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Theorem sspid 30800

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 3956	. . . 4 ⊢ ( +_𝑣 ‘𝑈) ⊆ ( +_𝑣 ‘𝑈)
2		ssid 3956	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑈) ⊆ ( ·_𝑠OLD ‘𝑈)
3		ssid 3956	. . . 4 ⊢ (norm_CV‘𝑈) ⊆ (norm_CV‘𝑈)
4	1, 2, 3	3pm3.2i 1340	. . 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 2736	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
7		eqid 2736	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
8		eqid 2736	. . 3 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
9		sspid.h	. . 3 ⊢ 𝐻 = (SubSp‘𝑈)
10	6, 6, 7, 7, 8, 8, 9	isssp 30799	. 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 1541 ∈ wcel 2113 ⊆ wss 3901 ‘cfv 6492 NrmCVeccnv 30659 +_𝑣 cpv 30660 ·_𝑠OLD cns 30662 norm_CVcnmcv 30665 SubSpcss 30796

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 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 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fo 6498 df-fv 6500 df-oprab 7362 df-1st 7933 df-2nd 7934 df-vc 30634 df-nv 30667 df-va 30670 df-sm 30672 df-nmcv 30675 df-ssp 30797

This theorem is referenced by: hhsssh 31344

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