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Theorem hmopidmpji 22726
Description: An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Halmos seems to omit the proof that  H is a closed subspace, which is not trivial as hmopidmchi 22725 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
hmopidmch.1  |-  T  e. 
HrmOp
hmopidmch.2  |-  ( T  o.  T )  =  T
Assertion
Ref Expression
hmopidmpji  |-  T  =  ( proj  h `  ran  T )
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.

Proof of Theorem hmopidmpji
StepHypRef Expression
1 hmopidmch.1 . . . . . 6  |-  T  e. 
HrmOp
2 hmoplin 22516 . . . . . 6  |-  ( T  e.  HrmOp  ->  T  e.  LinOp
)
31, 2ax-mp 10 . . . . 5  |-  T  e. 
LinOp
43lnopfi 22543 . . . 4  |-  T : ~H
--> ~H
5 ffn 5356 . . . 4  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
64, 5ax-mp 10 . . 3  |-  T  Fn  ~H
7 hmopidmch.2 . . . . 5  |-  ( T  o.  T )  =  T
81, 7hmopidmchi 22725 . . . 4  |-  ran  T  e.  CH
98pjfni 22274 . . 3  |-  ( proj 
h `  ran  T )  Fn  ~H
10 eqfnfv 5585 . . 3  |-  ( ( T  Fn  ~H  /\  ( proj  h `  ran  T )  Fn  ~H )  ->  ( T  =  (
proj  h `  ran  T
)  <->  A. x  e.  ~H  ( T `  x )  =  ( ( proj 
h `  ran  T ) `
 x ) ) )
116, 9, 10mp2an 655 . 2  |-  ( T  =  ( proj  h `  ran  T )  <->  A. x  e.  ~H  ( T `  x )  =  ( ( proj  h `  ran  T ) `  x ) )
12 fnfvelrn 5625 . . . . 5  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ran  T
)
136, 12mpan 653 . . . 4  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ran  T )
14 id 21 . . . . . 6  |-  ( x  e.  ~H  ->  x  e.  ~H )
154ffvelrni 5627 . . . . . 6  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
16 hvsubcl 21591 . . . . . 6  |-  ( ( x  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( x  -h  ( T `  x )
)  e.  ~H )
1714, 15, 16syl2anc 644 . . . . 5  |-  ( x  e.  ~H  ->  (
x  -h  ( T `
 x ) )  e.  ~H )
18 simpl 445 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  x  e.  ~H )
1915adantr 453 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( T `  x
)  e.  ~H )
204ffvelrni 5627 . . . . . . . . . 10  |-  ( y  e.  ~H  ->  ( T `  y )  e.  ~H )
2120adantl 454 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( T `  y
)  e.  ~H )
22 his2sub 21665 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  ( T `  x )  e.  ~H  /\  ( T `  y )  e.  ~H )  ->  (
( x  -h  ( T `  x )
)  .ih  ( T `  y ) )  =  ( ( x  .ih  ( T `  y ) )  -  ( ( T `  x ) 
.ih  ( T `  y ) ) ) )
2318, 19, 21, 22syl3anc 1184 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( x  -h  ( T `  x ) )  .ih  ( T `
 y ) )  =  ( ( x 
.ih  ( T `  y ) )  -  ( ( T `  x )  .ih  ( T `  y )
) ) )
24 hmop 22496 . . . . . . . . . . . 12  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H  /\  ( T `
 y )  e. 
~H )  ->  (
x  .ih  ( T `  ( T `  y
) ) )  =  ( ( T `  x )  .ih  ( T `  y )
) )
251, 24mp3an1 1266 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  y )  e.  ~H )  -> 
( x  .ih  ( T `  ( T `  y ) ) )  =  ( ( T `
 x )  .ih  ( T `  y ) ) )
2620, 25sylan2 462 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  ( T `  ( T `  y ) ) )  =  ( ( T `
 x )  .ih  ( T `  y ) ) )
277fveq1i 5488 . . . . . . . . . . . . 13  |-  ( ( T  o.  T ) `
 y )  =  ( T `  y
)
284, 4hocoi 22338 . . . . . . . . . . . . 13  |-  ( y  e.  ~H  ->  (
( T  o.  T
) `  y )  =  ( T `  ( T `  y ) ) )
2927, 28syl5reqr 2333 . . . . . . . . . . . 12  |-  ( y  e.  ~H  ->  ( T `  ( T `  y ) )  =  ( T `  y
) )
3029adantl 454 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( T `  ( T `  y )
)  =  ( T `
 y ) )
3130oveq2d 5837 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  ( T `  ( T `  y ) ) )  =  ( x  .ih  ( T `  y ) ) )
3226, 31eqtr3d 2320 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( T `  x )  .ih  ( T `  y )
)  =  ( x 
.ih  ( T `  y ) ) )
3332oveq2d 5837 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( x  .ih  ( T `  y ) )  -  ( ( T `  x ) 
.ih  ( T `  y ) ) )  =  ( ( x 
.ih  ( T `  y ) )  -  ( x  .ih  ( T `
 y ) ) ) )
34 hicl 21653 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  ( T `  y )  e.  ~H )  -> 
( x  .ih  ( T `  y )
)  e.  CC )
3520, 34sylan2 462 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  ( T `  y )
)  e.  CC )
3635subidd 9142 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( x  .ih  ( T `  y ) )  -  ( x 
.ih  ( T `  y ) ) )  =  0 )
3723, 33, 363eqtrd 2322 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( x  -h  ( T `  x ) )  .ih  ( T `
 y ) )  =  0 )
3837ralrimiva 2629 . . . . . 6  |-  ( x  e.  ~H  ->  A. y  e.  ~H  ( ( x  -h  ( T `  x ) )  .ih  ( T `  y ) )  =  0 )
39 oveq2 5829 . . . . . . . . 9  |-  ( z  =  ( T `  y )  ->  (
( x  -h  ( T `  x )
)  .ih  z )  =  ( ( x  -h  ( T `  x ) )  .ih  ( T `  y ) ) )
4039eqeq1d 2294 . . . . . . . 8  |-  ( z  =  ( T `  y )  ->  (
( ( x  -h  ( T `  x ) )  .ih  z )  =  0  <->  ( (
x  -h  ( T `
 x ) ) 
.ih  ( T `  y ) )  =  0 ) )
4140ralrn 5631 . . . . . . 7  |-  ( T  Fn  ~H  ->  ( A. z  e.  ran  T ( ( x  -h  ( T `  x ) )  .ih  z )  =  0  <->  A. y  e.  ~H  ( ( x  -h  ( T `  x ) )  .ih  ( T `  y ) )  =  0 ) )
426, 41ax-mp 10 . . . . . 6  |-  ( A. z  e.  ran  T ( ( x  -h  ( T `  x )
)  .ih  z )  =  0  <->  A. y  e.  ~H  ( ( x  -h  ( T `  x ) )  .ih  ( T `  y ) )  =  0 )
4338, 42sylibr 205 . . . . 5  |-  ( x  e.  ~H  ->  A. z  e.  ran  T ( ( x  -h  ( T `
 x ) ) 
.ih  z )  =  0 )
448chssii 21805 . . . . . 6  |-  ran  T  C_ 
~H
45 ocel 21854 . . . . . 6  |-  ( ran 
T  C_  ~H  ->  ( ( x  -h  ( T `  x )
)  e.  ( _|_ `  ran  T )  <->  ( (
x  -h  ( T `
 x ) )  e.  ~H  /\  A. z  e.  ran  T ( ( x  -h  ( T `  x )
)  .ih  z )  =  0 ) ) )
4644, 45ax-mp 10 . . . . 5  |-  ( ( x  -h  ( T `
 x ) )  e.  ( _|_ `  ran  T )  <->  ( ( x  -h  ( T `  x ) )  e. 
~H  /\  A. z  e.  ran  T ( ( x  -h  ( T `
 x ) ) 
.ih  z )  =  0 ) )
4717, 43, 46sylanbrc 647 . . . 4  |-  ( x  e.  ~H  ->  (
x  -h  ( T `
 x ) )  e.  ( _|_ `  ran  T ) )
488pjcompi 22245 . . . 4  |-  ( ( ( T `  x
)  e.  ran  T  /\  ( x  -h  ( T `  x )
)  e.  ( _|_ `  ran  T ) )  ->  ( ( proj 
h `  ran  T ) `
 ( ( T `
 x )  +h  ( x  -h  ( T `  x )
) ) )  =  ( T `  x
) )
4913, 47, 48syl2anc 644 . . 3  |-  ( x  e.  ~H  ->  (
( proj  h `  ran  T ) `  ( ( T `  x )  +h  ( x  -h  ( T `  x ) ) ) )  =  ( T `  x
) )
50 hvpncan3 21615 . . . . 5  |-  ( ( ( T `  x
)  e.  ~H  /\  x  e.  ~H )  ->  ( ( T `  x )  +h  (
x  -h  ( T `
 x ) ) )  =  x )
5115, 14, 50syl2anc 644 . . . 4  |-  ( x  e.  ~H  ->  (
( T `  x
)  +h  ( x  -h  ( T `  x ) ) )  =  x )
5251fveq2d 5491 . . 3  |-  ( x  e.  ~H  ->  (
( proj  h `  ran  T ) `  ( ( T `  x )  +h  ( x  -h  ( T `  x ) ) ) )  =  ( ( proj  h `  ran  T ) `  x
) )
5349, 52eqtr3d 2320 . 2  |-  ( x  e.  ~H  ->  ( T `  x )  =  ( ( proj 
h `  ran  T ) `
 x ) )
5411, 53mprgbir 2616 1  |-  T  =  ( proj  h `  ran  T )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1625    e. wcel 1687   A.wral 2546    C_ wss 3155   ran crn 4691    o. ccom 4694    Fn wfn 5218   -->wf 5219   ` cfv 5223  (class class class)co 5821   CCcc 8732   0cc0 8734    - cmin 9034   ~Hchil 21493    +h cva 21494    .ih csp 21496    -h cmv 21499   _|_cort 21504   proj  hcpjh 21511   LinOpclo 21521   HrmOpcho 21524
This theorem is referenced by:  hmopidmpj  22728
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-inf2 7339  ax-cc 8058  ax-dc 8069  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-pre-sup 8812  ax-addf 8813  ax-mulf 8814  ax-hilex 21573  ax-hfvadd 21574  ax-hvcom 21575  ax-hvass 21576  ax-hv0cl 21577  ax-hvaddid 21578  ax-hfvmul 21579  ax-hvmulid 21580  ax-hvmulass 21581  ax-hvdistr1 21582  ax-hvdistr2 21583  ax-hvmul0 21584  ax-hfi 21652  ax-his1 21655  ax-his2 21656  ax-his3 21657  ax-his4 21658  ax-hcompl 21775
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-int 3866  df-iun 3910  df-iin 3911  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-se 4354  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-isom 5232  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-of 6041  df-1st 6085  df-2nd 6086  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-1o 6476  df-2o 6477  df-oadd 6480  df-omul 6481  df-er 6657  df-map 6771  df-pm 6772  df-ixp 6815  df-en 6861  df-dom 6862  df-sdom 6863  df-fin 6864  df-fi 7162  df-sup 7191  df-oi 7222  df-card 7569  df-acn 7572  df-cda 7791  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-div 9421  df-nn 9744  df-2 9801  df-3 9802  df-4 9803  df-5 9804  df-6 9805  df-7 9806  df-8 9807  df-9 9808  df-10 9809  df-n0 9963  df-z 10022  df-dec 10122  df-uz 10228  df-q 10314  df-rp 10352  df-xneg 10449  df-xadd 10450  df-xmul 10451  df-ioo 10656  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-fl 10921  df-seq 11043  df-exp 11101  df-hash 11334  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-clim 11958  df-rlim 11959  df-sum 12155  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-starv 13219  df-sca 13220  df-vsca 13221  df-tset 13223  df-ple 13224  df-ds 13226  df-hom 13228  df-cco 13229  df-rest 13323  df-topn 13324  df-topgen 13340  df-pt 13341  df-prds 13344  df-xrs 13399  df-0g 13400  df-gsum 13401  df-qtop 13406  df-imas 13407  df-xps 13409  df-mre 13484  df-mrc 13485  df-acs 13487  df-mnd 14363  df-submnd 14412  df-mulg 14488  df-cntz 14789  df-cmn 15087  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-cld 16752  df-ntr 16753  df-cls 16754  df-nei 16831  df-cn 16953  df-cnp 16954  df-lm 16955  df-t1 17038  df-haus 17039  df-cmp 17110  df-tx 17253  df-hmeo 17442  df-fbas 17516  df-fg 17517  df-fil 17537  df-fm 17629  df-flim 17630  df-flf 17631  df-fcls 17632  df-xms 17881  df-ms 17882  df-tms 17883  df-cncf 18378  df-cfil 18677  df-cau 18678  df-cmet 18679  df-grpo 20852  df-gid 20853  df-ginv 20854  df-gdiv 20855  df-ablo 20943  df-subgo 20963  df-vc 21096  df-nv 21142  df-va 21145  df-ba 21146  df-sm 21147  df-0v 21148  df-vs 21149  df-nmcv 21150  df-ims 21151  df-dip 21268  df-ssp 21292  df-lno 21316  df-nmoo 21317  df-blo 21318  df-0o 21319  df-ph 21385  df-cbn 21436  df-hlo 21459  df-hnorm 21542  df-hba 21543  df-hvsub 21545  df-hlim 21546  df-hcau 21547  df-sh 21780  df-ch 21795  df-oc 21825  df-ch0 21826  df-shs 21881  df-pjh 21968  df-h0op 22322  df-nmop 22413  df-cnop 22414  df-lnop 22415  df-bdop 22416  df-unop 22417  df-hmop 22418
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