Theorem sspid 30874

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 3958	. . . 4 ⊢ ( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈)
2		ssid 3958	. . . 4 ⊢ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈)
3		ssid 3958	. . . 4 ⊢ (normCV‘𝑈) ⊆ (normCV‘𝑈)
4	1, 2, 3	3pm3.2i 1352	. . 3 ⊢ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈))
5	4	jctr 532	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ∧ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈))))
6		eqid 2761	. . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
7		eqid 2761	. . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
8		eqid 2761	. . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
9		sspid.h	. . 3 ⊢ 𝐻 = (SubSp‘𝑈)
10	6, 6, 7, 7, 8, 8, 9	isssp 30873	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ NrmCVec ∧ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈)))))
11	5, 10	mpbird 259	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ 𝐻)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ⊆ wss 3904  ‘cfv 6517  NrmCVeccnv 30733   +_𝑣 cpv 30734   ·_𝑠OLD cns 30736  norm_CVcnmcv 30739  SubSpcss 30870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fo 6523  df-fv 6525  df-oprab 7396  df-1st 7966  df-2nd 7967  df-vc 30708  df-nv 30741  df-va 30744  df-sm 30746  df-nmcv 30749  df-ssp 30871

This theorem is referenced by:  hhsssh  31418