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Theorem ocin 22790
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 22776 . . . . . . 7  |-  ( A  e.  SH  ->  (
x  e.  ( _|_ `  A )  <->  ( x  e.  ~H  /\  A. y  e.  A  ( x  .ih  y )  =  0 ) ) )
2 oveq2 6081 . . . . . . . . . 10  |-  ( y  =  x  ->  (
x  .ih  y )  =  ( x  .ih  x ) )
32eqeq1d 2443 . . . . . . . . 9  |-  ( y  =  x  ->  (
( x  .ih  y
)  =  0  <->  (
x  .ih  x )  =  0 ) )
43rspccv 3041 . . . . . . . 8  |-  ( A. y  e.  A  (
x  .ih  y )  =  0  ->  (
x  e.  A  -> 
( x  .ih  x
)  =  0 ) )
5 his6 22593 . . . . . . . . 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 22710 . . . . . 6  |-  ( A  e.  SH  ->  0h  e.  A )
12 oc0 22784 . . . . . 6  |-  ( A  e.  SH  ->  0h  e.  ( _|_ `  A
) )
1311, 12jca 519 . . . . 5  |-  ( A  e.  SH  ->  ( 0h  e.  A  /\  0h  e.  ( _|_ `  A
) ) )
14 eleq1 2495 . . . . . 6  |-  ( x  =  0h  ->  (
x  e.  A  <->  0h  e.  A ) )
15 eleq1 2495 . . . . . 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 3522 . . 3  |-  ( x  e.  ( A  i^i  ( _|_ `  A ) )  <->  ( x  e.  A  /\  x  e.  ( _|_ `  A
) ) )
20 elch0 22748 . . 3  |-  ( x  e.  0H  <->  x  =  0h )
2118, 19, 203bitr4g 280 . 2  |-  ( A  e.  SH  ->  (
x  e.  ( A  i^i  ( _|_ `  A
) )  <->  x  e.  0H ) )
2221eqrdv 2433 1  |-  ( A  e.  SH  ->  ( A  i^i  ( _|_ `  A
) )  =  0H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    i^i cin 3311   ` cfv 5446  (class class class)co 6073   0cc0 8982   ~Hchil 22414    .ih csp 22417   0hc0v 22419   SHcsh 22423   _|_cort 22425   0Hc0h 22430
This theorem is referenced by:  ocnel  22792  chocunii  22795  pjhtheu  22888  pjpreeq  22892  omlsi  22898  ococi  22899  pjoc1i  22925  orthin  22940  ssjo  22941  chocini  22948  chscllem3  23133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-hilex 22494  ax-hfvadd 22495  ax-hv0cl 22498  ax-hfvmul 22500  ax-hvmul0 22505  ax-hfi 22573  ax-his2 22577  ax-his3 22578  ax-his4 22579
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-sh 22701  df-oc 22746  df-ch0 22747
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