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Theorem pjspansn 24979
Description: A projection on the span of a singleton. (The proof ws shortened by Mario Carneiro, 15-Dec-2013.) (Contributed by NM, 28-May-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
pjspansn  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  ->  (
( proj h `  ( span `  { A } ) ) `  B )  =  ( ( ( B  .ih  A )  /  ( (
normh `  A ) ^
2 ) )  .h  A ) )

Proof of Theorem pjspansn
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 spansnch 24962 . . . 4  |-  ( A  e.  ~H  ->  ( span `  { A }
)  e.  CH )
213ad2ant1 1009 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  ->  ( span `  { A }
)  e.  CH )
3 simp2 989 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  ->  B  e.  ~H )
4 eqid 2442 . . . . 5  |-  ( (
proj h `  ( span `  { A } ) ) `  B )  =  ( ( proj h `  ( span `  { A } ) ) `  B )
5 pjeq 24801 . . . . 5  |-  ( ( ( span `  { A } )  e.  CH  /\  B  e.  ~H )  ->  ( ( ( proj h `  ( span `  { A } ) ) `  B )  =  ( ( proj h `  ( span `  { A } ) ) `  B )  <-> 
( ( ( proj h `  ( span `  { A } ) ) `  B )  e.  ( span `  { A } )  /\  E. y  e.  ( _|_ `  ( span `  { A } ) ) B  =  ( ( (
proj h `  ( span `  { A } ) ) `  B )  +h  y ) ) ) )
64, 5mpbii 211 . . . 4  |-  ( ( ( span `  { A } )  e.  CH  /\  B  e.  ~H )  ->  ( ( ( proj h `  ( span `  { A } ) ) `  B )  e.  ( span `  { A } )  /\  E. y  e.  ( _|_ `  ( span `  { A } ) ) B  =  ( ( (
proj h `  ( span `  { A } ) ) `  B )  +h  y ) ) )
76simprd 463 . . 3  |-  ( ( ( span `  { A } )  e.  CH  /\  B  e.  ~H )  ->  E. y  e.  ( _|_ `  ( span `  { A } ) ) B  =  ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y
) )
82, 3, 7syl2anc 661 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  ->  E. y  e.  ( _|_ `  ( span `  { A }
) ) B  =  ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y ) )
9 oveq1 6097 . . . . . . 7  |-  ( B  =  ( ( (
proj h `  ( span `  { A } ) ) `  B )  +h  y )  -> 
( B  .ih  A
)  =  ( ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y
)  .ih  A )
)
109ad2antll 728 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  ( y  e.  ( _|_ `  ( span `  { A } ) )  /\  B  =  ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y ) ) )  ->  ( B  .ih  A )  =  ( ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y )  .ih  A ) )
11 pjhcl 24803 . . . . . . . . . . 11  |-  ( ( ( span `  { A } )  e.  CH  /\  B  e.  ~H )  ->  ( ( proj h `  ( span `  { A } ) ) `  B )  e.  ~H )
122, 3, 11syl2anc 661 . . . . . . . . . 10  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  ->  (
( proj h `  ( span `  { A } ) ) `  B )  e.  ~H )
1312adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  ( ( proj h `  ( span `  { A } ) ) `  B )  e.  ~H )
14 choccl 24708 . . . . . . . . . . . 12  |-  ( (
span `  { A } )  e.  CH  ->  ( _|_ `  ( span `  { A }
) )  e.  CH )
151, 14syl 16 . . . . . . . . . . 11  |-  ( A  e.  ~H  ->  ( _|_ `  ( span `  { A } ) )  e. 
CH )
16153ad2ant1 1009 . . . . . . . . . 10  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  ->  ( _|_ `  ( span `  { A } ) )  e. 
CH )
17 chel 24632 . . . . . . . . . 10  |-  ( ( ( _|_ `  ( span `  { A }
) )  e.  CH  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  y  e.  ~H )
1816, 17sylan 471 . . . . . . . . 9  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  y  e.  ~H )
19 simpl1 991 . . . . . . . . 9  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  A  e.  ~H )
20 ax-his2 24484 . . . . . . . . 9  |-  ( ( ( ( proj h `  ( span `  { A } ) ) `  B )  e.  ~H  /\  y  e.  ~H  /\  A  e.  ~H )  ->  ( ( ( (
proj h `  ( span `  { A } ) ) `  B )  +h  y )  .ih  A )  =  ( ( ( ( proj h `  ( span `  { A } ) ) `  B )  .ih  A
)  +  ( y 
.ih  A ) ) )
2113, 18, 19, 20syl3anc 1218 . . . . . . . 8  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  ( ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y
)  .ih  A )  =  ( ( ( ( proj h `  ( span `  { A } ) ) `  B )  .ih  A
)  +  ( y 
.ih  A ) ) )
22 spansnsh 24963 . . . . . . . . . . . . 13  |-  ( A  e.  ~H  ->  ( span `  { A }
)  e.  SH )
2322adantr 465 . . . . . . . . . . . 12  |-  ( ( A  e.  ~H  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  ( span `  { A } )  e.  SH )
24 spansnid 24965 . . . . . . . . . . . . 13  |-  ( A  e.  ~H  ->  A  e.  ( span `  { A } ) )
2524adantr 465 . . . . . . . . . . . 12  |-  ( ( A  e.  ~H  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  A  e.  (
span `  { A } ) )
26 simpr 461 . . . . . . . . . . . 12  |-  ( ( A  e.  ~H  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  y  e.  ( _|_ `  ( span `  { A } ) ) )
27 shocorth 24694 . . . . . . . . . . . . 13  |-  ( (
span `  { A } )  e.  SH  ->  ( ( A  e.  ( span `  { A } )  /\  y  e.  ( _|_ `  ( span `  { A }
) ) )  -> 
( A  .ih  y
)  =  0 ) )
28273impib 1185 . . . . . . . . . . . 12  |-  ( ( ( span `  { A } )  e.  SH  /\  A  e.  ( span `  { A } )  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  ( A  .ih  y )  =  0 )
2923, 25, 26, 28syl3anc 1218 . . . . . . . . . . 11  |-  ( ( A  e.  ~H  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  ( A  .ih  y )  =  0 )
3015, 17sylan 471 . . . . . . . . . . . 12  |-  ( ( A  e.  ~H  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  y  e.  ~H )
31 orthcom 24509 . . . . . . . . . . . 12  |-  ( ( A  e.  ~H  /\  y  e.  ~H )  ->  ( ( A  .ih  y )  =  0  <-> 
( y  .ih  A
)  =  0 ) )
3230, 31syldan 470 . . . . . . . . . . 11  |-  ( ( A  e.  ~H  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  ( ( A 
.ih  y )  =  0  <->  ( y  .ih  A )  =  0 ) )
3329, 32mpbid 210 . . . . . . . . . 10  |-  ( ( A  e.  ~H  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  ( y  .ih  A )  =  0 )
34333ad2antl1 1150 . . . . . . . . 9  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  ( y  .ih  A )  =  0 )
3534oveq2d 6106 . . . . . . . 8  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  ( ( ( ( proj h `  ( span `  { A } ) ) `  B )  .ih  A
)  +  ( y 
.ih  A ) )  =  ( ( ( ( proj h `  ( span `  { A } ) ) `  B )  .ih  A
)  +  0 ) )
36 hicl 24481 . . . . . . . . . 10  |-  ( ( ( ( proj h `  ( span `  { A } ) ) `  B )  e.  ~H  /\  A  e.  ~H )  ->  ( ( ( proj h `  ( span `  { A } ) ) `  B ) 
.ih  A )  e.  CC )
3713, 19, 36syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  ( ( (
proj h `  ( span `  { A } ) ) `  B ) 
.ih  A )  e.  CC )
3837addid1d 9568 . . . . . . . 8  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  ( ( ( ( proj h `  ( span `  { A } ) ) `  B )  .ih  A
)  +  0 )  =  ( ( (
proj h `  ( span `  { A } ) ) `  B ) 
.ih  A ) )
3921, 35, 383eqtrd 2478 . . . . . . 7  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  y  e.  ( _|_ `  ( span `  { A } ) ) )  ->  ( ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y
)  .ih  A )  =  ( ( (
proj h `  ( span `  { A } ) ) `  B ) 
.ih  A ) )
4039adantrr 716 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  ( y  e.  ( _|_ `  ( span `  { A } ) )  /\  B  =  ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y ) ) )  ->  ( (
( ( proj h `  ( span `  { A } ) ) `  B )  +h  y
)  .ih  A )  =  ( ( (
proj h `  ( span `  { A } ) ) `  B ) 
.ih  A ) )
4110, 40eqtrd 2474 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  ( y  e.  ( _|_ `  ( span `  { A } ) )  /\  B  =  ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y ) ) )  ->  ( B  .ih  A )  =  ( ( ( proj h `  ( span `  { A } ) ) `  B )  .ih  A
) )
4241oveq1d 6105 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  ( y  e.  ( _|_ `  ( span `  { A } ) )  /\  B  =  ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y ) ) )  ->  ( ( B  .ih  A )  / 
( ( normh `  A
) ^ 2 ) )  =  ( ( ( ( proj h `  ( span `  { A } ) ) `  B )  .ih  A
)  /  ( (
normh `  A ) ^
2 ) ) )
4342oveq1d 6105 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  ( y  e.  ( _|_ `  ( span `  { A } ) )  /\  B  =  ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y ) ) )  ->  ( (
( B  .ih  A
)  /  ( (
normh `  A ) ^
2 ) )  .h  A )  =  ( ( ( ( (
proj h `  ( span `  { A } ) ) `  B ) 
.ih  A )  / 
( ( normh `  A
) ^ 2 ) )  .h  A ) )
44 simpl1 991 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  ( y  e.  ( _|_ `  ( span `  { A } ) )  /\  B  =  ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y ) ) )  ->  A  e.  ~H )
45 simpl3 993 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  ( y  e.  ( _|_ `  ( span `  { A } ) )  /\  B  =  ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y ) ) )  ->  A  =/=  0h )
46 axpjcl 24802 . . . . . 6  |-  ( ( ( span `  { A } )  e.  CH  /\  B  e.  ~H )  ->  ( ( proj h `  ( span `  { A } ) ) `  B )  e.  (
span `  { A } ) )
472, 3, 46syl2anc 661 . . . . 5  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  ->  (
( proj h `  ( span `  { A } ) ) `  B )  e.  (
span `  { A } ) )
4847adantr 465 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  ( y  e.  ( _|_ `  ( span `  { A } ) )  /\  B  =  ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y ) ) )  ->  ( ( proj h `  ( span `  { A } ) ) `  B )  e.  ( span `  { A } ) )
49 normcan 24978 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h  /\  (
( proj h `  ( span `  { A } ) ) `  B )  e.  (
span `  { A } ) )  -> 
( ( ( ( ( proj h `  ( span `  { A } ) ) `  B )  .ih  A
)  /  ( (
normh `  A ) ^
2 ) )  .h  A )  =  ( ( proj h `  ( span `  { A } ) ) `  B ) )
5044, 45, 48, 49syl3anc 1218 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  ( y  e.  ( _|_ `  ( span `  { A } ) )  /\  B  =  ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y ) ) )  ->  ( (
( ( ( proj h `  ( span `  { A } ) ) `  B ) 
.ih  A )  / 
( ( normh `  A
) ^ 2 ) )  .h  A )  =  ( ( proj h `  ( span `  { A } ) ) `  B ) )
5143, 50eqtr2d 2475 . 2  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  /\  ( y  e.  ( _|_ `  ( span `  { A } ) )  /\  B  =  ( ( ( proj h `  ( span `  { A } ) ) `  B )  +h  y ) ) )  ->  ( ( proj h `  ( span `  { A } ) ) `  B )  =  ( ( ( B  .ih  A )  /  ( ( normh `  A ) ^ 2 ) )  .h  A
) )
528, 51rexlimddv 2844 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  A  =/=  0h )  ->  (
( proj h `  ( span `  { A } ) ) `  B )  =  ( ( ( B  .ih  A )  /  ( (
normh `  A ) ^
2 ) )  .h  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2605   E.wrex 2715   {csn 3876   ` cfv 5417  (class class class)co 6090   CCcc 9279   0cc0 9281    + caddc 9284    / cdiv 9992   2c2 10370   ^cexp 11864   ~Hchil 24320    +h cva 24321    .h csm 24322    .ih csp 24323   normhcno 24324   0hc0v 24325   SHcsh 24329   CHcch 24330   _|_cort 24331   spancspn 24333   proj hcpjh 24338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cc 8603  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361  ax-hilex 24400  ax-hfvadd 24401  ax-hvcom 24402  ax-hvass 24403  ax-hv0cl 24404  ax-hvaddid 24405  ax-hfvmul 24406  ax-hvmulid 24407  ax-hvmulass 24408  ax-hvdistr1 24409  ax-hvdistr2 24410  ax-hvmul0 24411  ax-hfi 24480  ax-his1 24483  ax-his2 24484  ax-his3 24485  ax-his4 24486  ax-hcompl 24603
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  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-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-omul 6924  df-er 7100  df-map 7215  df-pm 7216  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-fi 7660  df-sup 7690  df-oi 7723  df-card 8108  df-acn 8111  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-q 10953  df-rp 10991  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-ioo 11303  df-ico 11305  df-icc 11306  df-fz 11437  df-fzo 11548  df-fl 11641  df-seq 11806  df-exp 11865  df-hash 12103  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-clim 12965  df-rlim 12966  df-sum 13163  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-sca 14253  df-vsca 14254  df-ip 14255  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-hom 14261  df-cco 14262  df-rest 14360  df-topn 14361  df-0g 14379  df-gsum 14380  df-topgen 14381  df-pt 14382  df-prds 14385  df-xrs 14439  df-qtop 14444  df-imas 14445  df-xps 14447  df-mre 14523  df-mrc 14524  df-acs 14526  df-mnd 15414  df-submnd 15464  df-mulg 15547  df-cntz 15834  df-cmn 16278  df-psmet 17808  df-xmet 17809  df-met 17810  df-bl 17811  df-mopn 17812  df-fbas 17813  df-fg 17814  df-cnfld 17818  df-top 18502  df-bases 18504  df-topon 18505  df-topsp 18506  df-cld 18622  df-ntr 18623  df-cls 18624  df-nei 18701  df-cn 18830  df-cnp 18831  df-lm 18832  df-haus 18918  df-tx 19134  df-hmeo 19327  df-fil 19418  df-fm 19510  df-flim 19511  df-flf 19512  df-xms 19894  df-ms 19895  df-tms 19896  df-cfil 20765  df-cau 20766  df-cmet 20767  df-grpo 23677  df-gid 23678  df-ginv 23679  df-gdiv 23680  df-ablo 23768  df-subgo 23788  df-vc 23923  df-nv 23969  df-va 23972  df-ba 23973  df-sm 23974  df-0v 23975  df-vs 23976  df-nmcv 23977  df-ims 23978  df-dip 24095  df-ssp 24119  df-ph 24212  df-cbn 24263  df-hnorm 24369  df-hba 24370  df-hvsub 24372  df-hlim 24373  df-hcau 24374  df-sh 24608  df-ch 24623  df-oc 24654  df-ch0 24655  df-shs 24710  df-span 24711  df-pjh 24797
This theorem is referenced by:  kbpj  25359
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