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Theorem cdleme11g 29143
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29148. (Contributed by NM, 14-Jun-2012.)
Hypotheses
Ref Expression
cdleme11.l  |-  .<_  =  ( le `  K )
cdleme11.j  |-  .\/  =  ( join `  K )
cdleme11.m  |-  ./\  =  ( meet `  K )
cdleme11.a  |-  A  =  ( Atoms `  K )
cdleme11.h  |-  H  =  ( LHyp `  K
)
cdleme11.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme11.c  |-  C  =  ( ( P  .\/  S )  ./\  W )
cdleme11.d  |-  D  =  ( ( P  .\/  T )  ./\  W )
cdleme11.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
Assertion
Ref Expression
cdleme11g  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  F )  =  ( Q  .\/  C ) )

Proof of Theorem cdleme11g
StepHypRef Expression
1 cdleme11.f . . . 4  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
21oveq2i 5721 . . 3  |-  ( Q 
.\/  F )  =  ( Q  .\/  (
( S  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
3 simp1l 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  K  e.  HL )
4 simp22l 1079 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
5 hllat 28242 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
63, 5syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  K  e.  Lat )
7 simp23 995 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  S  e.  A )
8 eqid 2253 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
9 cdleme11.a . . . . . . 7  |-  A  =  ( Atoms `  K )
108, 9atbase 28168 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
117, 10syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  S  e.  ( Base `  K )
)
12 simp1 960 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( K  e.  HL  /\  W  e.  H ) )
13 simp21 993 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  P  e.  A )
14 cdleme11.l . . . . . . 7  |-  .<_  =  ( le `  K )
15 cdleme11.j . . . . . . 7  |-  .\/  =  ( join `  K )
16 cdleme11.m . . . . . . 7  |-  ./\  =  ( meet `  K )
17 cdleme11.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
18 cdleme11.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
1914, 15, 16, 9, 17, 18, 8cdleme0aa 29088 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  U  e.  ( Base `  K )
)
2012, 13, 4, 19syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  U  e.  ( Base `  K )
)
218, 15latjcl 14000 . . . . 5  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  ( S  .\/  U )  e.  ( Base `  K
) )
226, 11, 20, 21syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( S  .\/  U )  e.  (
Base `  K )
)
238, 9atbase 28168 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
244, 23syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  ( Base `  K )
)
258, 9atbase 28168 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2613, 25syl 17 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  P  e.  ( Base `  K )
)
278, 15latjcl 14000 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  ( P  .\/  S )  e.  ( Base `  K
) )
286, 26, 11, 27syl3anc 1187 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  S )  e.  (
Base `  K )
)
29 simp1r 985 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  W  e.  H )
308, 17lhpbase 28876 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3129, 30syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  W  e.  ( Base `  K )
)
328, 16latmcl 14001 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  e.  ( Base `  K ) )
336, 28, 31, 32syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  S )  ./\  W )  e.  ( Base `  K ) )
348, 15latjcl 14000 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  (
( P  .\/  S
)  ./\  W )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) )  e.  (
Base `  K )
)
356, 24, 33, 34syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
)  e.  ( Base `  K ) )
368, 14, 15latlej1 14010 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  (
( P  .\/  S
)  ./\  W )  e.  ( Base `  K
) )  ->  Q  .<_  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
) )
376, 24, 33, 36syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  Q  .<_  ( Q  .\/  ( ( P  .\/  S ) 
./\  W ) ) )
388, 14, 15, 16, 9atmod1i1 28735 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  ( S  .\/  U
)  e.  ( Base `  K )  /\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) )  e.  (
Base `  K )
)  /\  Q  .<_  ( Q  .\/  ( ( P  .\/  S ) 
./\  W ) ) )  ->  ( Q  .\/  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )  =  ( ( Q  .\/  ( S  .\/  U ) )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
393, 4, 22, 35, 37, 38syl131anc 1200 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )  =  ( ( Q  .\/  ( S  .\/  U ) )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
402, 39syl5eq 2297 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  F )  =  ( ( Q  .\/  ( S  .\/  U ) ) 
./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
41 simp22 994 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
4214, 15, 16, 9, 17, 18cdleme0cq 29093 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( Q  .\/  U )  =  ( P  .\/  Q ) )
4312, 13, 41, 42syl12anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  U )  =  ( P  .\/  Q ) )
4443oveq2d 5726 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( S  .\/  ( Q  .\/  U
) )  =  ( S  .\/  ( P 
.\/  Q ) ) )
458, 15latj12 14046 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  S  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) ) )  -> 
( Q  .\/  ( S  .\/  U ) )  =  ( S  .\/  ( Q  .\/  U ) ) )
466, 24, 11, 20, 45syl13anc 1189 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( S  .\/  U
) )  =  ( S  .\/  ( Q 
.\/  U ) ) )
478, 15latj13 14048 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) ) )  -> 
( Q  .\/  ( P  .\/  S ) )  =  ( S  .\/  ( P  .\/  Q ) ) )
486, 24, 26, 11, 47syl13anc 1189 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( P  .\/  S
) )  =  ( S  .\/  ( P 
.\/  Q ) ) )
4944, 46, 483eqtr4d 2295 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( S  .\/  U
) )  =  ( Q  .\/  ( P 
.\/  S ) ) )
5049oveq1d 5725 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( Q  .\/  ( S  .\/  U ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
518, 14, 16latmle1 14026 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
526, 28, 31, 51syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
538, 14, 15latjlej2 14016 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( ( P 
.\/  S )  ./\  W )  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  Q  e.  ( Base `  K )
) )  ->  (
( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S
)  ->  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
)  .<_  ( Q  .\/  ( P  .\/  S ) ) ) )
546, 33, 28, 24, 53syl13anc 1189 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( (
( P  .\/  S
)  ./\  W )  .<_  ( P  .\/  S
)  ->  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
)  .<_  ( Q  .\/  ( P  .\/  S ) ) ) )
5552, 54mpd 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
)  .<_  ( Q  .\/  ( P  .\/  S ) ) )
568, 15latjcl 14000 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( P  .\/  S ) )  e.  (
Base `  K )
)
576, 24, 28, 56syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( P  .\/  S
) )  e.  (
Base `  K )
)
588, 14, 16latleeqm2 14030 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  ( ( P  .\/  S ) 
./\  W ) )  e.  ( Base `  K
)  /\  ( Q  .\/  ( P  .\/  S
) )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) )  .<_  ( Q 
.\/  ( P  .\/  S ) )  <->  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
) ) )
596, 35, 57, 58syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) )  .<_  ( Q 
.\/  ( P  .\/  S ) )  <->  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
) ) )
6055, 59mpbid 203 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
) )
61 cdleme11.c . . . 4  |-  C  =  ( ( P  .\/  S )  ./\  W )
6261oveq2i 5721 . . 3  |-  ( Q 
.\/  C )  =  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
)
6360, 62syl6eqr 2303 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( Q  .\/  C
) )
6440, 50, 633eqtrd 2289 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  F )  =  ( Q  .\/  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Latclat 13995   Atomscatm 28142   HLchlt 28229   LHypclh 28862
This theorem is referenced by:  cdleme11h  29144  cdleme11j  29145  cdleme15a  29152
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-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-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-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866
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