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Theorem hmopidmpji 27886
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 27885 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 27676 . . . . . 6  |-  ( T  e.  HrmOp  ->  T  e.  LinOp
)
31, 2ax-mp 5 . . . . 5  |-  T  e. 
LinOp
43lnopfi 27703 . . . 4  |-  T : ~H
--> ~H
5 ffn 5739 . . . 4  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
64, 5ax-mp 5 . . 3  |-  T  Fn  ~H
7 hmopidmch.2 . . . . 5  |-  ( T  o.  T )  =  T
81, 7hmopidmchi 27885 . . . 4  |-  ran  T  e.  CH
98pjfni 27435 . . 3  |-  ( proj h `  ran  T )  Fn  ~H
10 eqfnfv 5991 . . 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 686 . 2  |-  ( T  =  ( proj h `  ran  T )  <->  A. x  e.  ~H  ( T `  x )  =  ( ( proj h `  ran  T ) `  x
) )
12 fnfvelrn 6034 . . . . 5  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ran  T
)
136, 12mpan 684 . . . 4  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ran  T )
14 id 22 . . . . . 6  |-  ( x  e.  ~H  ->  x  e.  ~H )
154ffvelrni 6036 . . . . . 6  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
16 hvsubcl 26751 . . . . . 6  |-  ( ( x  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( x  -h  ( T `  x )
)  e.  ~H )
1714, 15, 16syl2anc 673 . . . . 5  |-  ( x  e.  ~H  ->  (
x  -h  ( T `
 x ) )  e.  ~H )
18 simpl 464 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  x  e.  ~H )
1915adantr 472 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( T `  x
)  e.  ~H )
204ffvelrni 6036 . . . . . . . . . 10  |-  ( y  e.  ~H  ->  ( T `  y )  e.  ~H )
2120adantl 473 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( T `  y
)  e.  ~H )
22 his2sub 26826 . . . . . . . . 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 1292 . . . . . . . 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 27656 . . . . . . . . . . . 12  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H  /\  ( T `
 y )  e. 
~H )  ->  (
x  .ih  ( T `  ( T `  y
) ) )  =  ( ( T `  x )  .ih  ( T `  y )
) )
251, 24mp3an1 1377 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  y )  e.  ~H )  -> 
( x  .ih  ( T `  ( T `  y ) ) )  =  ( ( T `
 x )  .ih  ( T `  y ) ) )
2620, 25sylan2 482 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  ( T `  ( T `  y ) ) )  =  ( ( T `
 x )  .ih  ( T `  y ) ) )
277fveq1i 5880 . . . . . . . . . . . . 13  |-  ( ( T  o.  T ) `
 y )  =  ( T `  y
)
284, 4hocoi 27498 . . . . . . . . . . . . 13  |-  ( y  e.  ~H  ->  (
( T  o.  T
) `  y )  =  ( T `  ( T `  y ) ) )
2927, 28syl5reqr 2520 . . . . . . . . . . . 12  |-  ( y  e.  ~H  ->  ( T `  ( T `  y ) )  =  ( T `  y
) )
3029adantl 473 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( T `  ( T `  y )
)  =  ( T `
 y ) )
3130oveq2d 6324 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  ( T `  ( T `  y ) ) )  =  ( x  .ih  ( T `  y ) ) )
3226, 31eqtr3d 2507 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( T `  x )  .ih  ( T `  y )
)  =  ( x 
.ih  ( T `  y ) ) )
3332oveq2d 6324 . . . . . . . 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 26814 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  ( T `  y )  e.  ~H )  -> 
( x  .ih  ( T `  y )
)  e.  CC )
3520, 34sylan2 482 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  ( T `  y )
)  e.  CC )
3635subidd 9993 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( x  .ih  ( T `  y ) )  -  ( x 
.ih  ( T `  y ) ) )  =  0 )
3723, 33, 363eqtrd 2509 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( x  -h  ( T `  x ) )  .ih  ( T `
 y ) )  =  0 )
3837ralrimiva 2809 . . . . . 6  |-  ( x  e.  ~H  ->  A. y  e.  ~H  ( ( x  -h  ( T `  x ) )  .ih  ( T `  y ) )  =  0 )
39 oveq2 6316 . . . . . . . . 9  |-  ( z  =  ( T `  y )  ->  (
( x  -h  ( T `  x )
)  .ih  z )  =  ( ( x  -h  ( T `  x ) )  .ih  ( T `  y ) ) )
4039eqeq1d 2473 . . . . . . . 8  |-  ( z  =  ( T `  y )  ->  (
( ( x  -h  ( T `  x ) )  .ih  z )  =  0  <->  ( (
x  -h  ( T `
 x ) ) 
.ih  ( T `  y ) )  =  0 ) )
4140ralrn 6040 . . . . . . 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 5 . . . . . 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 217 . . . . 5  |-  ( x  e.  ~H  ->  A. z  e.  ran  T ( ( x  -h  ( T `
 x ) ) 
.ih  z )  =  0 )
448chssii 26965 . . . . . 6  |-  ran  T  C_ 
~H
45 ocel 27015 . . . . . 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 5 . . . . 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 677 . . . 4  |-  ( x  e.  ~H  ->  (
x  -h  ( T `
 x ) )  e.  ( _|_ `  ran  T ) )
488pjcompi 27406 . . . 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 673 . . 3  |-  ( x  e.  ~H  ->  (
( proj h `  ran  T ) `  (
( T `  x
)  +h  ( x  -h  ( T `  x ) ) ) )  =  ( T `
 x ) )
50 hvpncan3 26776 . . . . 5  |-  ( ( ( T `  x
)  e.  ~H  /\  x  e.  ~H )  ->  ( ( T `  x )  +h  (
x  -h  ( T `
 x ) ) )  =  x )
5115, 14, 50syl2anc 673 . . . 4  |-  ( x  e.  ~H  ->  (
( T `  x
)  +h  ( x  -h  ( T `  x ) ) )  =  x )
5251fveq2d 5883 . . 3  |-  ( x  e.  ~H  ->  (
( proj h `  ran  T ) `  (
( T `  x
)  +h  ( x  -h  ( T `  x ) ) ) )  =  ( (
proj h `  ran  T
) `  x )
)
5349, 52eqtr3d 2507 . 2  |-  ( x  e.  ~H  ->  ( T `  x )  =  ( ( proj h `  ran  T ) `
 x ) )
5411, 53mprgbir 2771 1  |-  T  =  ( proj h `  ran  T )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756    C_ wss 3390   ran crn 4840    o. ccom 4843    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   0cc0 9557    - cmin 9880   ~Hchil 26653    +h cva 26654    .ih csp 26656    -h cmv 26659   _|_cort 26664   proj hcpjh 26671   LinOpclo 26681   HrmOpcho 26684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-dc 8894  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637  ax-hilex 26733  ax-hfvadd 26734  ax-hvcom 26735  ax-hvass 26736  ax-hv0cl 26737  ax-hvaddid 26738  ax-hfvmul 26739  ax-hvmulid 26740  ax-hvmulass 26741  ax-hvdistr1 26742  ax-hvdistr2 26743  ax-hvmul0 26744  ax-hfi 26813  ax-his1 26816  ax-his2 26817  ax-his3 26818  ax-his4 26819  ax-hcompl 26936
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-cn 20320  df-cnp 20321  df-lm 20322  df-t1 20407  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-fcls 21034  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-cfil 22303  df-cau 22304  df-cmet 22305  df-grpo 26000  df-gid 26001  df-ginv 26002  df-gdiv 26003  df-ablo 26091  df-subgo 26111  df-vc 26246  df-nv 26292  df-va 26295  df-ba 26296  df-sm 26297  df-0v 26298  df-vs 26299  df-nmcv 26300  df-ims 26301  df-dip 26418  df-ssp 26442  df-lno 26466  df-nmoo 26467  df-blo 26468  df-0o 26469  df-ph 26535  df-cbn 26586  df-hlo 26619  df-hnorm 26702  df-hba 26703  df-hvsub 26705  df-hlim 26706  df-hcau 26707  df-sh 26941  df-ch 26955  df-oc 26986  df-ch0 26987  df-shs 27042  df-pjh 27129  df-h0op 27482  df-nmop 27573  df-cnop 27574  df-lnop 27575  df-bdop 27576  df-unop 27577  df-hmop 27578
This theorem is referenced by:  hmopidmpj  27888
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