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Theorem occon 22781
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 3399 . . . . . 6  |-  ( A 
C_  B  ->  ( A. y  e.  B  ( x  .ih  y )  =  0  ->  A. y  e.  A  ( x  .ih  y )  =  0 ) )
21ralrimivw 2782 . . . . 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 3411 . . . . 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 204 . . . 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 453 . . 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 22774 . . . 4  |-  ( B 
C_  ~H  ->  ( _|_ `  B )  =  {
x  e.  ~H  |  A. y  e.  B  ( x  .ih  y )  =  0 } )
76ad2antlr 708 . . 3  |-  ( ( ( A  C_  ~H  /\  B  C_  ~H )  /\  A  C_  B )  ->  ( _|_ `  B
)  =  { x  e.  ~H  |  A. y  e.  B  ( x  .ih  y )  =  0 } )
8 ocval 22774 . . . 4  |-  ( A 
C_  ~H  ->  ( _|_ `  A )  =  {
x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 } )
98ad2antrr 707 . . 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 3383 . 2  |-  ( ( ( A  C_  ~H  /\  B  C_  ~H )  /\  A  C_  B )  ->  ( _|_ `  B
)  C_  ( _|_ `  A ) )
1110ex 424 1  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  C_  B  ->  ( _|_ `  B )  C_  ( _|_ `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652   A.wral 2697   {crab 2701    C_ wss 3312   ` cfv 5446  (class class class)co 6073   0cc0 8982   ~Hchil 22414    .ih csp 22417   _|_cort 22425
This theorem is referenced by:  occon2  22782  occon3  22791  ococin  22902  ssjo  22941  chsscon3i  22955  shjshsi  22986
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-hilex 22494
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  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-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-oc 22746
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