Theorem sspid 30687

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 3960	. . . 4 ⊢ ( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈)
2		ssid 3960	. . . 4 ⊢ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈)
3		ssid 3960	. . . 4 ⊢ (normCV‘𝑈) ⊆ (normCV‘𝑈)
4	1, 2, 3	3pm3.2i 1340	. . 3 ⊢ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈))
5	4	jctr 524	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ∧ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈))))
6		eqid 2729	. . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
7		eqid 2729	. . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
8		eqid 2729	. . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
9		sspid.h	. . 3 ⊢ 𝐻 = (SubSp‘𝑈)
10	6, 6, 7, 7, 8, 8, 9	isssp 30686	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ NrmCVec ∧ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈)))))
11	5, 10	mpbird 257	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ 𝐻)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3905  ‘cfv 6486  NrmCVeccnv 30546   +_𝑣 cpv 30547   ·_𝑠OLD cns 30549  norm_CVcnmcv 30552  SubSpcss 30683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fo 6492  df-fv 6494  df-oprab 7357  df-1st 7931  df-2nd 7932  df-vc 30521  df-nv 30554  df-va 30557  df-sm 30559  df-nmcv 30562  df-ssp 30684

This theorem is referenced by:  hhsssh  31231