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Theorem occon 26322
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 3478 . . . . . 6  |-  ( A 
C_  B  ->  ( A. y  e.  B  ( x  .ih  y )  =  0  ->  A. y  e.  A  ( x  .ih  y )  =  0 ) )
21ralrimivw 2797 . . . . 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 3490 . . . . 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 212 . . . 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 464 . . 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 26315 . . . 4  |-  ( B 
C_  ~H  ->  ( _|_ `  B )  =  {
x  e.  ~H  |  A. y  e.  B  ( x  .ih  y )  =  0 } )
76ad2antlr 724 . . 3  |-  ( ( ( A  C_  ~H  /\  B  C_  ~H )  /\  A  C_  B )  ->  ( _|_ `  B
)  =  { x  e.  ~H  |  A. y  e.  B  ( x  .ih  y )  =  0 } )
8 ocval 26315 . . . 4  |-  ( A 
C_  ~H  ->  ( _|_ `  A )  =  {
x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 } )
98ad2antrr 723 . . 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 3460 . 2  |-  ( ( ( A  C_  ~H  /\  B  C_  ~H )  /\  A  C_  B )  ->  ( _|_ `  B
)  C_  ( _|_ `  A ) )
1110ex 432 1  |-  ( ( A  C_  ~H  /\  B  C_ 
~H )  ->  ( A  C_  B  ->  ( _|_ `  B )  C_  ( _|_ `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399   A.wral 2732   {crab 2736    C_ wss 3389   ` cfv 5496  (class class class)co 6196   0cc0 9403   ~Hchil 25953    .ih csp 25956   _|_cort 25964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-hilex 26033
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-oc 26287
This theorem is referenced by:  occon2  26323  occon3  26332  ococin  26443  ssjo  26482  chsscon3i  26496  shjshsi  26527
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