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Theorem ocin 21891
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 21877 . . . . . . 7  |-  ( A  e.  SH  ->  (
x  e.  ( _|_ `  A )  <->  ( x  e.  ~H  /\  A. y  e.  A  ( x  .ih  y )  =  0 ) ) )
2 oveq2 5882 . . . . . . . . . 10  |-  ( y  =  x  ->  (
x  .ih  y )  =  ( x  .ih  x ) )
32eqeq1d 2304 . . . . . . . . 9  |-  ( y  =  x  ->  (
( x  .ih  y
)  =  0  <->  (
x  .ih  x )  =  0 ) )
43rspccv 2894 . . . . . . . 8  |-  ( A. y  e.  A  (
x  .ih  y )  =  0  ->  (
x  e.  A  -> 
( x  .ih  x
)  =  0 ) )
5 his6 21694 . . . . . . . . 9  |-  ( x  e.  ~H  ->  (
( x  .ih  x
)  =  0  <->  x  =  0h ) )
65biimpd 198 . . . . . . . 8  |-  ( x  e.  ~H  ->  (
( x  .ih  x
)  =  0  ->  x  =  0h )
)
74, 6sylan9r 639 . . . . . . 7  |-  ( ( x  e.  ~H  /\  A. y  e.  A  ( x  .ih  y )  =  0 )  -> 
( x  e.  A  ->  x  =  0h )
)
81, 7syl6bi 219 . . . . . 6  |-  ( A  e.  SH  ->  (
x  e.  ( _|_ `  A )  ->  (
x  e.  A  ->  x  =  0h )
) )
98com23 72 . . . . 5  |-  ( A  e.  SH  ->  (
x  e.  A  -> 
( x  e.  ( _|_ `  A )  ->  x  =  0h ) ) )
109imp3a 420 . . . 4  |-  ( A  e.  SH  ->  (
( x  e.  A  /\  x  e.  ( _|_ `  A ) )  ->  x  =  0h ) )
11 sh0 21811 . . . . . 6  |-  ( A  e.  SH  ->  0h  e.  A )
12 oc0 21885 . . . . . 6  |-  ( A  e.  SH  ->  0h  e.  ( _|_ `  A
) )
1311, 12jca 518 . . . . 5  |-  ( A  e.  SH  ->  ( 0h  e.  A  /\  0h  e.  ( _|_ `  A
) ) )
14 eleq1 2356 . . . . . 6  |-  ( x  =  0h  ->  (
x  e.  A  <->  0h  e.  A ) )
15 eleq1 2356 . . . . . 6  |-  ( x  =  0h  ->  (
x  e.  ( _|_ `  A )  <->  0h  e.  ( _|_ `  A ) ) )
1614, 15anbi12d 691 . . . . 5  |-  ( x  =  0h  ->  (
( x  e.  A  /\  x  e.  ( _|_ `  A ) )  <-> 
( 0h  e.  A  /\  0h  e.  ( _|_ `  A ) ) ) )
1713, 16syl5ibrcom 213 . . . 4  |-  ( A  e.  SH  ->  (
x  =  0h  ->  ( x  e.  A  /\  x  e.  ( _|_ `  A ) ) ) )
1810, 17impbid 183 . . 3  |-  ( A  e.  SH  ->  (
( x  e.  A  /\  x  e.  ( _|_ `  A ) )  <-> 
x  =  0h )
)
19 elin 3371 . . 3  |-  ( x  e.  ( A  i^i  ( _|_ `  A ) )  <->  ( x  e.  A  /\  x  e.  ( _|_ `  A
) ) )
20 elch0 21849 . . 3  |-  ( x  e.  0H  <->  x  =  0h )
2118, 19, 203bitr4g 279 . 2  |-  ( A  e.  SH  ->  (
x  e.  ( A  i^i  ( _|_ `  A
) )  <->  x  e.  0H ) )
2221eqrdv 2294 1  |-  ( A  e.  SH  ->  ( A  i^i  ( _|_ `  A
) )  =  0H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    i^i cin 3164   ` cfv 5271  (class class class)co 5874   0cc0 8753   ~Hchil 21515    .ih csp 21518   0hc0v 21520   SHcsh 21524   _|_cort 21526   0Hc0h 21531
This theorem is referenced by:  ocnel  21893  chocunii  21896  pjhtheu  21989  pjpreeq  21993  omlsi  21999  ococi  22000  pjoc1i  22026  orthin  22041  ssjo  22042  chocini  22049  chscllem3  22234
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-hilex 21595  ax-hfvadd 21596  ax-hv0cl 21599  ax-hfvmul 21601  ax-hvmul0 21606  ax-hfi 21674  ax-his2 21678  ax-his3 21679  ax-his4 21680
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sh 21802  df-oc 21847  df-ch0 21848
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