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Theorem oplecon3 30011
Description: Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)
Hypotheses
Ref Expression
opcon3.b  |-  B  =  ( Base `  K
)
opcon3.l  |-  .<_  =  ( le `  K )
opcon3.o  |-  ._|_  =  ( oc `  K )
Assertion
Ref Expression
oplecon3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
(  ._|_  `  Y )  .<_  (  ._|_  `  X ) ) )

Proof of Theorem oplecon3
StepHypRef Expression
1 opcon3.b . . . 4  |-  B  =  ( Base `  K
)
2 opcon3.l . . . 4  |-  .<_  =  ( le `  K )
3 opcon3.o . . . 4  |-  ._|_  =  ( oc `  K )
4 eqid 2296 . . . 4  |-  ( join `  K )  =  (
join `  K )
5 eqid 2296 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
6 eqid 2296 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 eqid 2296 . . . 4  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7oposlem 29995 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( (  ._|_  `  X )  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X  .<_  Y  ->  (  ._|_  `  Y )  .<_  (  ._|_  `  X )
) )  /\  ( X ( join `  K
) (  ._|_  `  X
) )  =  ( 1. `  K )  /\  ( X (
meet `  K )
(  ._|_  `  X )
)  =  ( 0.
`  K ) ) )
98simp1d 967 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  X
)  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X 
.<_  Y  ->  (  ._|_  `  Y )  .<_  (  ._|_  `  X ) ) ) )
109simp3d 969 1  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
(  ._|_  `  Y )  .<_  (  ._|_  `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   occoc 13232   joincjn 14094   meetcmee 14095   0.cp0 14159   1.cp1 14160   OPcops 29984
This theorem is referenced by:  oplecon3b  30012
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
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-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-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-oposet 29988
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