Theorem sspid 31333

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 3930	. . . 4 ⊢ ( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈)
2		ssid 3930	. . . 4 ⊢ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈)
3		ssid 3930	. . . 4 ⊢ (normCV‘𝑈) ⊆ (normCV‘𝑈)
4	1, 2, 3	3pm3.2i 1358	. . 3 ⊢ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈))
5	4	jctr 534	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ∧ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈))))
6		eqid 2760	. . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
7		eqid 2760	. . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
8		eqid 2760	. . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
9		sspid.h	. . 3 ⊢ 𝐻 = (SubSp‘𝑈)
10	6, 6, 7, 7, 8, 8, 9	isssp 31332	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ NrmCVec ∧ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈)))))
11	5, 10	mpbird 260	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ 𝐻)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895  ‘cfv 6539  NrmCVeccnv 31192   +_𝑣 cpv 31193   ·_𝑠OLD cns 31195  norm_CVcnmcv 31198  SubSpcss 31329

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fo 6545  df-fv 6547  df-oprab 7425  df-1st 8002  df-2nd 8003  df-vc 31167  df-nv 31200  df-va 31203  df-sm 31205  df-nmcv 31208  df-ssp 31330

This theorem is used by:  hhsssh  31877