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Theorem isopiN 29916
Description: Properties that determine an orthoposet (constructed structure version). (Contributed by NM, 13-Sep-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
isopi.0  |-  K  e. 
Poset
isopi.b  |-  B  =  ( Base `  K
)
isopi.l  |-  .<_  =  ( le `  K )
isopi.o  |-  ._|_  =  ( oc `  K )
isopi.j  |-  .\/  =  ( join `  K )
isopi.m  |-  ./\  =  ( meet `  K )
isopi.z  |-  .0.  =  ( 0. `  K )
isopi.u  |-  .1.  =  ( 1. `  K )
isopi.8  |-  .0.  e.  B
isopi.9  |-  .1.  e.  B
isopi.9a  |-  ( x  e.  B  ->  (  ._|_  `  x )  e.  B )
isopi.10  |-  ( x  e.  B  ->  (  ._|_  `  (  ._|_  `  x
) )  =  x )
isopi.11  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .<_  y  -> 
(  ._|_  `  y )  .<_  (  ._|_  `  x ) ) )
isopi.12  |-  ( x  e.  B  ->  (
x  .\/  (  ._|_  `  x ) )  =  .1.  )
isopi.13  |-  ( x  e.  B  ->  (
x  ./\  (  ._|_  `  x ) )  =  .0.  )
Assertion
Ref Expression
isopiN  |-  K  e.  OP
Distinct variable groups:    x, y, B    x,  ._|_ , y    x, K, y
Allowed substitution hints:    .1. ( x, y)    .\/ ( x, y)    .<_ ( x, y)    ./\ (
x, y)    .0. ( x, y)

Proof of Theorem isopiN
StepHypRef Expression
1 isopi.0 . . 3  |-  K  e. 
Poset
2 isopi.8 . . 3  |-  .0.  e.  B
3 isopi.9 . . 3  |-  .1.  e.  B
41, 2, 33pm3.2i 1132 . 2  |-  ( K  e.  Poset  /\  .0.  e.  B  /\  .1.  e.  B
)
5 isopi.9a . . . . . 6  |-  ( x  e.  B  ->  (  ._|_  `  x )  e.  B )
65adantr 452 . . . . 5  |-  ( ( x  e.  B  /\  y  e.  B )  ->  (  ._|_  `  x )  e.  B )
7 isopi.10 . . . . . 6  |-  ( x  e.  B  ->  (  ._|_  `  (  ._|_  `  x
) )  =  x )
87adantr 452 . . . . 5  |-  ( ( x  e.  B  /\  y  e.  B )  ->  (  ._|_  `  (  ._|_  `  x ) )  =  x )
9 isopi.11 . . . . 5  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .<_  y  -> 
(  ._|_  `  y )  .<_  (  ._|_  `  x ) ) )
106, 8, 93jca 1134 . . . 4  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( (  ._|_  `  x
)  e.  B  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x  /\  ( x 
.<_  y  ->  (  ._|_  `  y )  .<_  (  ._|_  `  x ) ) ) )
11 isopi.12 . . . . 5  |-  ( x  e.  B  ->  (
x  .\/  (  ._|_  `  x ) )  =  .1.  )
1211adantr 452 . . . 4  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .\/  (  ._|_  `  x ) )  =  .1.  )
13 isopi.13 . . . . 5  |-  ( x  e.  B  ->  (
x  ./\  (  ._|_  `  x ) )  =  .0.  )
1413adantr 452 . . . 4  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  ./\  (  ._|_  `  x ) )  =  .0.  )
1510, 12, 143jca 1134 . . 3  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( ( (  ._|_  `  x )  e.  B  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x  /\  (
x  .<_  y  ->  (  ._|_  `  y )  .<_  (  ._|_  `  x )
) )  /\  (
x  .\/  (  ._|_  `  x ) )  =  .1.  /\  ( x 
./\  (  ._|_  `  x
) )  =  .0.  ) )
1615rgen2a 2764 . 2  |-  A. x  e.  B  A. y  e.  B  ( (
(  ._|_  `  x )  e.  B  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x  /\  ( x  .<_  y  ->  (  ._|_  `  y
)  .<_  (  ._|_  `  x
) ) )  /\  ( x  .\/  (  ._|_  `  x ) )  =  .1.  /\  ( x 
./\  (  ._|_  `  x
) )  =  .0.  )
17 isopi.b . . 3  |-  B  =  ( Base `  K
)
18 isopi.l . . 3  |-  .<_  =  ( le `  K )
19 isopi.o . . 3  |-  ._|_  =  ( oc `  K )
20 isopi.j . . 3  |-  .\/  =  ( join `  K )
21 isopi.m . . 3  |-  ./\  =  ( meet `  K )
22 isopi.z . . 3  |-  .0.  =  ( 0. `  K )
23 isopi.u . . 3  |-  .1.  =  ( 1. `  K )
2417, 18, 19, 20, 21, 22, 23isopos 29915 . 2  |-  ( K  e.  OP  <->  ( ( K  e.  Poset  /\  .0.  e.  B  /\  .1.  e.  B )  /\  A. x  e.  B  A. y  e.  B  (
( (  ._|_  `  x
)  e.  B  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x  /\  ( x 
.<_  y  ->  (  ._|_  `  y )  .<_  (  ._|_  `  x ) ) )  /\  ( x  .\/  (  ._|_  `  x )
)  =  .1.  /\  ( x  ./\  (  ._|_  `  x ) )  =  .0.  ) ) )
254, 16, 24mpbir2an 887 1  |-  K  e.  OP
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   occoc 13529   Posetcpo 14389   joincjn 14393   meetcmee 14394   0.cp0 14458   1.cp1 14459   OPcops 29907
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
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-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-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-oposet 29911
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