Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dibord Structured version   Unicode version

Theorem dibord 31894
Description: The isomorphism B for a lattice  K is order-preserving in the region under co-atom  W. (Contributed by NM, 24-Feb-2014.)
Hypotheses
Ref Expression
dib11.b  |-  B  =  ( Base `  K
)
dib11.l  |-  .<_  =  ( le `  K )
dib11.h  |-  H  =  ( LHyp `  K
)
dib11.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibord  |-  ( ( ( 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 dibord
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 dib11.b . . . . 5  |-  B  =  ( Base `  K
)
2 dib11.l . . . . 5  |-  .<_  =  ( le `  K )
3 dib11.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 eqid 2435 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
5 eqid 2435 . . . . 5  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )
6 eqid 2435 . . . . 5  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dib11.i . . . . 5  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 31879 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } ) )
983adant3 977 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( I `  X
)  =  ( ( ( ( DIsoA `  K
) `  W ) `  X )  X.  {
( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } ) )
101, 2, 3, 4, 5, 6, 7dibval2 31879 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  (
I `  Y )  =  ( ( ( ( DIsoA `  K ) `  W ) `  Y
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } ) )
11103adant2 976 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( I `  Y
)  =  ( ( ( ( DIsoA `  K
) `  W ) `  Y )  X.  {
( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } ) )
129, 11sseq12d 3369 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( I `  X )  C_  (
I `  Y )  <->  ( ( ( ( DIsoA `  K ) `  W
) `  X )  X.  { ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } )  C_  ( ( ( (
DIsoA `  K ) `  W ) `  Y
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } ) ) )
131, 2, 3, 7dibn0 31888 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =/=  (/) )
14133adant3 977 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( I `  X
)  =/=  (/) )
159, 14eqnetrrd 2618 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( ( (
DIsoA `  K ) `  W ) `  X
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } )  =/=  (/) )
16 ssxpb 5295 . . 3  |-  ( ( ( ( ( DIsoA `  K ) `  W
) `  X )  X.  { ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } )  =/=  (/)  ->  ( ( ( ( ( DIsoA `  K
) `  W ) `  X )  X.  {
( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } )  C_  ( ( ( (
DIsoA `  K ) `  W ) `  Y
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } )  <->  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  C_  ( (
( DIsoA `  K ) `  W ) `  Y
)  /\  { (
f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } 
C_  { ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) ) } ) ) )
1715, 16syl 16 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } )  C_  ( (
( ( DIsoA `  K
) `  W ) `  Y )  X.  {
( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } )  <->  ( (
( ( DIsoA `  K
) `  W ) `  X )  C_  (
( ( DIsoA `  K
) `  W ) `  Y )  /\  {
( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) }  C_  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } ) ) )
18 ssid 3359 . . . 4  |-  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } 
C_  { ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) ) }
1918biantru 492 . . 3  |-  ( ( ( ( DIsoA `  K
) `  W ) `  X )  C_  (
( ( DIsoA `  K
) `  W ) `  Y )  <->  ( (
( ( DIsoA `  K
) `  W ) `  X )  C_  (
( ( DIsoA `  K
) `  W ) `  Y )  /\  {
( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) }  C_  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } ) )
201, 2, 3, 6diaord 31782 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( ( (
DIsoA `  K ) `  W ) `  X
)  C_  ( (
( DIsoA `  K ) `  W ) `  Y
)  <->  X  .<_  Y ) )
2119, 20syl5bbr 251 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( ( ( ( DIsoA `  K ) `  W ) `  X
)  C_  ( (
( DIsoA `  K ) `  W ) `  Y
)  /\  { (
f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } 
C_  { ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) ) } )  <->  X  .<_  Y ) )
2212, 17, 213bitrd 271 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    =/= wne 2598    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204    e. cmpt 4258    _I cid 4485    X. cxp 4868    |` cres 4872   ` cfv 5446   Basecbs 13461   lecple 13528   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   DIsoAcdia 31763   DIsoBcdib 31873
This theorem is referenced by:  dib11N  31895  cdlemn2a  31931  dihord1  31953  dihord3  31992  dihord5b  31994
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-disoa 31764  df-dib 31874
  Copyright terms: Public domain W3C validator