Theorem hhsssh2 9140

Description: The predicate "H is a subspace of Hilbert space."

Hypothesis

Ref	Expression
hhsssh2.1	|- W = <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.

Assertion

Ref	Expression
hhsssh2	|- (H e. SH <-> (W e. NrmCVec /\ H (_ H~))

Proof of Theorem hhsssh2

Step	Hyp	Ref	Expression
1		eqid 1475	. . 3 |- <.<. +h , .h >., normh>. = <.<. +h , .h >., normh>.
2		hhsssh2.1	. . 3 |- W = <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.
3	1, 2	hhsssh 9139	. 2 |- (H e. SH <-> (W e. (SubSp` <.<. +h , .h >., normh>.) /\ H (_ H~))
4	1	hhnv 9032	. . . . 5 |- <.<. +h , .h >., normh>. e. NrmCVec
5	1	hhva 9033	. . . . . 6 |- +h = (+v` <.<. +h , .h >., normh>.)
6	2	fveq2i 3727	. . . . . . 7 |- (+v` W) = (+v` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.)
7		eqid 1475	. . . . . . . . 9 |- (+v` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.) = (+v` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.)
8	7	vafval 8222	. . . . . . . 8 |- (+v` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.) = (1st` (1st` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.))
9		opex 2782	. . . . . . . . . . 11 |- <.( +h |` (H X. H)), ( .h |` (CC X. H))>. e. V
10	9	op1st 4085	. . . . . . . . . 10 |- (1st` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.) = <.( +h |` (H X. H)), ( .h |` (CC X. H))>.
11	10	fveq2i 3727	. . . . . . . . 9 |- (1st` (1st` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.)) = (1st` <.( +h |` (H X. H)), ( .h |` (CC X. H))>.)
12		hilabl 9027	. . . . . . . . . . 11 |- +h e. Abel
13		resexg 3394	. . . . . . . . . . 11 |- ( +h e. Abel -> ( +h |` (H X. H)) e. V)
14	12, 13	ax-mp 7	. . . . . . . . . 10 |- ( +h |` (H X. H)) e. V
15	14	op1st 4085	. . . . . . . . 9 |- (1st` <.( +h |` (H X. H)), ( .h |` (CC X. H))>.) = ( +h |` (H X. H))
16	11, 15	eqtr 1495	. . . . . . . 8 |- (1st` (1st` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.)) = ( +h |` (H X. H))
17	8, 16	eqtr 1495	. . . . . . 7 |- (+v` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.) = ( +h |` (H X. H))
18	6, 17	eqtr2 1496	. . . . . 6 |- ( +h |` (H X. H)) = (+v` W)
19	1	hhsm 9036	. . . . . 6 |- .h = (.s` <.<. +h , .h >., normh>.)
20	2	fveq2i 3727	. . . . . . 7 |- (.s` W) = (.s` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.)
21		eqid 1475	. . . . . . . . 9 |- (.s` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.) = (.s` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.)
22	21	smfval 8224	. . . . . . . 8 |- (.s` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.) = (2nd` (1st` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.))
23	10	fveq2i 3727	. . . . . . . . 9 |- (2nd` (1st` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.)) = (2nd` <.( +h |` (H X. H)), ( .h |` (CC X. H))>.)
24		hvmulex 8881	. . . . . . . . . . 11 |- .h e. V
25		resexg 3394	. . . . . . . . . . 11 |- ( .h e. V -> ( .h |` (CC X. H)) e. V)
26	24, 25	ax-mp 7	. . . . . . . . . 10 |- ( .h |` (CC X. H)) e. V
27	14, 26	op2nd 4086	. . . . . . . . 9 |- (2nd` <.( +h |` (H X. H)), ( .h |` (CC X. H))>.) = ( .h |` (CC X. H))
28	23, 27	eqtr 1495	. . . . . . . 8 |- (2nd` (1st` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.)) = ( .h |` (CC X. H))
29	22, 28	eqtr 1495	. . . . . . 7 |- (.s` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.) = ( .h |` (CC X. H))
30	20, 29	eqtr2 1496	. . . . . 6 |- ( .h |` (CC X. H)) = (.s` W)
31	1	hhnm 9038	. . . . . 6 |- normh = (norm` <.<. +h , .h >., normh>.)
32	2	fveq2i 3727	. . . . . . 7 |- (norm` W) = (norm` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.)
33		eqid 1475	. . . . . . . . 9 |- (norm` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.) = (norm` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.)
34	33	nmfval 8226	. . . . . . . 8 |- (norm` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.) = (2nd` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.)
35		normf 8989	. . . . . . . . . . 11 |- normh:H~-->RR
36		ax-hilex 8869	. . . . . . . . . . 11 |- H~ e. V
37		fex 3652	. . . . . . . . . . 11 |- ((normh:H~-->RR /\ H~ e. V) -> normh e. V)
38	35, 36, 37	mp2an 697	. . . . . . . . . 10 |- normh e. V
39		resexg 3394	. . . . . . . . . 10 |- (normh e. V -> (normh |` H) e. V)
40	38, 39	ax-mp 7	. . . . . . . . 9 |- (normh |` H) e. V
41	9, 40	op2nd 4086	. . . . . . . 8 |- (2nd` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.) = (normh |` H)
42	34, 41	eqtr 1495	. . . . . . 7 |- (norm` <.<.( +h |` (H X. H)), ( .h |` (CC X. H))>., (normh |` H)>.) = (normh |` H)
43	32, 42	eqtr2 1496	. . . . . 6 |- (normh |` H) = (norm` W)
44		eqid 1475	. . . . . 6 |- (SubSp` <.<. +h , .h >., normh>.) = (SubSp` <.<. +h , .h >., normh>.)
45	5, 18, 19, 30, 31, 43, 44	isssp 8383	. . . . 5 |- (<.<. +h , .h >., normh>. e. NrmCVec -> (W e. (SubSp` <.<. +h , .h >., normh>.) <-> (W e. NrmCVec /\ (( +h |` (H X. H)) (_ +h /\ ( .h |` (CC X. H)) (_ .h /\ (normh |` H) (_ normh))))
46	4, 45	ax-mp 7	. . . 4 |- (W e. (SubSp` <.<. +h , .h >., normh>.) <-> (W e. NrmCVec /\ (( +h |` (H X. H)) (_ +h /\ ( .h |` (CC X. H)) (_ .h /\ (normh |` H) (_ normh)))
47		resss 3383	. . . . 5 |- ( +h |` (H X. H)) (_ +h
48		resss 3383	. . . . 5 |- ( .h |` (CC X. H)) (_ .h
49		resss 3383	. . . . 5 |- (normh |` H) (_ normh
50	47, 48, 49	3pm3.2i 818	. . . 4 |- (( +h |` (H X. H)) (_ +h /\ ( .h |` (CC X. H)) (_ .h /\ (normh |` H) (_ normh)
51	46, 50	mpbiran2 729	. . 3 |- (W e. (SubSp` <.<. +h , .h >., normh>.) <-> W e. NrmCVec)
52	51	anbi1i 481	. 2 |- ((W e. (SubSp` <.<. +h , .h >., normh>.) /\ H (_ H~) <-> (W e. NrmCVec /\ H (_ H~))
53	3, 52	bitr 173	1 |- (H e. SH <-> (W e. NrmCVec /\ H (_ H~))

Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  Vcvv 1811   (_ wss 2047  <.cop 2411   X. cxp 3168   |` cres 3172  -->wf 3178  ` cfv 3182  1stc1st 4077  2ndc2nd 4078  CCcc 5232  RRcr 5233  Abelcabl 8099  NrmCVeccnv 8203  +vcpv 8204  .scns 8206  normcnm 8209  SubSpcss 8380  H~chil 8788   +h cva 8789   .h csm 8790  normhcno 8794  SHcsh 8797

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625  ax-hilex 8869  ax-hfvadd 8870  ax-hvcom 8871  ax-hvass 8872  ax-hv0cl 8873  ax-hvaddid 8874  ax-hfvmul 8875  ax-hvmulid 8876  ax-hvmulass 8877  ax-hvdistr1 8878  ax-hvdistr2 8879  ax-hvmul0 8880  ax-hfi 8946  ax-his1 8949  ax-his2 8950  ax-his3 8951  ax-his4 8952

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-sup 4574  df-ni