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Theorem 3atlem6 30299
Description: Lemma for 3at 30301. (Contributed by NM, 23-Jun-2012.)
Hypotheses
Ref Expression
3at.l  |-  .<_  =  ( le `  K )
3at.j  |-  .\/  =  ( join `  K )
3at.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
3atlem6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U ) )

Proof of Theorem 3atlem6
StepHypRef Expression
1 simp11 985 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  K  e.  HL )
2 simp121 1087 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  P  e.  A )
3 simp122 1088 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  Q  e.  A )
4 simp123 1089 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  R  e.  A )
5 3at.j . . . 4  |-  .\/  =  ( join `  K )
6 3at.a . . . 4  |-  A  =  ( Atoms `  K )
75, 6hlatj32 30183 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( P 
.\/  R )  .\/  Q ) )
81, 2, 3, 4, 7syl13anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( P 
.\/  R )  .\/  Q ) )
92, 4, 33jca 1132 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( P  e.  A  /\  R  e.  A  /\  Q  e.  A )
)
10 simp13 987 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)
11 simp21 988 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  -.  R  .<_  ( P  .\/  Q ) )
12 simp22 989 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  P  =/=  Q )
1312necomd 2542 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  Q  =/=  P )
14 3at.l . . . . . 6  |-  .<_  =  ( le `  K )
1514, 5, 6hlatexch1 30206 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  Q  =/=  P )  ->  ( Q  .<_  ( P  .\/  R
)  ->  R  .<_  ( P  .\/  Q ) ) )
161, 3, 4, 2, 13, 15syl131anc 1195 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( Q  .<_  ( P  .\/  R )  ->  R  .<_  ( P  .\/  Q ) ) )
1711, 16mtod 168 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  -.  Q  .<_  ( P  .\/  R ) )
18 hllat 30175 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
191, 18syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  K  e.  Lat )
20 eqid 2296 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2120, 6atbase 30101 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
224, 21syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  R  e.  ( Base `  K
) )
2320, 6atbase 30101 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
242, 23syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  P  e.  ( Base `  K
) )
2520, 6atbase 30101 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
263, 25syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  Q  e.  ( Base `  K
) )
2720, 14, 5latnlej1l 14191 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  R  =/=  P )
2827necomd 2542 . . . 4  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  P  =/=  R )
2919, 22, 24, 26, 11, 28syl131anc 1195 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  P  =/=  R )
30 simp23 990 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  Q  .<_  ( P  .\/  U
) )
31 simp133 1092 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  U  e.  A )
3214, 5, 6hlatexchb1 30204 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  U  e.  A  /\  P  e.  A
)  /\  Q  =/=  P )  ->  ( Q  .<_  ( P  .\/  U
)  <->  ( P  .\/  Q )  =  ( P 
.\/  U ) ) )
331, 3, 31, 2, 13, 32syl131anc 1195 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( Q  .<_  ( P  .\/  U )  <->  ( P  .\/  Q )  =  ( P 
.\/  U ) ) )
3430, 33mpbid 201 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( P  .\/  Q )  =  ( P  .\/  U
) )
3534breq2d 4051 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( R  .<_  ( P  .\/  Q )  <->  R  .<_  ( P 
.\/  U ) ) )
3611, 35mtbid 291 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  -.  R  .<_  ( P  .\/  U ) )
37 simp3 957 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )
388, 37eqbrtrrd 4061 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  R
)  .\/  Q )  .<_  ( ( S  .\/  T )  .\/  U ) )
3914, 5, 63atlem5 30298 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A  /\  Q  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  Q  .<_  ( P  .\/  R )  /\  P  =/=  R  /\  -.  R  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  R )  .\/  Q )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  R
)  .\/  Q )  =  ( ( S 
.\/  T )  .\/  U ) )
401, 9, 10, 17, 29, 36, 38, 39syl331anc 1207 . 2  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  R
)  .\/  Q )  =  ( ( S 
.\/  T )  .\/  U ) )
418, 40eqtrd 2328 1  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q  /\  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   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:  3atlem7  30300
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-plt 14108  df-lub 14124  df-join 14126  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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