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Theorem dib11N 34180
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 3457 . 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 34179 . . . 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 34179 . . . . 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 1203 . . . 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 709 . . 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 1021 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  K  e.  HL )
11 hllat 32381 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
1210, 11syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  K  e.  Lat )
13 simp2l 1023 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  X  e.  B )
14 simp3l 1025 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  Y  e.  B )
152, 3latasymb 16008 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y  /\  Y  .<_  X )  <-> 
X  =  Y ) )
1612, 13, 14, 15syl3anc 1230 . . 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 253 . 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 257 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    C_ wss 3414   class class class wbr 4395   ` cfv 5569   Basecbs 14841   lecple 14916   Latclat 15999   HLchlt 32368   LHypclh 33001   DIsoBcdib 34158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-riotaBAD 31977
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-undef 7005  df-map 7459  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515  df-lplanes 32516  df-lvols 32517  df-lines 32518  df-psubsp 32520  df-pmap 32521  df-padd 32813  df-lhyp 33005  df-laut 33006  df-ldil 33121  df-ltrn 33122  df-trl 33177  df-disoa 34049  df-dib 34159
This theorem is referenced by:  dibf11N  34181
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