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Theorem sspid 26964

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 3587	. . . 4 ⊢ ( +_𝑣 ‘𝑈) ⊆ ( +_𝑣 ‘𝑈)
2		ssid 3587	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑈) ⊆ ( ·_𝑠OLD ‘𝑈)
3		ssid 3587	. . . 4 ⊢ (norm_CV‘𝑈) ⊆ (norm_CV‘𝑈)
4	1, 2, 3	3pm3.2i 1232	. . 3 ⊢ (( +_𝑣 ‘𝑈) ⊆ ( +_𝑣 ‘𝑈) ∧ ( ·_𝑠OLD ‘𝑈) ⊆ ( ·_𝑠OLD ‘𝑈) ∧ (norm_CV‘𝑈) ⊆ (norm_CV‘𝑈))
5	4	jctr 563	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ∧ (( +_𝑣 ‘𝑈) ⊆ ( +_𝑣 ‘𝑈) ∧ ( ·_𝑠OLD ‘𝑈) ⊆ ( ·_𝑠OLD ‘𝑈) ∧ (norm_CV‘𝑈) ⊆ (norm_CV‘𝑈))))
6		eqid 2610	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
7		eqid 2610	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
8		eqid 2610	. . 3 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
9		sspid.h	. . 3 ⊢ 𝐻 = (SubSp‘𝑈)
10	6, 6, 7, 7, 8, 8, 9	isssp 26963	. 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 246	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5804 NrmCVeccnv 26823 +_𝑣 cpv 26824 ·_𝑠OLD cns 26826 norm_CVcnmcv 26829 SubSpcss 26960

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-oprab 6553 df-1st 7059 df-2nd 7060 df-vc 26798 df-nv 26831 df-va 26834 df-sm 26836 df-nmcv 26839 df-ssp 26961

This theorem is referenced by: hhsssh 27510

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