Theorem sspid 27580

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 e. NrmCVec -> U e. H )

Proof of Theorem sspid

Step	Hyp	Ref	Expression
1		ssid 3624	. . . 4 |- ( +v ` U ) C_ ( +v ` U )
2		ssid 3624	. . . 4 |- ( .sOLD ` U ) C_ ( .sOLD ` U )
3		ssid 3624	. . . 4 |- ( normCV ` U ) C_ ( normCV ` U )
4	1, 2, 3	3pm3.2i 1239	. . 3 |- ( ( +v ` U ) C_ ( +v ` U ) /\ ( .sOLD ` U ) C_ ( .sOLD ` U ) /\ ( normCV ` U ) C_ ( normCV ` U ) )
5	4	jctr 565	. 2 |- ( U e. NrmCVec -> ( U e. NrmCVec /\ ( ( +v ` U ) C_ ( +v ` U ) /\ ( .sOLD ` U ) C_ ( .sOLD ` U ) /\ ( normCV ` U ) C_ ( normCV ` U ) ) ) )
6		eqid 2622	. . 3 |- ( +v ` U ) = ( +v ` U )
7		eqid 2622	. . 3 |- ( .sOLD ` U ) = ( .sOLD ` U )
8		eqid 2622	. . 3 |- ( normCV ` U ) = ( normCV ` U )
9		sspid.h	. . 3 |- H = ( SubSp ` U )
10	6, 6, 7, 7, 8, 8, 9	isssp 27579	. 2 |- ( U e. NrmCVec -> ( U e. H <-> ( U e. NrmCVec /\ ( ( +v ` U ) C_ ( +v ` U ) /\ ( .sOLD ` U ) C_ ( .sOLD ` U ) /\ ( normCV ` U ) C_ ( normCV ` U ) ) ) ) )
11	5, 10	mpbird 247	1 |- ( U e. NrmCVec -> U e. H )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   ` cfv 5888   NrmCVeccnv 27439   +vcpv 27440   .sOLDcns 27442   norm_CVcnmcv 27445   SubSpcss 27576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-oprab 6654  df-1st 7168  df-2nd 7169  df-vc 27414  df-nv 27447  df-va 27450  df-sm 27452  df-nmcv 27455  df-ssp 27577

This theorem is referenced by:  hhsssh  28126