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Theorem dib11N 31276
Description: The isomorphism B for a lattice  K is one-to-one in the region under co-atom  W. (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.)
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
dib11N  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( I `  X )  =  ( I `  Y )  <-> 
X  =  Y ) )

Proof of Theorem dib11N
StepHypRef Expression
1 eqss 3307 . 2  |-  ( ( I `  X )  =  ( I `  Y )  <->  ( (
I `  X )  C_  ( I `  Y
)  /\  ( I `  Y )  C_  (
I `  X )
) )
2 dib11.b . . . . 5  |-  B  =  ( Base `  K
)
3 dib11.l . . . . 5  |-  .<_  =  ( le `  K )
4 dib11.h . . . . 5  |-  H  =  ( LHyp `  K
)
5 dib11.i . . . . 5  |-  I  =  ( ( DIsoB `  K
) `  W )
62, 3, 4, 5dibord 31275 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( I `  X )  C_  (
I `  Y )  <->  X 
.<_  Y ) )
72, 3, 4, 5dibord 31275 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Y  e.  B  /\  Y  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  -> 
( ( I `  Y )  C_  (
I `  X )  <->  Y 
.<_  X ) )
873com23 1159 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( I `  Y )  C_  (
I `  X )  <->  Y 
.<_  X ) )
96, 8anbi12d 692 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( ( I `
 X )  C_  ( I `  Y
)  /\  ( I `  Y )  C_  (
I `  X )
)  <->  ( X  .<_  Y  /\  Y  .<_  X ) ) )
10 simp1l 981 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  K  e.  HL )
11 hllat 29479 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
1210, 11syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  K  e.  Lat )
13 simp2l 983 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  X  e.  B )
14 simp3l 985 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  Y  e.  B )
152, 3latasymb 14411 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y  /\  Y  .<_  X )  <-> 
X  =  Y ) )
1612, 13, 14, 15syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( X  .<_  Y  /\  Y  .<_  X )  <-> 
X  =  Y ) )
179, 16bitrd 245 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( ( I `
 X )  C_  ( I `  Y
)  /\  ( I `  Y )  C_  (
I `  X )
)  <->  X  =  Y
) )
181, 17syl5bb 249 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  -> 
( ( I `  X )  =  ( I `  Y )  <-> 
X  =  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3264   class class class wbr 4154   ` cfv 5395   Basecbs 13397   lecple 13464   Latclat 14402   HLchlt 29466   LHypclh 30099   DIsoBcdib 31254
This theorem is referenced by:  dibf11N  31277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-map 6957  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-llines 29613  df-lplanes 29614  df-lvols 29615  df-lines 29616  df-psubsp 29618  df-pmap 29619  df-padd 29911  df-lhyp 30103  df-laut 30104  df-ldil 30219  df-ltrn 30220  df-trl 30274  df-disoa 31145  df-dib 31255
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