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Theorem diaord 31530
Description: The partial isomorphism A for a lattice  K is order-preserving in the region under co-atom  W. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 26-Nov-2013.)
Hypotheses
Ref Expression
dia11.b  |-  B  =  ( Base `  K
)
dia11.l  |-  .<_  =  ( le `  K )
dia11.h  |-  H  =  ( LHyp `  K
)
dia11.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
diaord  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( I `  X )  C_  (
I `  Y )  <->  X 
.<_  Y ) )

Proof of Theorem diaord
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 dia11.b . . . . 5  |-  B  =  ( Base `  K
)
2 dia11.l . . . . 5  |-  .<_  =  ( le `  K )
3 dia11.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 eqid 2404 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
5 eqid 2404 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
6 dia11.i . . . . 5  |-  I  =  ( ( DIsoA `  K
) `  W )
71, 2, 3, 4, 5, 6diaval 31515 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  { f  e.  ( ( LTrn `  K
) `  W )  |  ( ( ( trL `  K ) `
 W ) `  f )  .<_  X }
)
873adant3 977 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( I `  X
)  =  { f  e.  ( ( LTrn `  K ) `  W
)  |  ( ( ( trL `  K
) `  W ) `  f )  .<_  X }
)
91, 2, 3, 4, 5, 6diaval 31515 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  (
I `  Y )  =  { f  e.  ( ( LTrn `  K
) `  W )  |  ( ( ( trL `  K ) `
 W ) `  f )  .<_  Y }
)
1093adant2 976 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( I `  Y
)  =  { f  e.  ( ( LTrn `  K ) `  W
)  |  ( ( ( trL `  K
) `  W ) `  f )  .<_  Y }
)
118, 10sseq12d 3337 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( I `  X )  C_  (
I `  Y )  <->  { f  e.  ( (
LTrn `  K ) `  W )  |  ( ( ( trL `  K
) `  W ) `  f )  .<_  X }  C_ 
{ f  e.  ( ( LTrn `  K
) `  W )  |  ( ( ( trL `  K ) `
 W ) `  f )  .<_  Y }
) )
12 eqid 2404 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
131, 2, 12, 3, 4, 5trlord 31051 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( X  .<_  Y  <->  A. f  e.  ( ( LTrn `  K
) `  W )
( ( ( ( trL `  K ) `
 W ) `  f )  .<_  X  -> 
( ( ( trL `  K ) `  W
) `  f )  .<_  Y ) ) )
14 ss2rab 3379 . . 3  |-  ( { f  e.  ( (
LTrn `  K ) `  W )  |  ( ( ( trL `  K
) `  W ) `  f )  .<_  X }  C_ 
{ f  e.  ( ( LTrn `  K
) `  W )  |  ( ( ( trL `  K ) `
 W ) `  f )  .<_  Y }  <->  A. f  e.  ( (
LTrn `  K ) `  W ) ( ( ( ( trL `  K
) `  W ) `  f )  .<_  X  -> 
( ( ( trL `  K ) `  W
) `  f )  .<_  Y ) )
1513, 14syl6rbbr 256 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( { f  e.  ( ( LTrn `  K
) `  W )  |  ( ( ( trL `  K ) `
 W ) `  f )  .<_  X }  C_ 
{ f  e.  ( ( LTrn `  K
) `  W )  |  ( ( ( trL `  K ) `
 W ) `  f )  .<_  Y }  <->  X 
.<_  Y ) )
1611, 15bitrd 245 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( I `  X )  C_  (
I `  Y )  <->  X 
.<_  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670    C_ wss 3280   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640   DIsoAcdia 31511
This theorem is referenced by:  dia11N  31531  dia2dimlem10  31556  dibord  31642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-disoa 31512
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