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Theorem diaord 31783
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 2436 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
5 eqid 2436 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
6 dia11.i . . . . 5  |-  I  =  ( ( DIsoA `  K
) `  W )
71, 2, 3, 4, 5, 6diaval 31768 . . . 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 31768 . . . 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 3370 . 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 2436 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
131, 2, 12, 3, 4, 5trlord 31304 . . 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 3412 . . 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 1652    e. wcel 1725   A.wral 2698   {crab 2702    C_ wss 3313   class class class wbr 4205   ` cfv 5447   Basecbs 13462   lecple 13529   Atomscatm 29999   HLchlt 30086   LHypclh 30719   LTrncltrn 30836   trLctrl 30893   DIsoAcdia 31764
This theorem is referenced by:  dia11N  31784  dia2dimlem10  31809  dibord  31895
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-iin 4089  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-undef 6536  df-riota 6542  df-map 7013  df-poset 14396  df-plt 14408  df-lub 14424  df-glb 14425  df-join 14426  df-meet 14427  df-p0 14461  df-p1 14462  df-lat 14468  df-clat 14530  df-oposet 29912  df-ol 29914  df-oml 29915  df-covers 30002  df-ats 30003  df-atl 30034  df-cvlat 30058  df-hlat 30087  df-llines 30233  df-lplanes 30234  df-lvols 30235  df-lines 30236  df-psubsp 30238  df-pmap 30239  df-padd 30531  df-lhyp 30723  df-laut 30724  df-ldil 30839  df-ltrn 30840  df-trl 30894  df-disoa 31765
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