𝑈)
3		ssid 4018	. . . 4 ⊢ (norm_CV‘𝑈) ⊆ (norm_CV‘𝑈)
4	1, 2, 3	3pm3.2i 1338	. . 3 ⊢ (( +_𝑣 ‘𝑈) ⊆ ( +_𝑣 ‘𝑈) ∧ ( ·_𝑠OLD ‘𝑈) ⊆ ( ·_𝑠OLD ‘𝑈) ∧ (norm_CV‘𝑈) ⊆ (norm_CV‘𝑈))
5	4	jctr 524	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ∧ (( +_𝑣 ‘𝑈) ⊆ ( +_𝑣 ‘𝑈) ∧ ( ·_𝑠OLD ‘𝑈) ⊆ ( ·_𝑠OLD ‘𝑈) ∧ (norm_CV‘𝑈) ⊆ (norm_CV‘𝑈))))
6		eqid 2735	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
7		eqid 2735	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
8		eqid 2735	. . 3 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
9		sspid.h	. . 3 ⊢ 𝐻 = (SubSp‘𝑈)
10	6, 6, 7, 7, 8, 8, 9	isssp 30753	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ NrmCVec ∧ (( +_𝑣 ‘𝑈) ⊆ ( +_𝑣 ‘𝑈) ∧ ( ·_𝑠OLD ‘𝑈) ⊆ ( ·_𝑠OLD ‘𝑈) ∧ (norm_CV‘𝑈) ⊆ (norm_CV‘𝑈)))))
11	5, 10	mpbird 257	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 NrmCVeccnv 30613 +_𝑣 cpv 30614 ·_𝑠OLD cns 30616 norm_CVcnmcv 30619 SubSpcss 30750

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-oprab 7435 df-1st 8013 df-2nd 8014 df-vc 30588 df-nv 30621 df-va 30624 df-sm 30626 df-nmcv 30629 df-ssp 30751

This theorem is referenced by: hhsssh 31298

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