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Theorem cdleme11g 30901
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 30906. (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 6083 . . 3  |-  ( Q 
.\/  F )  =  ( Q  .\/  (
( S  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
3 simp1l 981 . . . 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 1076 . . . 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 30000 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
63, 5syl 16 . . . . 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 992 . . . . . 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 2435 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
9 cdleme11.a . . . . . . 7  |-  A  =  ( Atoms `  K )
108, 9atbase 29926 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
117, 10syl 16 . . . . 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 957 . . . . . 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 990 . . . . . 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 30846 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  U  e.  ( Base `  K )
)
2012, 13, 4, 19syl3anc 1184 . . . . 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 14467 . . . . 5  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  ( S  .\/  U )  e.  ( Base `  K
) )
226, 11, 20, 21syl3anc 1184 . . . 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 29926 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
244, 23syl 16 . . . . 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 29926 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2613, 25syl 16 . . . . . . 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 14467 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  ( P  .\/  S )  e.  ( Base `  K
) )
286, 26, 11, 27syl3anc 1184 . . . . . 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 982 . . . . . . 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 30634 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3129, 30syl 16 . . . . . 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 14468 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  e.  ( Base `  K ) )
336, 28, 31, 32syl3anc 1184 . . . . 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 14467 . . . . 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 1184 . . . 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 14477 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  (
( P  .\/  S
)  ./\  W )  e.  ( Base `  K
) )  ->  Q  .<_  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
) )
376, 24, 33, 36syl3anc 1184 . . . 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 30493 . . . 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 1197 . . 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 2479 . 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 991 . . . . . 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 30851 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( Q  .\/  U )  =  ( P  .\/  Q ) )
4312, 13, 41, 42syl12anc 1182 . . . . 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 6088 . . . 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 14513 . . . . 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 1186 . . . 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 14515 . . . . 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 1186 . . . 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 2477 . . 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 6087 . 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 14493 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
526, 28, 31, 51syl3anc 1184 . . . . 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 14483 . . . . . 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 1186 . . . . 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 15 . . . 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 14467 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( P  .\/  S ) )  e.  (
Base `  K )
)
576, 24, 28, 56syl3anc 1184 . . . . 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 14497 . . . . 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 1184 . . . 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 202 . . 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 6083 . . 3  |-  ( Q 
.\/  C )  =  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
)
6360, 62syl6eqr 2485 . 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 2471 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 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   Latclat 14462   Atomscatm 29900   HLchlt 29987   LHypclh 30620
This theorem is referenced by:  cdleme11h  30902  cdleme11j  30903  cdleme15a  30910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624
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