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Theorem ocin 26037
Description: Intersection of a Hilbert subspace and its complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
ocin  |-  ( A  e.  SH  ->  ( A  i^i  ( _|_ `  A
) )  =  0H )

Proof of Theorem ocin
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 shocel 26023 . . . . . . 7  |-  ( A  e.  SH  ->  (
x  e.  ( _|_ `  A )  <->  ( x  e.  ~H  /\  A. y  e.  A  ( x  .ih  y )  =  0 ) ) )
2 oveq2 6303 . . . . . . . . . 10  |-  ( y  =  x  ->  (
x  .ih  y )  =  ( x  .ih  x ) )
32eqeq1d 2469 . . . . . . . . 9  |-  ( y  =  x  ->  (
( x  .ih  y
)  =  0  <->  (
x  .ih  x )  =  0 ) )
43rspccv 3216 . . . . . . . 8  |-  ( A. y  e.  A  (
x  .ih  y )  =  0  ->  (
x  e.  A  -> 
( x  .ih  x
)  =  0 ) )
5 his6 25839 . . . . . . . . 9  |-  ( x  e.  ~H  ->  (
( x  .ih  x
)  =  0  <->  x  =  0h ) )
65biimpd 207 . . . . . . . 8  |-  ( x  e.  ~H  ->  (
( x  .ih  x
)  =  0  ->  x  =  0h )
)
74, 6sylan9r 658 . . . . . . 7  |-  ( ( x  e.  ~H  /\  A. y  e.  A  ( x  .ih  y )  =  0 )  -> 
( x  e.  A  ->  x  =  0h )
)
81, 7syl6bi 228 . . . . . 6  |-  ( A  e.  SH  ->  (
x  e.  ( _|_ `  A )  ->  (
x  e.  A  ->  x  =  0h )
) )
98com23 78 . . . . 5  |-  ( A  e.  SH  ->  (
x  e.  A  -> 
( x  e.  ( _|_ `  A )  ->  x  =  0h ) ) )
109impd 431 . . . 4  |-  ( A  e.  SH  ->  (
( x  e.  A  /\  x  e.  ( _|_ `  A ) )  ->  x  =  0h ) )
11 sh0 25956 . . . . . 6  |-  ( A  e.  SH  ->  0h  e.  A )
12 oc0 26031 . . . . . 6  |-  ( A  e.  SH  ->  0h  e.  ( _|_ `  A
) )
1311, 12jca 532 . . . . 5  |-  ( A  e.  SH  ->  ( 0h  e.  A  /\  0h  e.  ( _|_ `  A
) ) )
14 eleq1 2539 . . . . . 6  |-  ( x  =  0h  ->  (
x  e.  A  <->  0h  e.  A ) )
15 eleq1 2539 . . . . . 6  |-  ( x  =  0h  ->  (
x  e.  ( _|_ `  A )  <->  0h  e.  ( _|_ `  A ) ) )
1614, 15anbi12d 710 . . . . 5  |-  ( x  =  0h  ->  (
( x  e.  A  /\  x  e.  ( _|_ `  A ) )  <-> 
( 0h  e.  A  /\  0h  e.  ( _|_ `  A ) ) ) )
1713, 16syl5ibrcom 222 . . . 4  |-  ( A  e.  SH  ->  (
x  =  0h  ->  ( x  e.  A  /\  x  e.  ( _|_ `  A ) ) ) )
1810, 17impbid 191 . . 3  |-  ( A  e.  SH  ->  (
( x  e.  A  /\  x  e.  ( _|_ `  A ) )  <-> 
x  =  0h )
)
19 elin 3692 . . 3  |-  ( x  e.  ( A  i^i  ( _|_ `  A ) )  <->  ( x  e.  A  /\  x  e.  ( _|_ `  A
) ) )
20 elch0 25995 . . 3  |-  ( x  e.  0H  <->  x  =  0h )
2118, 19, 203bitr4g 288 . 2  |-  ( A  e.  SH  ->  (
x  e.  ( A  i^i  ( _|_ `  A
) )  <->  x  e.  0H ) )
2221eqrdv 2464 1  |-  ( A  e.  SH  ->  ( A  i^i  ( _|_ `  A
) )  =  0H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817    i^i cin 3480   ` cfv 5594  (class class class)co 6295   0cc0 9504   ~Hchil 25659    .ih csp 25662   0hc0v 25664   SHcsh 25668   _|_cort 25670   0Hc0h 25675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-hilex 25739  ax-hfvadd 25740  ax-hv0cl 25743  ax-hfvmul 25745  ax-hvmul0 25750  ax-hfi 25819  ax-his2 25823  ax-his3 25824  ax-his4 25825
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-ltxr 9645  df-sh 25947  df-oc 25993  df-ch0 25994
This theorem is referenced by:  ocnel  26039  chocunii  26042  pjhtheu  26135  pjpreeq  26139  omlsi  26145  ococi  26146  pjoc1i  26172  orthin  26187  ssjo  26188  chocini  26195  chscllem3  26380
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