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Theorem cdleme20c 29630
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 984 . . . . . . . 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 1077 . . . . . . . 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 1079 . . . . . . . . 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 2256 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
5 cdleme19.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
6 cdleme19.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
74, 5, 6hlatjcl 28686 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
81, 2, 3, 7syl3anc 1187 . . . . . . . 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 985 . . . . . . . . 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 29317 . . . . . . . . 9  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
129, 11syl 17 . . . . . . . 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 28694 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  R  .<_  ( R  .\/  S ) )
151, 2, 3, 14syl3anc 1187 . . . . . . . 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 29180 . . . . . . . 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 1200 . . . . . . 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 993 . . . . . . . . 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 2256 . . . . . . . . . 10  |-  ( 1.
`  K )  =  ( 1. `  K
)
2113, 5, 20, 6, 10lhpjat1 29339 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( W  .\/  R
)  =  ( 1.
`  K ) )
221, 9, 19, 21syl21anc 1186 . . . . . . . 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 5773 . . . . . . 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 28681 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OL )
251, 24syl 17 . . . . . . . 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 28547 . . . . . . . 8  |-  ( ( K  e.  OL  /\  ( R  .\/  S )  e.  ( Base `  K
) )  ->  (
( R  .\/  S
)  ./\  ( 1. `  K ) )  =  ( R  .\/  S
) )
2725, 8, 26syl2anc 645 . . . . . . 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 2293 . . . . . 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 5772 . . . . 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 1080 . . . . . . 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 989 . . . . . . 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 988 . . . . . . 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 2256 . . . . . . . 8  |-  ( ( R  .\/  S ) 
./\  W )  =  ( ( R  .\/  S )  ./\  W )
3413, 5, 16, 6, 10, 33cdlemeda 29617 . . . . . . 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 1213 . . . . . 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 995 . . . . . 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 28689 . . . . . 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 1189 . . . . 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 2288 . . . 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 5772 . . 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 28686 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  T  e.  A )  ->  ( R  .\/  T
)  e.  ( Base `  K ) )
421, 2, 36, 41syl3anc 1187 . . . 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 28683 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
441, 43syl 17 . . . . 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 14110 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( R  .\/  S )  ./\  W )  .<_  W )
4644, 8, 12, 45syl3anc 1187 . . . 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 29176 . . . 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 1200 . . 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 2291 . 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 5769 . 2  |-  ( D 
.\/  Y )  =  ( ( ( R 
.\/  S )  ./\  W )  .\/  ( ( R  .\/  T ) 
./\  W ) )
5349, 52syl6reqr 2307 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   Basecbs 13075   lecple 13142   joincjn 14005   meetcmee 14006   1.cp1 14071   Latclat 14078   OLcol 28494   Atomscatm 28583   HLchlt 28670   LHypclh 29303
This theorem is referenced by:  cdleme20d  29631
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 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307
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