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Theorem oplecon3 30059
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 2438 . . . 4  |-  ( join `  K )  =  (
join `  K )
5 eqid 2438 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
6 eqid 2438 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 eqid 2438 . . . 4  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7oposlem 30043 . . 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 970 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  X
)  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X 
.<_  Y  ->  (  ._|_  `  Y )  .<_  (  ._|_  `  X ) ) ) )
109simp3d 972 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 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   occoc 13539   joincjn 14403   meetcmee 14404   0.cp0 14468   1.cp1 14469   OPcops 30032
This theorem is referenced by:  oplecon3b  30060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4340
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-oposet 30036
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