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Theorem cdleme20c 30573
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 979 . . . . . . . 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 1072 . . . . . . . 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 1074 . . . . . . . . 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 2285 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
5 cdleme19.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
6 cdleme19.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
74, 5, 6hlatjcl 29629 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
81, 2, 3, 7syl3anc 1182 . . . . . . . 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 980 . . . . . . . . 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 30260 . . . . . . . . 9  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
129, 11syl 15 . . . . . . . 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 29637 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  R  .<_  ( R  .\/  S ) )
151, 2, 3, 14syl3anc 1182 . . . . . . . 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 30123 . . . . . . . 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 1195 . . . . . . 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 988 . . . . . . . . 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 2285 . . . . . . . . . 10  |-  ( 1.
`  K )  =  ( 1. `  K
)
2113, 5, 20, 6, 10lhpjat1 30282 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( W  .\/  R
)  =  ( 1.
`  K ) )
221, 9, 19, 21syl21anc 1181 . . . . . . . 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 5876 . . . . . . 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 29624 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OL )
251, 24syl 15 . . . . . . . 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 29490 . . . . . . . 8  |-  ( ( K  e.  OL  /\  ( R  .\/  S )  e.  ( Base `  K
) )  ->  (
( R  .\/  S
)  ./\  ( 1. `  K ) )  =  ( R  .\/  S
) )
2725, 8, 26syl2anc 642 . . . . . . 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 2322 . . . . . 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 5875 . . . . 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 1075 . . . . . . 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 984 . . . . . . 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 983 . . . . . . 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 2285 . . . . . . . 8  |-  ( ( R  .\/  S ) 
./\  W )  =  ( ( R  .\/  S )  ./\  W )
3413, 5, 16, 6, 10, 33cdlemeda 30560 . . . . . . 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 1208 . . . . . 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 990 . . . . . 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 29632 . . . . . 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 1184 . . . . 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 2317 . . . 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 5875 . . 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 29629 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  T  e.  A )  ->  ( R  .\/  T
)  e.  ( Base `  K ) )
421, 2, 36, 41syl3anc 1182 . . . 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 29626 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
441, 43syl 15 . . . . 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 14185 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( R  .\/  S )  ./\  W )  .<_  W )
4644, 8, 12, 45syl3anc 1182 . . . 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 30119 . . . 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 1195 . . 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 2320 . 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 5872 . 2  |-  ( D 
.\/  Y )  =  ( ( ( R 
.\/  S )  ./\  W )  .\/  ( ( R  .\/  T ) 
./\  W ) )
5349, 52syl6reqr 2336 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 358    /\ w3a 934    = wceq 1625    e. wcel 1686   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   Basecbs 13150   lecple 13217   joincjn 14080   meetcmee 14081   1.cp1 14146   Latclat 14153   OLcol 29437   Atomscatm 29526   HLchlt 29613   LHypclh 30246
This theorem is referenced by:  cdleme20d  30574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250
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