Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  4atlem9 Unicode version

Theorem 4atlem9 30414
Description: Lemma for 4at 30424. Substitute  W for  S. (Contributed by NM, 9-Jul-2012.)
Hypotheses
Ref Expression
4at.l  |-  .<_  =  ( le `  K )
4at.j  |-  .\/  =  ( join `  K )
4at.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
4atlem9  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( S  .<_  ( ( P  .\/  Q )  .\/  ( R 
.\/  W ) )  <-> 
( ( P  .\/  Q )  .\/  ( R 
.\/  S ) )  =  ( ( P 
.\/  Q )  .\/  ( R  .\/  W ) ) ) )

Proof of Theorem 4atlem9
StepHypRef Expression
1 simp11 985 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  K  e.  HL )
2 simp22 989 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  S  e.  A )
3 simp23 990 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  W  e.  A )
4 hllat 30175 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
51, 4syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  K  e.  Lat )
6 simp1 955 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) )
7 eqid 2296 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
8 4at.j . . . . . 6  |-  .\/  =  ( join `  K )
9 4at.a . . . . . 6  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 30178 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
116, 10syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
12 simp21 988 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  R  e.  A )
137, 9atbase 30101 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1412, 13syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  R  e.  ( Base `  K )
)
157, 8latjcl 14172 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
165, 11, 14, 15syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
17 simp3 957 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )
18 4at.l . . . 4  |-  .<_  =  ( le `  K )
197, 18, 8, 9hlexchb2 30196 . . 3  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  W  e.  A  /\  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K
) )  /\  -.  S  .<_  ( ( P 
.\/  Q )  .\/  R ) )  ->  ( S  .<_  ( W  .\/  ( ( P  .\/  Q )  .\/  R ) )  <->  ( S  .\/  ( ( P  .\/  Q )  .\/  R ) )  =  ( W 
.\/  ( ( P 
.\/  Q )  .\/  R ) ) ) )
201, 2, 3, 16, 17, 19syl131anc 1195 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( S  .<_  ( W  .\/  (
( P  .\/  Q
)  .\/  R )
)  <->  ( S  .\/  ( ( P  .\/  Q )  .\/  R ) )  =  ( W 
.\/  ( ( P 
.\/  Q )  .\/  R ) ) ) )
2118, 8, 94atlem4d 30413 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  W  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  ( R  .\/  W ) )  =  ( W  .\/  (
( P  .\/  Q
)  .\/  R )
) )
226, 12, 3, 21syl12anc 1180 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( ( P  .\/  Q )  .\/  ( R  .\/  W ) )  =  ( W 
.\/  ( ( P 
.\/  Q )  .\/  R ) ) )
2322breq2d 4051 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( S  .<_  ( ( P  .\/  Q )  .\/  ( R 
.\/  W ) )  <-> 
S  .<_  ( W  .\/  ( ( P  .\/  Q )  .\/  R ) ) ) )
2418, 8, 94atlem4d 30413 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  ( R  .\/  S ) )  =  ( S  .\/  (
( P  .\/  Q
)  .\/  R )
) )
256, 12, 2, 24syl12anc 1180 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( ( P  .\/  Q )  .\/  ( R  .\/  S ) )  =  ( S 
.\/  ( ( P 
.\/  Q )  .\/  R ) ) )
2625, 22eqeq12d 2310 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( (
( P  .\/  Q
)  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  Q )  .\/  ( R 
.\/  W ) )  <-> 
( S  .\/  (
( P  .\/  Q
)  .\/  R )
)  =  ( W 
.\/  ( ( P 
.\/  Q )  .\/  R ) ) ) )
2720, 23, 263bitr4d 276 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( S  .<_  ( ( P  .\/  Q )  .\/  ( R 
.\/  W ) )  <-> 
( ( P  .\/  Q )  .\/  ( R 
.\/  S ) )  =  ( ( P 
.\/  Q )  .\/  ( R  .\/  W ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Latclat 14167   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  4atlem10b  30416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-lub 14124  df-join 14126  df-lat 14168  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
  Copyright terms: Public domain W3C validator