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Theorem occon 21882
Description: Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)
Assertion
Ref Expression
occon  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  C_  B  ->  ( _|_ `  B )  C_  ( _|_ `  A ) ) )

Proof of Theorem occon
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssralv 3250 . . . . . 6  |-  ( A 
C_  B  ->  ( A. y  e.  B  ( x  .ih  y )  =  0  ->  A. y  e.  A  ( x  .ih  y )  =  0 ) )
21ralrimivw 2640 . . . . 5  |-  ( A 
C_  B  ->  A. x  e.  ~H  ( A. y  e.  B  ( x  .ih  y )  =  0  ->  A. y  e.  A  ( x  .ih  y )  =  0 ) )
3 ss2rab 3262 . . . . 5  |-  ( { x  e.  ~H  |  A. y  e.  B  ( x  .ih  y )  =  0 }  C_  { x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 }  <->  A. x  e.  ~H  ( A. y  e.  B  ( x  .ih  y )  =  0  ->  A. y  e.  A  ( x  .ih  y )  =  0 ) )
42, 3sylibr 203 . . . 4  |-  ( A 
C_  B  ->  { x  e.  ~H  |  A. y  e.  B  ( x  .ih  y )  =  0 }  C_  { x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 } )
54adantl 452 . . 3  |-  ( ( ( A  C_  ~H  /\  B  C_  ~H )  /\  A  C_  B )  ->  { x  e. 
~H  |  A. y  e.  B  ( x  .ih  y )  =  0 }  C_  { x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 } )
6 ocval 21875 . . . 4  |-  ( B 
C_  ~H  ->  ( _|_ `  B )  =  {
x  e.  ~H  |  A. y  e.  B  ( x  .ih  y )  =  0 } )
76ad2antlr 707 . . 3  |-  ( ( ( A  C_  ~H  /\  B  C_  ~H )  /\  A  C_  B )  ->  ( _|_ `  B
)  =  { x  e.  ~H  |  A. y  e.  B  ( x  .ih  y )  =  0 } )
8 ocval 21875 . . . 4  |-  ( A 
C_  ~H  ->  ( _|_ `  A )  =  {
x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 } )
98ad2antrr 706 . . 3  |-  ( ( ( A  C_  ~H  /\  B  C_  ~H )  /\  A  C_  B )  ->  ( _|_ `  A
)  =  { x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 } )
105, 7, 93sstr4d 3234 . 2  |-  ( ( ( A  C_  ~H  /\  B  C_  ~H )  /\  A  C_  B )  ->  ( _|_ `  B
)  C_  ( _|_ `  A ) )
1110ex 423 1  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  C_  B  ->  ( _|_ `  B )  C_  ( _|_ `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632   A.wral 2556   {crab 2560    C_ wss 3165   ` cfv 5271  (class class class)co 5874   0cc0 8753   ~Hchil 21515    .ih csp 21518   _|_cort 21526
This theorem is referenced by:  occon2  21883  occon3  21892  ococin  22003  ssjo  22042  chsscon3i  22056  shjshsi  22087
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  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-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-oc 21847
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