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Theorem dihord3 31447
Description: The isomorphism H for a lattice  K is order-preserving in the region under co-atom  W. (Contributed by NM, 6-Mar-2014.)
Hypotheses
Ref Expression
dihord3.b  |-  B  =  ( Base `  K
)
dihord3.l  |-  .<_  =  ( le `  K )
dihord3.h  |-  H  =  ( LHyp `  K
)
dihord3.i  |-  I  =  ( ( DIsoH `  K
) `  W )
Assertion
Ref Expression
dihord3  |-  ( ( ( 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 dihord3
StepHypRef Expression
1 dihord3.b . . . . 5  |-  B  =  ( Base `  K
)
2 dihord3.l . . . . 5  |-  .<_  =  ( le `  K )
3 dihord3.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 dihord3.i . . . . 5  |-  I  =  ( ( DIsoH `  K
) `  W )
5 eqid 2283 . . . . 5  |-  ( (
DIsoB `  K ) `  W )  =  ( ( DIsoB `  K ) `  W )
61, 2, 3, 4, 5dihvalb 31427 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( (
DIsoB `  K ) `  W ) `  X
) )
763adant3 975 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( I `  X
)  =  ( ( ( DIsoB `  K ) `  W ) `  X
) )
81, 2, 3, 4, 5dihvalb 31427 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  (
I `  Y )  =  ( ( (
DIsoB `  K ) `  W ) `  Y
) )
983adant2 974 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( I `  Y
)  =  ( ( ( DIsoB `  K ) `  W ) `  Y
) )
107, 9sseq12d 3207 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( I `  X )  C_  (
I `  Y )  <->  ( ( ( DIsoB `  K
) `  W ) `  X )  C_  (
( ( DIsoB `  K
) `  W ) `  Y ) ) )
111, 2, 3, 5dibord 31349 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( ( (
DIsoB `  K ) `  W ) `  X
)  C_  ( (
( DIsoB `  K ) `  W ) `  Y
)  <->  X  .<_  Y ) )
1210, 11bitrd 244 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   HLchlt 29540   LHypclh 30173   DIsoBcdib 31328   DIsoHcdih 31418
This theorem is referenced by:  dihord  31454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-disoa 31219  df-dib 31329  df-dih 31419
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