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Theorem dia11N 30488
Description: The partial isomorphism A for a lattice  K is one-to-one in the region under co-atom  W. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 25-Nov-2013.) (New usage is discouraged.)
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
dia11N  |-  ( ( ( 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 dia11N
StepHypRef Expression
1 eqss 3169 . 2  |-  ( ( I `  X )  =  ( I `  Y )  <->  ( (
I `  X )  C_  ( I `  Y
)  /\  ( I `  Y )  C_  (
I `  X )
) )
2 dia11.b . . . . 5  |-  B  =  ( Base `  K
)
3 dia11.l . . . . 5  |-  .<_  =  ( le `  K )
4 dia11.h . . . . 5  |-  H  =  ( LHyp `  K
)
5 dia11.i . . . . 5  |-  I  =  ( ( DIsoA `  K
) `  W )
62, 3, 4, 5diaord 30487 . . . 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, 5diaord 30487 . . . . 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 1162 . . . 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 694 . . 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 984 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  K  e.  HL )
11 hllat 28803 . . . . 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 986 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  X  e.  B )
14 simp3l 988 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  Y  e.  B )
152, 3latasymb 14123 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y  /\  Y  .<_  X )  <-> 
X  =  Y ) )
1612, 13, 14, 15syl3anc 1187 . . 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 246 . 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 250 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 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    C_ wss 3127   class class class wbr 3997   ` cfv 4673   Basecbs 13111   lecple 13178   Latclat 14114   HLchlt 28790   LHypclh 29423   DIsoAcdia 30468
This theorem is referenced by:  diaf11N  30489
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-map 6742  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-llines 28937  df-lplanes 28938  df-lvols 28939  df-lines 28940  df-psubsp 28942  df-pmap 28943  df-padd 29235  df-lhyp 29427  df-laut 29428  df-ldil 29543  df-ltrn 29544  df-trl 29598  df-disoa 30469
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