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Theorem hmopidmpji 23643
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 23642 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 )

Proof of Theorem hmopidmpji
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hmopidmch.1 . . . . . 6  |-  T  e. 
HrmOp
2 hmoplin 23433 . . . . . 6  |-  ( T  e.  HrmOp  ->  T  e.  LinOp
)
31, 2ax-mp 8 . . . . 5  |-  T  e. 
LinOp
43lnopfi 23460 . . . 4  |-  T : ~H
--> ~H
5 ffn 5582 . . . 4  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
64, 5ax-mp 8 . . 3  |-  T  Fn  ~H
7 hmopidmch.2 . . . . 5  |-  ( T  o.  T )  =  T
81, 7hmopidmchi 23642 . . . 4  |-  ran  T  e.  CH
98pjfni 23191 . . 3  |-  ( proj 
h `  ran  T )  Fn  ~H
10 eqfnfv 5818 . . 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 654 . 2  |-  ( T  =  ( proj  h `  ran  T )  <->  A. x  e.  ~H  ( T `  x )  =  ( ( proj  h `  ran  T ) `  x ) )
12 fnfvelrn 5858 . . . . 5  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ran  T
)
136, 12mpan 652 . . . 4  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ran  T )
14 id 20 . . . . . 6  |-  ( x  e.  ~H  ->  x  e.  ~H )
154ffvelrni 5860 . . . . . 6  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
16 hvsubcl 22508 . . . . . 6  |-  ( ( x  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( x  -h  ( T `  x )
)  e.  ~H )
1714, 15, 16syl2anc 643 . . . . 5  |-  ( x  e.  ~H  ->  (
x  -h  ( T `
 x ) )  e.  ~H )
18 simpl 444 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  x  e.  ~H )
1915adantr 452 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( T `  x
)  e.  ~H )
204ffvelrni 5860 . . . . . . . . . 10  |-  ( y  e.  ~H  ->  ( T `  y )  e.  ~H )
2120adantl 453 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( T `  y
)  e.  ~H )
22 his2sub 22582 . . . . . . . . 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 23413 . . . . . . . . . . . 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 461 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  ( T `  ( T `  y ) ) )  =  ( ( T `
 x )  .ih  ( T `  y ) ) )
277fveq1i 5720 . . . . . . . . . . . . 13  |-  ( ( T  o.  T ) `
 y )  =  ( T `  y
)
284, 4hocoi 23255 . . . . . . . . . . . . 13  |-  ( y  e.  ~H  ->  (
( T  o.  T
) `  y )  =  ( T `  ( T `  y ) ) )
2927, 28syl5reqr 2482 . . . . . . . . . . . 12  |-  ( y  e.  ~H  ->  ( T `  ( T `  y ) )  =  ( T `  y
) )
3029adantl 453 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( T `  ( T `  y )
)  =  ( T `
 y ) )
3130oveq2d 6088 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  ( T `  ( T `  y ) ) )  =  ( x  .ih  ( T `  y ) ) )
3226, 31eqtr3d 2469 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( T `  x )  .ih  ( T `  y )
)  =  ( x 
.ih  ( T `  y ) ) )
3332oveq2d 6088 . . . . . . . 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 22570 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  ( T `  y )  e.  ~H )  -> 
( x  .ih  ( T `  y )
)  e.  CC )
3520, 34sylan2 461 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  ( T `  y )
)  e.  CC )
3635subidd 9388 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( x  .ih  ( T `  y ) )  -  ( x 
.ih  ( T `  y ) ) )  =  0 )
3723, 33, 363eqtrd 2471 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( x  -h  ( T `  x ) )  .ih  ( T `
 y ) )  =  0 )
3837ralrimiva 2781 . . . . . 6  |-  ( x  e.  ~H  ->  A. y  e.  ~H  ( ( x  -h  ( T `  x ) )  .ih  ( T `  y ) )  =  0 )
39 oveq2 6080 . . . . . . . . 9  |-  ( z  =  ( T `  y )  ->  (
( x  -h  ( T `  x )
)  .ih  z )  =  ( ( x  -h  ( T `  x ) )  .ih  ( T `  y ) ) )
4039eqeq1d 2443 . . . . . . . 8  |-  ( z  =  ( T `  y )  ->  (
( ( x  -h  ( T `  x ) )  .ih  z )  =  0  <->  ( (
x  -h  ( T `
 x ) ) 
.ih  ( T `  y ) )  =  0 ) )
4140ralrn 5864 . . . . . . 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 8 . . . . . 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 204 . . . . 5  |-  ( x  e.  ~H  ->  A. z  e.  ran  T ( ( x  -h  ( T `
 x ) ) 
.ih  z )  =  0 )
448chssii 22722 . . . . . 6  |-  ran  T  C_ 
~H
45 ocel 22771 . . . . . 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 8 . . . . 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 646 . . . 4  |-  ( x  e.  ~H  ->  (
x  -h  ( T `
 x ) )  e.  ( _|_ `  ran  T ) )
488pjcompi 23162 . . . 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 643 . . 3  |-  ( x  e.  ~H  ->  (
( proj  h `  ran  T ) `  ( ( T `  x )  +h  ( x  -h  ( T `  x ) ) ) )  =  ( T `  x
) )
50 hvpncan3 22532 . . . . 5  |-  ( ( ( T `  x
)  e.  ~H  /\  x  e.  ~H )  ->  ( ( T `  x )  +h  (
x  -h  ( T `
 x ) ) )  =  x )
5115, 14, 50syl2anc 643 . . . 4  |-  ( x  e.  ~H  ->  (
( T `  x
)  +h  ( x  -h  ( T `  x ) ) )  =  x )
5251fveq2d 5723 . . 3  |-  ( x  e.  ~H  ->  (
( proj  h `  ran  T ) `  ( ( T `  x )  +h  ( x  -h  ( T `  x ) ) ) )  =  ( ( proj  h `  ran  T ) `  x
) )
5349, 52eqtr3d 2469 . 2  |-  ( x  e.  ~H  ->  ( T `  x )  =  ( ( proj 
h `  ran  T ) `
 x ) )
5411, 53mprgbir 2768 1  |-  T  =  ( proj  h `  ran  T )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   ran crn 4870    o. ccom 4873    Fn wfn 5440   -->wf 5441   ` cfv 5445  (class class class)co 6072   CCcc 8977   0cc0 8979    - cmin 9280   ~Hchil 22410    +h cva 22411    .ih csp 22413    -h cmv 22416   _|_cort 22421   proj  hcpjh 22428   LinOpclo 22438   HrmOpcho 22441
This theorem is referenced by:  hmopidmpj  23645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cc 8304  ax-dc 8315  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-addf 9058  ax-mulf 9059  ax-hilex 22490  ax-hfvadd 22491  ax-hvcom 22492  ax-hvass 22493  ax-hv0cl 22494  ax-hvaddid 22495  ax-hfvmul 22496  ax-hvmulid 22497  ax-hvmulass 22498  ax-hvdistr1 22499  ax-hvdistr2 22500  ax-hvmul0 22501  ax-hfi 22569  ax-his1 22572  ax-his2 22573  ax-his3 22574  ax-his4 22575  ax-hcompl 22692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-omul 6720  df-er 6896  df-map 7011  df-pm 7012  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-fi 7407  df-sup 7437  df-oi 7468  df-card 7815  df-acn 7818  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-ioo 10909  df-ico 10911  df-icc 10912  df-fz 11033  df-fzo 11124  df-fl 11190  df-seq 11312  df-exp 11371  df-hash 11607  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-clim 12270  df-rlim 12271  df-sum 12468  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-starv 13532  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-unif 13540  df-hom 13541  df-cco 13542  df-rest 13638  df-topn 13639  df-topgen 13655  df-pt 13656  df-prds 13659  df-xrs 13714  df-0g 13715  df-gsum 13716  df-qtop 13721  df-imas 13722  df-xps 13724  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-submnd 14727  df-mulg 14803  df-cntz 15104  df-cmn 15402  df-psmet 16682  df-xmet 16683  df-met 16684  df-bl 16685  df-mopn 16686  df-fbas 16687  df-fg 16688  df-cnfld 16692  df-top 16951  df-bases 16953  df-topon 16954  df-topsp 16955  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-cn 17279  df-cnp 17280  df-lm 17281  df-t1 17366  df-haus 17367  df-cmp 17438  df-tx 17582  df-hmeo 17775  df-fil 17866  df-fm 17958  df-flim 17959  df-flf 17960  df-fcls 17961  df-xms 18338  df-ms 18339  df-tms 18340  df-cncf 18896  df-cfil 19196  df-cau 19197  df-cmet 19198  df-grpo 21767  df-gid 21768  df-ginv 21769  df-gdiv 21770  df-ablo 21858  df-subgo 21878  df-vc 22013  df-nv 22059  df-va 22062  df-ba 22063  df-sm 22064  df-0v 22065  df-vs 22066  df-nmcv 22067  df-ims 22068  df-dip 22185  df-ssp 22209  df-lno 22233  df-nmoo 22234  df-blo 22235  df-0o 22236  df-ph 22302  df-cbn 22353  df-hlo 22376  df-hnorm 22459  df-hba 22460  df-hvsub 22462  df-hlim 22463  df-hcau 22464  df-sh 22697  df-ch 22712  df-oc 22742  df-ch0 22743  df-shs 22798  df-pjh 22885  df-h0op 23239  df-nmop 23330  df-cnop 23331  df-lnop 23332  df-bdop 23333  df-unop 23334  df-hmop 23335
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