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Theorem cdleme11g 29605
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29610. (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 5789 . . 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 28704 . . . . . 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 2256 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
9 cdleme11.a . . . . . . 7  |-  A  =  ( Atoms `  K )
108, 9atbase 28630 . . . . . 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 29550 . . . . . 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 14104 . . . . 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 28630 . . . . . 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 28630 . . . . . . . 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 14104 . . . . . . 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 29338 . . . . . . 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 14105 . . . . . 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 14104 . . . . 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 14114 . . . . 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 29197 . . . 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 2300 . 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 29555 . . . . . 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 5794 . . . 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 14150 . . . . 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 14152 . . . . 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 2298 . . 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 5793 . 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 14130 . . . . . 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 14120 . . . . . 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 14104 . . . . . 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 14134 . . . . 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 5789 . . 3  |-  ( Q 
.\/  C )  =  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
)
6360, 62syl6eqr 2306 . 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 2292 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 2419   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   Basecbs 13096   lecple 13163   joincjn 14026   meetcmee 14027   Latclat 14099   Atomscatm 28604   HLchlt 28691   LHypclh 29324
This theorem is referenced by:  cdleme11h  29606  cdleme11j  29607  cdleme15a  29614
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 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328
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