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Theorem hmopidmpji 27187
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 27186 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 26977 . . . . . 6  |-  ( T  e.  HrmOp  ->  T  e.  LinOp
)
31, 2ax-mp 5 . . . . 5  |-  T  e. 
LinOp
43lnopfi 27004 . . . 4  |-  T : ~H
--> ~H
5 ffn 5639 . . . 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 27186 . . . 4  |-  ran  T  e.  CH
98pjfni 26736 . . 3  |-  ( proj h `  ran  T )  Fn  ~H
10 eqfnfv 5883 . . 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 670 . 2  |-  ( T  =  ( proj h `  ran  T )  <->  A. x  e.  ~H  ( T `  x )  =  ( ( proj h `  ran  T ) `  x
) )
12 fnfvelrn 5930 . . . . 5  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ran  T
)
136, 12mpan 668 . . . 4  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ran  T )
14 id 22 . . . . . 6  |-  ( x  e.  ~H  ->  x  e.  ~H )
154ffvelrni 5932 . . . . . 6  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
16 hvsubcl 26051 . . . . . 6  |-  ( ( x  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( x  -h  ( T `  x )
)  e.  ~H )
1714, 15, 16syl2anc 659 . . . . 5  |-  ( x  e.  ~H  ->  (
x  -h  ( T `
 x ) )  e.  ~H )
18 simpl 455 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  x  e.  ~H )
1915adantr 463 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( T `  x
)  e.  ~H )
204ffvelrni 5932 . . . . . . . . . 10  |-  ( y  e.  ~H  ->  ( T `  y )  e.  ~H )
2120adantl 464 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( T `  y
)  e.  ~H )
22 his2sub 26126 . . . . . . . . 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 1226 . . . . . . . 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 26957 . . . . . . . . . . . 12  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H  /\  ( T `
 y )  e. 
~H )  ->  (
x  .ih  ( T `  ( T `  y
) ) )  =  ( ( T `  x )  .ih  ( T `  y )
) )
251, 24mp3an1 1309 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  ( T `  y )  e.  ~H )  -> 
( x  .ih  ( T `  ( T `  y ) ) )  =  ( ( T `
 x )  .ih  ( T `  y ) ) )
2620, 25sylan2 472 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  ( T `  ( T `  y ) ) )  =  ( ( T `
 x )  .ih  ( T `  y ) ) )
277fveq1i 5775 . . . . . . . . . . . . 13  |-  ( ( T  o.  T ) `
 y )  =  ( T `  y
)
284, 4hocoi 26799 . . . . . . . . . . . . 13  |-  ( y  e.  ~H  ->  (
( T  o.  T
) `  y )  =  ( T `  ( T `  y ) ) )
2927, 28syl5reqr 2438 . . . . . . . . . . . 12  |-  ( y  e.  ~H  ->  ( T `  ( T `  y ) )  =  ( T `  y
) )
3029adantl 464 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( T `  ( T `  y )
)  =  ( T `
 y ) )
3130oveq2d 6212 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  ( T `  ( T `  y ) ) )  =  ( x  .ih  ( T `  y ) ) )
3226, 31eqtr3d 2425 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( T `  x )  .ih  ( T `  y )
)  =  ( x 
.ih  ( T `  y ) ) )
3332oveq2d 6212 . . . . . . . 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 26114 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  ( T `  y )  e.  ~H )  -> 
( x  .ih  ( T `  y )
)  e.  CC )
3520, 34sylan2 472 . . . . . . . . 9  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  .ih  ( T `  y )
)  e.  CC )
3635subidd 9832 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( x  .ih  ( T `  y ) )  -  ( x 
.ih  ( T `  y ) ) )  =  0 )
3723, 33, 363eqtrd 2427 . . . . . . 7  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( ( x  -h  ( T `  x ) )  .ih  ( T `
 y ) )  =  0 )
3837ralrimiva 2796 . . . . . 6  |-  ( x  e.  ~H  ->  A. y  e.  ~H  ( ( x  -h  ( T `  x ) )  .ih  ( T `  y ) )  =  0 )
39 oveq2 6204 . . . . . . . . 9  |-  ( z  =  ( T `  y )  ->  (
( x  -h  ( T `  x )
)  .ih  z )  =  ( ( x  -h  ( T `  x ) )  .ih  ( T `  y ) ) )
4039eqeq1d 2384 . . . . . . . 8  |-  ( z  =  ( T `  y )  ->  (
( ( x  -h  ( T `  x ) )  .ih  z )  =  0  <->  ( (
x  -h  ( T `
 x ) ) 
.ih  ( T `  y ) )  =  0 ) )
4140ralrn 5936 . . . . . . 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 212 . . . . 5  |-  ( x  e.  ~H  ->  A. z  e.  ran  T ( ( x  -h  ( T `
 x ) ) 
.ih  z )  =  0 )
448chssii 26266 . . . . . 6  |-  ran  T  C_ 
~H
45 ocel 26316 . . . . . 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 662 . . . 4  |-  ( x  e.  ~H  ->  (
x  -h  ( T `
 x ) )  e.  ( _|_ `  ran  T ) )
488pjcompi 26707 . . . 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 659 . . 3  |-  ( x  e.  ~H  ->  (
( proj h `  ran  T ) `  (
( T `  x
)  +h  ( x  -h  ( T `  x ) ) ) )  =  ( T `
 x ) )
50 hvpncan3 26076 . . . . 5  |-  ( ( ( T `  x
)  e.  ~H  /\  x  e.  ~H )  ->  ( ( T `  x )  +h  (
x  -h  ( T `
 x ) ) )  =  x )
5115, 14, 50syl2anc 659 . . . 4  |-  ( x  e.  ~H  ->  (
( T `  x
)  +h  ( x  -h  ( T `  x ) ) )  =  x )
5251fveq2d 5778 . . 3  |-  ( x  e.  ~H  ->  (
( proj h `  ran  T ) `  (
( T `  x
)  +h  ( x  -h  ( T `  x ) ) ) )  =  ( (
proj h `  ran  T
) `  x )
)
5349, 52eqtr3d 2425 . 2  |-  ( x  e.  ~H  ->  ( T `  x )  =  ( ( proj h `  ran  T ) `
 x ) )
5411, 53mprgbir 2746 1  |-  T  =  ( proj h `  ran  T )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732    C_ wss 3389   ran crn 4914    o. ccom 4917    Fn wfn 5491   -->wf 5492   ` cfv 5496  (class class class)co 6196   CCcc 9401   0cc0 9403    - cmin 9718   ~Hchil 25953    +h cva 25954    .ih csp 25956    -h cmv 25959   _|_cort 25964   proj hcpjh 25971   LinOpclo 25981   HrmOpcho 25984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cc 8728  ax-dc 8739  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482  ax-mulf 9483  ax-hilex 26033  ax-hfvadd 26034  ax-hvcom 26035  ax-hvass 26036  ax-hv0cl 26037  ax-hvaddid 26038  ax-hfvmul 26039  ax-hvmulid 26040  ax-hvmulass 26041  ax-hvdistr1 26042  ax-hvdistr2 26043  ax-hvmul0 26044  ax-hfi 26113  ax-his1 26116  ax-his2 26117  ax-his3 26118  ax-his4 26119  ax-hcompl 26236
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-omul 7053  df-er 7229  df-map 7340  df-pm 7341  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-fi 7786  df-sup 7816  df-oi 7850  df-card 8233  df-acn 8236  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-ico 11456  df-icc 11457  df-fz 11594  df-fzo 11718  df-fl 11828  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-rlim 13314  df-sum 13511  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-hom 14726  df-cco 14727  df-rest 14830  df-topn 14831  df-0g 14849  df-gsum 14850  df-topgen 14851  df-pt 14852  df-prds 14855  df-xrs 14909  df-qtop 14914  df-imas 14915  df-xps 14917  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-mulg 16177  df-cntz 16472  df-cmn 16917  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-fbas 18529  df-fg 18530  df-cnfld 18534  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-cld 19605  df-ntr 19606  df-cls 19607  df-nei 19685  df-cn 19814  df-cnp 19815  df-lm 19816  df-t1 19901  df-haus 19902  df-cmp 19973  df-tx 20148  df-hmeo 20341  df-fil 20432  df-fm 20524  df-flim 20525  df-flf 20526  df-fcls 20527  df-xms 20908  df-ms 20909  df-tms 20910  df-cncf 21467  df-cfil 21779  df-cau 21780  df-cmet 21781  df-grpo 25310  df-gid 25311  df-ginv 25312  df-gdiv 25313  df-ablo 25401  df-subgo 25421  df-vc 25556  df-nv 25602  df-va 25605  df-ba 25606  df-sm 25607  df-0v 25608  df-vs 25609  df-nmcv 25610  df-ims 25611  df-dip 25728  df-ssp 25752  df-lno 25776  df-nmoo 25777  df-blo 25778  df-0o 25779  df-ph 25845  df-cbn 25896  df-hlo 25919  df-hnorm 26002  df-hba 26003  df-hvsub 26005  df-hlim 26006  df-hcau 26007  df-sh 26241  df-ch 26256  df-oc 26287  df-ch0 26288  df-shs 26343  df-pjh 26430  df-h0op 26783  df-nmop 26874  df-cnop 26875  df-lnop 26876  df-bdop 26877  df-unop 26878  df-hmop 26879
This theorem is referenced by:  hmopidmpj  27189
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