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Theorem diaord 29926
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
StepHypRef Expression
1 dia11.b . . . . 5  |-  B  =  ( Base `  K
)
2 dia11.l . . . . 5  |-  .<_  =  ( le `  K )
3 dia11.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 eqid 2253 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
5 eqid 2253 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
6 dia11.i . . . . 5  |-  I  =  ( ( DIsoA `  K
) `  W )
71, 2, 3, 4, 5, 6diaval 29911 . . . 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 980 . . 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 29911 . . . 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 979 . . 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 3128 . 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 2253 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
131, 2, 12, 3, 4, 5trlord 29447 . . 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 3170 . . 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 257 . 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 246 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 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2509   {crab 2512    C_ wss 3078   class class class wbr 3920   ` cfv 4592   Basecbs 13022   lecple 13089   Atomscatm 28142   HLchlt 28229   LHypclh 28862   LTrncltrn 28979   trLctrl 29036   DIsoAcdia 29907
This theorem is referenced by:  dia11N  29927  dia2dimlem10  29952  dibord  30038
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-llines 28376  df-lplanes 28377  df-lvols 28378  df-lines 28379  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866  df-laut 28867  df-ldil 28982  df-ltrn 28983  df-trl 29037  df-disoa 29908
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