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Theorem cdleme20c 34261
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 1012 . . . . . . . 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 1105 . . . . . . . 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 1107 . . . . . . . . 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 2451 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
5 cdleme19.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
6 cdleme19.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
74, 5, 6hlatjcl 33317 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
81, 2, 3, 7syl3anc 1219 . . . . . . . 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 1013 . . . . . . . . 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 33948 . . . . . . . . 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 33325 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  R  .<_  ( R  .\/  S ) )
151, 2, 3, 14syl3anc 1219 . . . . . . . 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 33811 . . . . . . . 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 1232 . . . . . . 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 1021 . . . . . . . . 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 2451 . . . . . . . . . 10  |-  ( 1.
`  K )  =  ( 1. `  K
)
2113, 5, 20, 6, 10lhpjat1 33970 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( W  .\/  R
)  =  ( 1.
`  K ) )
221, 9, 19, 21syl21anc 1218 . . . . . . . 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 6206 . . . . . . 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 33312 . . . . . . . . 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 33178 . . . . . . . 8  |-  ( ( K  e.  OL  /\  ( R  .\/  S )  e.  ( Base `  K
) )  ->  (
( R  .\/  S
)  ./\  ( 1. `  K ) )  =  ( R  .\/  S
) )
2725, 8, 26syl2anc 661 . . . . . . 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 2497 . . . . . 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 6205 . . . . 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 1108 . . . . . . 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 1017 . . . . . . 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 1016 . . . . . . 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 2451 . . . . . . . 8  |-  ( ( R  .\/  S ) 
./\  W )  =  ( ( R  .\/  S )  ./\  W )
3413, 5, 16, 6, 10, 33cdlemeda 34248 . . . . . . 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 1245 . . . . . 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 1023 . . . . . 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 33320 . . . . . 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 1221 . . . . 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 2492 . . . 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 6205 . . 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 33317 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  T  e.  A )  ->  ( R  .\/  T
)  e.  ( Base `  K ) )
421, 2, 36, 41syl3anc 1219 . . . 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 33314 . . . . . 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 15349 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( R  .\/  S )  ./\  W )  .<_  W )
4644, 8, 12, 45syl3anc 1219 . . . 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 33807 . . . 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 1232 . . 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 2495 . 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 6202 . 2  |-  ( D 
.\/  Y )  =  ( ( ( R 
.\/  S )  ./\  W )  .\/  ( ( R  .\/  T ) 
./\  W ) )
5349, 52syl6reqr 2511 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   Basecbs 14276   lecple 14347   joincjn 15216   meetcmee 15217   1.cp1 15310   Latclat 15317   OLcol 33125   Atomscatm 33214   HLchlt 33301   LHypclh 33934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-poset 15218  df-plt 15230  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-p1 15312  df-lat 15318  df-clat 15380  df-oposet 33127  df-ol 33129  df-oml 33130  df-covers 33217  df-ats 33218  df-atl 33249  df-cvlat 33273  df-hlat 33302  df-psubsp 33453  df-pmap 33454  df-padd 33746  df-lhyp 33938
This theorem is referenced by:  cdleme20d  34262
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