Theorem sspid 8318

Description: A normed complex vector space is a subspace of itself.

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 2070	. . . 4 ⊢ ( +v ‘U) ⊆ ( +v ‘U)
2		ssid 2070	. . . 4 ⊢ ( ·s ‘U) ⊆ ( ·s ‘U)
3		ssid 2070	. . . 4 ⊢ (norm ‘U) ⊆ (norm ‘U)
4	1, 2, 3	3pm3.2i 816	. . 3 ⊢ (( +v ‘U) ⊆ ( +v ‘U) ⋀ ( ·s ‘U) ⊆ ( ·s ‘U) ⋀ (norm ‘U) ⊆ (norm ‘U))
5	4	jctr 291	. 2 ⊢ (U ∈ NrmCVec → (U ∈ NrmCVec ⋀ (( +v ‘U) ⊆ ( +v ‘U) ⋀ ( ·s ‘U) ⊆ ( ·s ‘U) ⋀ (norm ‘U) ⊆ (norm ‘U))))
6		eqid 1468	. . 3 ⊢ ( +v ‘U) = ( +v ‘U)
7		eqid 1468	. . 3 ⊢ ( ·s ‘U) = ( ·s ‘U)
8		eqid 1468	. . 3 ⊢ (norm ‘U) = (norm ‘U)
9		sspid.h	. . 3 ⊢ H = (SubSp ‘U)
10	6, 6, 7, 7, 8, 8, 9	isssp 8317	. 2 ⊢ (U ∈ NrmCVec → (U ∈ H ↔ (U ∈ NrmCVec ⋀ (( +v ‘U) ⊆ ( +v ‘U) ⋀ ( ·s ‘U) ⊆ ( ·s ‘U) ⋀ (norm ‘U) ⊆ (norm ‘U)))))
11	5, 10	mpbird 196	1 ⊢ (U ∈ NrmCVec → U ∈ H)

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 773   = wceq 953   ∈ wcel 955   ⊆ wss 2037   ‘cfv 3172  NrmCVeccnv 8141   +_v cpv 8142   ·_s cns 8144  normcnm 8147  SubSpcss 8314

This theorem is referenced by:  hhsssh 9059

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fo 3186  df-fv 3188  df-oprab 3951  df-1st 4063  df-2nd 4064  df-nv 8149  df-va 8152  df-sm 8154  df-nm 8157  df-ssp 8315

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