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Theorem cdleme20c 31170
Description: Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, second line.  D,  F,  Y,  G represent s2, f(s), t2, f(t). (Contributed by NM, 15-Nov-2012.)
Hypotheses
Ref Expression
cdleme19.l  |-  .<_  =  ( le `  K )
cdleme19.j  |-  .\/  =  ( join `  K )
cdleme19.m  |-  ./\  =  ( meet `  K )
cdleme19.a  |-  A  =  ( Atoms `  K )
cdleme19.h  |-  H  =  ( LHyp `  K
)
cdleme19.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme19.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme19.g  |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
cdleme19.d  |-  D  =  ( ( R  .\/  S )  ./\  W )
cdleme19.y  |-  Y  =  ( ( R  .\/  T )  ./\  W )
cdleme20.v  |-  V  =  ( ( S  .\/  T )  ./\  W )
Assertion
Ref Expression
cdleme20c  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( D  .\/  Y )  =  ( ( ( R 
.\/  S )  .\/  T )  ./\  W )
)

Proof of Theorem cdleme20c
StepHypRef Expression
1 simp1l 982 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  HL )
2 simp21l 1075 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  R  e.  A )
3 simp22l 1077 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  S  e.  A )
4 eqid 2438 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
5 cdleme19.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
6 cdleme19.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
74, 5, 6hlatjcl 30226 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
81, 2, 3, 7syl3anc 1185 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( R  .\/  S )  e.  ( Base `  K
) )
9 simp1r 983 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  W  e.  H )
10 cdleme19.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
114, 10lhpbase 30857 . . . . . . . . 9  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
129, 11syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  W  e.  ( Base `  K
) )
13 cdleme19.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
1413, 5, 6hlatlej1 30234 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  R  .<_  ( R  .\/  S ) )
151, 2, 3, 14syl3anc 1185 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  R  .<_  ( R  .\/  S
) )
16 cdleme19.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
174, 13, 5, 16, 6atmod2i1 30720 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  ( R  .\/  S
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  R  .<_  ( R  .\/  S
) )  ->  (
( ( R  .\/  S )  ./\  W )  .\/  R )  =  ( ( R  .\/  S
)  ./\  ( W  .\/  R ) ) )
181, 2, 8, 12, 15, 17syl131anc 1198 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( ( R  .\/  S )  ./\  W )  .\/  R )  =  ( ( R  .\/  S
)  ./\  ( W  .\/  R ) ) )
19 simp21 991 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
20 eqid 2438 . . . . . . . . . 10  |-  ( 1.
`  K )  =  ( 1. `  K
)
2113, 5, 20, 6, 10lhpjat1 30879 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( W  .\/  R
)  =  ( 1.
`  K ) )
221, 9, 19, 21syl21anc 1184 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( W  .\/  R )  =  ( 1. `  K
) )
2322oveq2d 6099 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( R  .\/  S
)  ./\  ( W  .\/  R ) )  =  ( ( R  .\/  S )  ./\  ( 1. `  K ) ) )
24 hlol 30221 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OL )
251, 24syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  OL )
264, 16, 20olm11 30087 . . . . . . . 8  |-  ( ( K  e.  OL  /\  ( R  .\/  S )  e.  ( Base `  K
) )  ->  (
( R  .\/  S
)  ./\  ( 1. `  K ) )  =  ( R  .\/  S
) )
2725, 8, 26syl2anc 644 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( R  .\/  S
)  ./\  ( 1. `  K ) )  =  ( R  .\/  S
) )
2818, 23, 273eqtrrd 2475 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( R  .\/  S )  =  ( ( ( R 
.\/  S )  ./\  W )  .\/  R ) )
2928oveq1d 6098 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( R  .\/  S
)  .\/  T )  =  ( ( ( ( R  .\/  S
)  ./\  W )  .\/  R )  .\/  T
) )
30 simp22r 1078 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  -.  S  .<_  W )
31 simp3r 987 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  R  .<_  ( P  .\/  Q
) )
32 simp3l 986 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  -.  S  .<_  ( P  .\/  Q ) )
33 eqid 2438 . . . . . . . 8  |-  ( ( R  .\/  S ) 
./\  W )  =  ( ( R  .\/  S )  ./\  W )
3413, 5, 16, 6, 10, 33cdlemeda 31157 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( R  .\/  S )  ./\  W )  e.  A )
351, 9, 3, 30, 2, 31, 32, 34syl223anc 1211 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( R  .\/  S
)  ./\  W )  e.  A )
36 simp23 993 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  T  e.  A )
375, 6hlatjass 30229 . . . . . 6  |-  ( ( K  e.  HL  /\  ( ( ( R 
.\/  S )  ./\  W )  e.  A  /\  R  e.  A  /\  T  e.  A )
)  ->  ( (
( ( R  .\/  S )  ./\  W )  .\/  R )  .\/  T
)  =  ( ( ( R  .\/  S
)  ./\  W )  .\/  ( R  .\/  T
) ) )
381, 35, 2, 36, 37syl13anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( ( ( R 
.\/  S )  ./\  W )  .\/  R ) 
.\/  T )  =  ( ( ( R 
.\/  S )  ./\  W )  .\/  ( R 
.\/  T ) ) )
3929, 38eqtrd 2470 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( R  .\/  S
)  .\/  T )  =  ( ( ( R  .\/  S ) 
./\  W )  .\/  ( R  .\/  T ) ) )
4039oveq1d 6098 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( ( R  .\/  S )  .\/  T ) 
./\  W )  =  ( ( ( ( R  .\/  S ) 
./\  W )  .\/  ( R  .\/  T ) )  ./\  W )
)
414, 5, 6hlatjcl 30226 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  T  e.  A )  ->  ( R  .\/  T
)  e.  ( Base `  K ) )
421, 2, 36, 41syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( R  .\/  T )  e.  ( Base `  K
) )
43 hllat 30223 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
441, 43syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  Lat )
454, 13, 16latmle2 14508 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( R  .\/  S )  ./\  W )  .<_  W )
4644, 8, 12, 45syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( R  .\/  S
)  ./\  W )  .<_  W )
474, 13, 5, 16, 6atmod1i1 30716 . . . 4  |-  ( ( K  e.  HL  /\  ( ( ( R 
.\/  S )  ./\  W )  e.  A  /\  ( R  .\/  T )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  /\  ( ( R  .\/  S )  ./\  W )  .<_  W )  ->  ( ( ( R 
.\/  S )  ./\  W )  .\/  ( ( R  .\/  T ) 
./\  W ) )  =  ( ( ( ( R  .\/  S
)  ./\  W )  .\/  ( R  .\/  T
) )  ./\  W
) )
481, 35, 42, 12, 46, 47syl131anc 1198 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( ( R  .\/  S )  ./\  W )  .\/  ( ( R  .\/  T )  ./\  W )
)  =  ( ( ( ( R  .\/  S )  ./\  W )  .\/  ( R  .\/  T
) )  ./\  W
) )
4940, 48eqtr4d 2473 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  (
( ( R  .\/  S )  .\/  T ) 
./\  W )  =  ( ( ( R 
.\/  S )  ./\  W )  .\/  ( ( R  .\/  T ) 
./\  W ) ) )
50 cdleme19.d . . 3  |-  D  =  ( ( R  .\/  S )  ./\  W )
51 cdleme19.y . . 3  |-  Y  =  ( ( R  .\/  T )  ./\  W )
5250, 51oveq12i 6095 . 2  |-  ( D 
.\/  Y )  =  ( ( ( R 
.\/  S )  ./\  W )  .\/  ( ( R  .\/  T ) 
./\  W ) )
5349, 52syl6reqr 2489 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( D  .\/  Y )  =  ( ( ( R 
.\/  S )  .\/  T )  ./\  W )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   1.cp1 14469   Latclat 14476   OLcol 30034   Atomscatm 30123   HLchlt 30210   LHypclh 30843
This theorem is referenced by:  cdleme20d  31171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-psubsp 30362  df-pmap 30363  df-padd 30655  df-lhyp 30847
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