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Theorem sspid 30700

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 3957	. . . 4 ⊢ ( +_𝑣 ‘𝑈) ⊆ ( +_𝑣 ‘𝑈)
2		ssid 3957	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑈) ⊆ ( ·_𝑠OLD ‘𝑈)
3		ssid 3957	. . . 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 2731	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
7		eqid 2731	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
8		eqid 2731	. . 3 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
9		sspid.h	. . 3 ⊢ 𝐻 = (SubSp‘𝑈)
10	6, 6, 7, 7, 8, 8, 9	isssp 30699	. 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 2111 ⊆ wss 3902 ‘cfv 6481 NrmCVeccnv 30559 +_𝑣 cpv 30560 ·_𝑠OLD cns 30562 norm_CVcnmcv 30565 SubSpcss 30696

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-fo 6487 df-fv 6489 df-oprab 7350 df-1st 7921 df-2nd 7922 df-vc 30534 df-nv 30567 df-va 30570 df-sm 30572 df-nmcv 30575 df-ssp 30697

This theorem is referenced by: hhsssh 31244

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