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Theorem ocin 21875
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 21861 . . . . . . 7  |-  ( A  e.  SH  ->  (
x  e.  ( _|_ `  A )  <->  ( x  e.  ~H  /\  A. y  e.  A  ( x  .ih  y )  =  0 ) ) )
2 oveq2 5866 . . . . . . . . . 10  |-  ( y  =  x  ->  (
x  .ih  y )  =  ( x  .ih  x ) )
32eqeq1d 2291 . . . . . . . . 9  |-  ( y  =  x  ->  (
( x  .ih  y
)  =  0  <->  (
x  .ih  x )  =  0 ) )
43rspccv 2881 . . . . . . . 8  |-  ( A. y  e.  A  (
x  .ih  y )  =  0  ->  (
x  e.  A  -> 
( x  .ih  x
)  =  0 ) )
5 his6 21678 . . . . . . . . 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 21795 . . . . . 6  |-  ( A  e.  SH  ->  0h  e.  A )
12 oc0 21869 . . . . . 6  |-  ( A  e.  SH  ->  0h  e.  ( _|_ `  A
) )
1311, 12jca 518 . . . . 5  |-  ( A  e.  SH  ->  ( 0h  e.  A  /\  0h  e.  ( _|_ `  A
) ) )
14 eleq1 2343 . . . . . 6  |-  ( x  =  0h  ->  (
x  e.  A  <->  0h  e.  A ) )
15 eleq1 2343 . . . . . 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 3358 . . 3  |-  ( x  e.  ( A  i^i  ( _|_ `  A ) )  <->  ( x  e.  A  /\  x  e.  ( _|_ `  A
) ) )
20 elch0 21833 . . 3  |-  ( x  e.  0H  <->  x  =  0h )
2118, 19, 203bitr4g 279 . 2  |-  ( A  e.  SH  ->  (
x  e.  ( A  i^i  ( _|_ `  A
) )  <->  x  e.  0H ) )
2221eqrdv 2281 1  |-  ( A  e.  SH  ->  ( A  i^i  ( _|_ `  A
) )  =  0H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151   ` cfv 5255  (class class class)co 5858   0cc0 8737   ~Hchil 21499    .ih csp 21502   0hc0v 21504   SHcsh 21508   _|_cort 21510   0Hc0h 21515
This theorem is referenced by:  ocnel  21877  chocunii  21880  pjhtheu  21973  pjpreeq  21977  omlsi  21983  ococi  21984  pjoc1i  22010  orthin  22025  ssjo  22026  chocini  22033  chscllem3  22218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-hilex 21579  ax-hfvadd 21580  ax-hv0cl 21583  ax-hfvmul 21585  ax-hvmul0 21590  ax-hfi 21658  ax-his2 21662  ax-his3 21663  ax-his4 21664
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sh 21786  df-oc 21831  df-ch0 21832
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