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Theorem sspid 28135

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 3848	. . . 4 ⊢ ( +_𝑣 ‘𝑈) ⊆ ( +_𝑣 ‘𝑈)
2		ssid 3848	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑈) ⊆ ( ·_𝑠OLD ‘𝑈)
3		ssid 3848	. . . 4 ⊢ (norm_CV‘𝑈) ⊆ (norm_CV‘𝑈)
4	1, 2, 3	3pm3.2i 1444	. . 3 ⊢ (( +_𝑣 ‘𝑈) ⊆ ( +_𝑣 ‘𝑈) ∧ ( ·_𝑠OLD ‘𝑈) ⊆ ( ·_𝑠OLD ‘𝑈) ∧ (norm_CV‘𝑈) ⊆ (norm_CV‘𝑈))
5	4	jctr 522	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ∧ (( +_𝑣 ‘𝑈) ⊆ ( +_𝑣 ‘𝑈) ∧ ( ·_𝑠OLD ‘𝑈) ⊆ ( ·_𝑠OLD ‘𝑈) ∧ (norm_CV‘𝑈) ⊆ (norm_CV‘𝑈))))
6		eqid 2825	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
7		eqid 2825	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
8		eqid 2825	. . 3 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
9		sspid.h	. . 3 ⊢ 𝐻 = (SubSp‘𝑈)
10	6, 6, 7, 7, 8, 8, 9	isssp 28134	. 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 249	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ⊆ wss 3798 ‘cfv 6123 NrmCVeccnv 27994 +_𝑣 cpv 27995 ·_𝑠OLD cns 27997 norm_CVcnmcv 28000 SubSpcss 28131

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-fo 6129 df-fv 6131 df-oprab 6909 df-1st 7428 df-2nd 7429 df-vc 27969 df-nv 28002 df-va 28005 df-sm 28007 df-nmcv 28010 df-ssp 28132

This theorem is referenced by: hhsssh 28681

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