Theorem sspval 27578

Description: The set of all subspaces of a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
sspval.g	|- G = ( +v ` U )
sspval.s	|- S = ( .sOLD ` U )
sspval.n	|- N = ( normCV ` U )
sspval.h	|- H = ( SubSp ` U )

Assertion

Ref	Expression
sspval	|- ( U e. NrmCVec -> H = { w e. NrmCVec | ( ( +v ` w ) C_ G /\ ( .sOLD ` w ) C_ S /\ ( normCV ` w ) C_ N ) } )

Distinct variable groups:   w, G   w, N   w, S   w, U

Allowed substitution hint:   H( w)

Proof of Theorem sspval

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sspval.h	. 2 |- H = ( SubSp ` U )
2		fveq2 6191	. . . . . . 7 |- ( u = U -> ( +v ` u ) = ( +v ` U ) )
3		sspval.g	. . . . . . 7 |- G = ( +v ` U )
4	2, 3	syl6eqr 2674	. . . . . 6 |- ( u = U -> ( +v ` u ) = G )
5	4	sseq2d 3633	. . . . 5 |- ( u = U -> ( ( +v ` w ) C_ ( +v ` u ) <-> ( +v ` w ) C_ G ) )
6		fveq2 6191	. . . . . . 7 |- ( u = U -> ( .sOLD ` u ) = ( .sOLD ` U ) )
7		sspval.s	. . . . . . 7 |- S = ( .sOLD ` U )
8	6, 7	syl6eqr 2674	. . . . . 6 |- ( u = U -> ( .sOLD ` u ) = S )
9	8	sseq2d 3633	. . . . 5 |- ( u = U -> ( ( .sOLD ` w ) C_ ( .sOLD ` u ) <-> ( .sOLD ` w ) C_ S ) )
10		fveq2 6191	. . . . . . 7 |- ( u = U -> ( normCV ` u ) = ( normCV ` U ) )
11		sspval.n	. . . . . . 7 |- N = ( normCV ` U )
12	10, 11	syl6eqr 2674	. . . . . 6 |- ( u = U -> ( normCV ` u ) = N )
13	12	sseq2d 3633	. . . . 5 |- ( u = U -> ( ( normCV ` w ) C_ ( normCV ` u ) <-> ( normCV ` w ) C_ N ) )
14	5, 9, 13	3anbi123d 1399	. . . 4 |- ( u = U -> ( ( ( +v ` w ) C_ ( +v ` u ) /\ ( .sOLD ` w ) C_ ( .sOLD ` u ) /\ ( normCV ` w ) C_ ( normCV ` u ) ) <-> ( ( +v ` w ) C_ G /\ ( .sOLD ` w ) C_ S /\ ( normCV ` w ) C_ N ) ) )
15	14	rabbidv 3189	. . 3 |- ( u = U -> { w e. NrmCVec | ( ( +v ` w ) C_ ( +v ` u ) /\ ( .sOLD ` w ) C_ ( .sOLD ` u ) /\ ( normCV ` w ) C_ ( normCV ` u ) ) } = { w e. NrmCVec | ( ( +v ` w ) C_ G /\ ( .sOLD ` w ) C_ S /\ ( normCV ` w ) C_ N ) } )
16		df-ssp 27577	. . 3 |- SubSp = ( u e. NrmCVec |-> { w e. NrmCVec | ( ( +v ` w ) C_ ( +v ` u ) /\ ( .sOLD ` w ) C_ ( .sOLD ` u ) /\ ( normCV ` w ) C_ ( normCV ` u ) ) } )
17		fvex 6201	. . . . . . . 8 |- ( +v ` U ) e. _V
18	3, 17	eqeltri 2697	. . . . . . 7 |- G e. _V
19	18	pwex 4848	. . . . . 6 |- ~P G e. _V
20		fvex 6201	. . . . . . . 8 |- ( .sOLD ` U ) e. _V
21	7, 20	eqeltri 2697	. . . . . . 7 |- S e. _V
22	21	pwex 4848	. . . . . 6 |- ~P S e. _V
23	19, 22	xpex 6962	. . . . 5 |- ( ~P G X. ~P S ) e. _V
24		fvex 6201	. . . . . . 7 |- ( normCV ` U ) e. _V
25	11, 24	eqeltri 2697	. . . . . 6 |- N e. _V
26	25	pwex 4848	. . . . 5 |- ~P N e. _V
27	23, 26	xpex 6962	. . . 4 |- ( ( ~P G X. ~P S ) X. ~P N ) e. _V
28		rabss 3679	. . . . 5 |- ( { w e. NrmCVec | ( ( +v ` w ) C_ G /\ ( .sOLD ` w ) C_ S /\ ( normCV ` w ) C_ N ) } C_ ( ( ~P G X. ~P S ) X. ~P N ) <-> A. w e. NrmCVec ( ( ( +v ` w ) C_ G /\ ( .sOLD ` w ) C_ S /\ ( normCV ` w ) C_ N ) -> w e. ( ( ~P G X. ~P S ) X. ~P N ) ) )
29		fvex 6201	. . . . . . . . . 10 |- ( +v ` w ) e. _V
30	29	elpw 4164	. . . . . . . . 9 |- ( ( +v ` w ) e. ~P G <-> ( +v ` w ) C_ G )
31		fvex 6201	. . . . . . . . . 10 |- ( .sOLD ` w ) e. _V
32	31	elpw 4164	. . . . . . . . 9 |- ( ( .sOLD ` w ) e. ~P S <-> ( .sOLD ` w ) C_ S )
33		opelxpi 5148	. . . . . . . . 9 |- ( ( ( +v ` w ) e. ~P G /\ ( .sOLD ` w ) e. ~P S ) -> <. ( +v ` w ) , ( .sOLD ` w ) >. e. ( ~P G X. ~P S ) )
34	30, 32, 33	syl2anbr 497	. . . . . . . 8 |- ( ( ( +v ` w ) C_ G /\ ( .sOLD ` w ) C_ S ) -> <. ( +v ` w ) , ( .sOLD ` w ) >. e. ( ~P G X. ~P S ) )
35		fvex 6201	. . . . . . . . . 10 |- ( normCV ` w ) e. _V
36	35	elpw 4164	. . . . . . . . 9 |- ( ( normCV ` w ) e. ~P N <-> ( normCV ` w ) C_ N )
37	36	biimpri 218	. . . . . . . 8 |- ( ( normCV ` w ) C_ N -> ( normCV ` w ) e. ~P N )
38		opelxpi 5148	. . . . . . . 8 |- ( ( <. ( +v ` w ) , ( .sOLD ` w ) >. e. ( ~P G X. ~P S ) /\ ( normCV ` w ) e. ~P N ) -> <. <. ( +v ` w ) , ( .sOLD ` w ) >. , ( normCV ` w ) >. e. ( ( ~P G X. ~P S ) X. ~P N ) )
39	34, 37, 38	syl2an 494	. . . . . . 7 |- ( ( ( ( +v ` w ) C_ G /\ ( .sOLD ` w ) C_ S ) /\ ( normCV ` w ) C_ N ) -> <. <. ( +v ` w ) , ( .sOLD ` w ) >. , ( normCV ` w ) >. e. ( ( ~P G X. ~P S ) X. ~P N ) )
40	39	3impa 1259	. . . . . 6 |- ( ( ( +v ` w ) C_ G /\ ( .sOLD ` w ) C_ S /\ ( normCV ` w ) C_ N ) -> <. <. ( +v ` w ) , ( .sOLD ` w ) >. , ( normCV ` w ) >. e. ( ( ~P G X. ~P S ) X. ~P N ) )
41		eqid 2622	. . . . . . . 8 |- ( +v ` w ) = ( +v ` w )
42		eqid 2622	. . . . . . . 8 |- ( .sOLD ` w ) = ( .sOLD ` w )
43		eqid 2622	. . . . . . . 8 |- ( normCV ` w ) = ( normCV ` w )
44	41, 42, 43	nvop 27531	. . . . . . 7 |- ( w e. NrmCVec -> w = <. <. ( +v ` w ) , ( .sOLD ` w ) >. , ( normCV ` w ) >. )
45	44	eleq1d 2686	. . . . . 6 |- ( w e. NrmCVec -> ( w e. ( ( ~P G X. ~P S ) X. ~P N ) <-> <. <. ( +v ` w ) , ( .sOLD ` w ) >. , ( normCV ` w ) >. e. ( ( ~P G X. ~P S ) X. ~P N ) ) )
46	40, 45	syl5ibr 236	. . . . 5 |- ( w e. NrmCVec -> ( ( ( +v ` w ) C_ G /\ ( .sOLD ` w ) C_ S /\ ( normCV ` w ) C_ N ) -> w e. ( ( ~P G X. ~P S ) X. ~P N ) ) )
47	28, 46	mprgbir 2927	. . . 4 |- { w e. NrmCVec | ( ( +v ` w ) C_ G /\ ( .sOLD ` w ) C_ S /\ ( normCV ` w ) C_ N ) } C_ ( ( ~P G X. ~P S ) X. ~P N )
48	27, 47	ssexi 4803	. . 3 |- { w e. NrmCVec | ( ( +v ` w ) C_ G /\ ( .sOLD ` w ) C_ S /\ ( normCV ` w ) C_ N ) } e. _V
49	15, 16, 48	fvmpt 6282	. 2 |- ( U e. NrmCVec -> ( SubSp ` U ) = { w e. NrmCVec | ( ( +v ` w ) C_ G /\ ( .sOLD ` w ) C_ S /\ ( normCV ` w ) C_ N ) } )
50	1, 49	syl5eq 2668	1 |- ( U e. NrmCVec -> H = { w e. NrmCVec | ( ( +v ` w ) C_ G /\ ( .sOLD ` w ) C_ S /\ ( normCV ` w ) C_ N ) } )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   <.cop 4183   X. cxp 5112   ` 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:  isssp  27579