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Theorem ocorth 24826
Description: Members of a subset and its complement are orthogonal. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
Assertion
Ref Expression
ocorth  |-  ( H 
C_  ~H  ->  ( ( A  e.  H  /\  B  e.  ( _|_ `  H ) )  -> 
( A  .ih  B
)  =  0 ) )

Proof of Theorem ocorth
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ocel 24816 . . . . . 6  |-  ( H 
C_  ~H  ->  ( B  e.  ( _|_ `  H
)  <->  ( B  e. 
~H  /\  A. x  e.  H  ( B  .ih  x )  =  0 ) ) )
21simplbda 624 . . . . 5  |-  ( ( H  C_  ~H  /\  B  e.  ( _|_ `  H
) )  ->  A. x  e.  H  ( B  .ih  x )  =  0 )
32adantl 466 . . . 4  |-  ( ( ( H  C_  ~H  /\  A  e.  H )  /\  ( H  C_  ~H  /\  B  e.  ( _|_ `  H ) ) )  ->  A. x  e.  H  ( B  .ih  x )  =  0 )
4 oveq2 6195 . . . . . . . 8  |-  ( x  =  A  ->  ( B  .ih  x )  =  ( B  .ih  A
) )
54eqeq1d 2453 . . . . . . 7  |-  ( x  =  A  ->  (
( B  .ih  x
)  =  0  <->  ( B  .ih  A )  =  0 ) )
65rspcv 3162 . . . . . 6  |-  ( A  e.  H  ->  ( A. x  e.  H  ( B  .ih  x )  =  0  ->  ( B  .ih  A )  =  0 ) )
76ad2antlr 726 . . . . 5  |-  ( ( ( H  C_  ~H  /\  A  e.  H )  /\  ( H  C_  ~H  /\  B  e.  ( _|_ `  H ) ) )  ->  ( A. x  e.  H  ( B  .ih  x )  =  0  ->  ( B  .ih  A )  =  0 ) )
8 ssel2 3446 . . . . . 6  |-  ( ( H  C_  ~H  /\  A  e.  H )  ->  A  e.  ~H )
9 ocss 24820 . . . . . . 7  |-  ( H 
C_  ~H  ->  ( _|_ `  H )  C_  ~H )
109sselda 3451 . . . . . 6  |-  ( ( H  C_  ~H  /\  B  e.  ( _|_ `  H
) )  ->  B  e.  ~H )
11 orthcom 24642 . . . . . 6  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  <->  ( B  .ih  A )  =  0 ) )
128, 10, 11syl2an 477 . . . . 5  |-  ( ( ( H  C_  ~H  /\  A  e.  H )  /\  ( H  C_  ~H  /\  B  e.  ( _|_ `  H ) ) )  ->  (
( A  .ih  B
)  =  0  <->  ( B  .ih  A )  =  0 ) )
137, 12sylibrd 234 . . . 4  |-  ( ( ( H  C_  ~H  /\  A  e.  H )  /\  ( H  C_  ~H  /\  B  e.  ( _|_ `  H ) ) )  ->  ( A. x  e.  H  ( B  .ih  x )  =  0  ->  ( A  .ih  B )  =  0 ) )
143, 13mpd 15 . . 3  |-  ( ( ( H  C_  ~H  /\  A  e.  H )  /\  ( H  C_  ~H  /\  B  e.  ( _|_ `  H ) ) )  ->  ( A  .ih  B )  =  0 )
1514anandis 826 . 2  |-  ( ( H  C_  ~H  /\  ( A  e.  H  /\  B  e.  ( _|_ `  H ) ) )  ->  ( A  .ih  B )  =  0 )
1615ex 434 1  |-  ( H 
C_  ~H  ->  ( ( A  e.  H  /\  B  e.  ( _|_ `  H ) )  -> 
( A  .ih  B
)  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2793    C_ wss 3423   ` cfv 5513  (class class class)co 6187   0cc0 9380   ~Hchil 24453    .ih csp 24456   _|_cort 24464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457  ax-hilex 24533  ax-hfvadd 24534  ax-hv0cl 24537  ax-hfvmul 24539  ax-hvmul0 24544  ax-hfi 24613  ax-his1 24616  ax-his2 24617  ax-his3 24618
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-po 4736  df-so 4737  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-div 10092  df-2 10478  df-cj 12687  df-re 12688  df-im 12689  df-sh 24741  df-oc 24787
This theorem is referenced by:  shocorth  24827  ococss  24828  riesz3i  25598
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