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Theorem dibord 35113
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 2451 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
5 eqid 2451 . . . . 5  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )
6 eqid 2451 . . . . 5  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dib11.i . . . . 5  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 35098 . . . 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 1008 . . 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 35098 . . . 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 1007 . . 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 3486 . 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 35107 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =/=  (/) )
14133adant3 1008 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( I `  X
)  =/=  (/) )
159, 14eqnetrrd 2742 . . 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 5373 . . 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 3476 . . . 4  |-  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } 
C_  { ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) ) }
1918biantru 505 . . 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 35001 . . 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 259 . 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 279 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644    C_ wss 3429   (/)c0 3738   {csn 3978   class class class wbr 4393    |-> cmpt 4451    _I cid 4732    X. cxp 4939    |` cres 4943   ` cfv 5519   Basecbs 14285   lecple 14356   HLchlt 33304   LHypclh 33937   LTrncltrn 34054   DIsoAcdia 34982   DIsoBcdib 35092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-riotaBAD 32913
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-undef 6895  df-map 7319  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-p1 15321  df-lat 15327  df-clat 15389  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-llines 33451  df-lplanes 33452  df-lvols 33453  df-lines 33454  df-psubsp 33456  df-pmap 33457  df-padd 33749  df-lhyp 33941  df-laut 33942  df-ldil 34057  df-ltrn 34058  df-trl 34112  df-disoa 34983  df-dib 35093
This theorem is referenced by:  dib11N  35114  cdlemn2a  35150  dihord1  35172  dihord3  35211  dihord5b  35213
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