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Theorem ocin 22639
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 22625 . . . . . . 7  |-  ( A  e.  SH  ->  (
x  e.  ( _|_ `  A )  <->  ( x  e.  ~H  /\  A. y  e.  A  ( x  .ih  y )  =  0 ) ) )
2 oveq2 6021 . . . . . . . . . 10  |-  ( y  =  x  ->  (
x  .ih  y )  =  ( x  .ih  x ) )
32eqeq1d 2388 . . . . . . . . 9  |-  ( y  =  x  ->  (
( x  .ih  y
)  =  0  <->  (
x  .ih  x )  =  0 ) )
43rspccv 2985 . . . . . . . 8  |-  ( A. y  e.  A  (
x  .ih  y )  =  0  ->  (
x  e.  A  -> 
( x  .ih  x
)  =  0 ) )
5 his6 22442 . . . . . . . . 9  |-  ( x  e.  ~H  ->  (
( x  .ih  x
)  =  0  <->  x  =  0h ) )
65biimpd 199 . . . . . . . 8  |-  ( x  e.  ~H  ->  (
( x  .ih  x
)  =  0  ->  x  =  0h )
)
74, 6sylan9r 640 . . . . . . 7  |-  ( ( x  e.  ~H  /\  A. y  e.  A  ( x  .ih  y )  =  0 )  -> 
( x  e.  A  ->  x  =  0h )
)
81, 7syl6bi 220 . . . . . 6  |-  ( A  e.  SH  ->  (
x  e.  ( _|_ `  A )  ->  (
x  e.  A  ->  x  =  0h )
) )
98com23 74 . . . . 5  |-  ( A  e.  SH  ->  (
x  e.  A  -> 
( x  e.  ( _|_ `  A )  ->  x  =  0h ) ) )
109imp3a 421 . . . 4  |-  ( A  e.  SH  ->  (
( x  e.  A  /\  x  e.  ( _|_ `  A ) )  ->  x  =  0h ) )
11 sh0 22559 . . . . . 6  |-  ( A  e.  SH  ->  0h  e.  A )
12 oc0 22633 . . . . . 6  |-  ( A  e.  SH  ->  0h  e.  ( _|_ `  A
) )
1311, 12jca 519 . . . . 5  |-  ( A  e.  SH  ->  ( 0h  e.  A  /\  0h  e.  ( _|_ `  A
) ) )
14 eleq1 2440 . . . . . 6  |-  ( x  =  0h  ->  (
x  e.  A  <->  0h  e.  A ) )
15 eleq1 2440 . . . . . 6  |-  ( x  =  0h  ->  (
x  e.  ( _|_ `  A )  <->  0h  e.  ( _|_ `  A ) ) )
1614, 15anbi12d 692 . . . . 5  |-  ( x  =  0h  ->  (
( x  e.  A  /\  x  e.  ( _|_ `  A ) )  <-> 
( 0h  e.  A  /\  0h  e.  ( _|_ `  A ) ) ) )
1713, 16syl5ibrcom 214 . . . 4  |-  ( A  e.  SH  ->  (
x  =  0h  ->  ( x  e.  A  /\  x  e.  ( _|_ `  A ) ) ) )
1810, 17impbid 184 . . 3  |-  ( A  e.  SH  ->  (
( x  e.  A  /\  x  e.  ( _|_ `  A ) )  <-> 
x  =  0h )
)
19 elin 3466 . . 3  |-  ( x  e.  ( A  i^i  ( _|_ `  A ) )  <->  ( x  e.  A  /\  x  e.  ( _|_ `  A
) ) )
20 elch0 22597 . . 3  |-  ( x  e.  0H  <->  x  =  0h )
2118, 19, 203bitr4g 280 . 2  |-  ( A  e.  SH  ->  (
x  e.  ( A  i^i  ( _|_ `  A
) )  <->  x  e.  0H ) )
2221eqrdv 2378 1  |-  ( A  e.  SH  ->  ( A  i^i  ( _|_ `  A
) )  =  0H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642    i^i cin 3255   ` cfv 5387  (class class class)co 6013   0cc0 8916   ~Hchil 22263    .ih csp 22266   0hc0v 22268   SHcsh 22272   _|_cort 22274   0Hc0h 22279
This theorem is referenced by:  ocnel  22641  chocunii  22644  pjhtheu  22737  pjpreeq  22741  omlsi  22747  ococi  22748  pjoc1i  22774  orthin  22789  ssjo  22790  chocini  22797  chscllem3  22982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-hilex 22343  ax-hfvadd 22344  ax-hv0cl 22347  ax-hfvmul 22349  ax-hvmul0 22354  ax-hfi 22422  ax-his2 22426  ax-his3 22427  ax-his4 22428
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-ltxr 9051  df-sh 22550  df-oc 22595  df-ch0 22596
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