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Theorem 4atlem12a 29726
Description: Lemma for 4at 29729. Substitute  T for  P. (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
4atlem12a  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  ( P  .<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  <-> 
( ( P  .\/  U )  .\/  ( V 
.\/  W ) )  =  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) )

Proof of Theorem 4atlem12a
StepHypRef Expression
1 simp11 987 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  K  e.  HL )
2 simp12 988 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  P  e.  A )
3 simp13 989 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  T  e.  A )
4 hllat 29480 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
51, 4syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  K  e.  Lat )
6 simp21 990 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  U  e.  A )
7 simp22 991 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  V  e.  A )
8 eqid 2389 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 4at.j . . . . . 6  |-  .\/  =  ( join `  K )
10 4at.a . . . . . 6  |-  A  =  ( Atoms `  K )
118, 9, 10hlatjcl 29483 . . . . 5  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
121, 6, 7, 11syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  ( U  .\/  V )  e.  (
Base `  K )
)
13 simp23 992 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  W  e.  A )
148, 10atbase 29406 . . . . 5  |-  ( W  e.  A  ->  W  e.  ( Base `  K
) )
1513, 14syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  W  e.  ( Base `  K )
)
168, 9latjcl 14408 . . . 4  |-  ( ( K  e.  Lat  /\  ( U  .\/  V )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( U  .\/  V )  .\/  W )  e.  ( Base `  K ) )
175, 12, 15, 16syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  ( ( U  .\/  V )  .\/  W )  e.  ( Base `  K ) )
18 simp3 959 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )
19 4at.l . . . 4  |-  .<_  =  ( le `  K )
208, 19, 9, 10hlexchb2 29501 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  T  e.  A  /\  ( ( U  .\/  V )  .\/  W )  e.  ( Base `  K
) )  /\  -.  P  .<_  ( ( U 
.\/  V )  .\/  W ) )  ->  ( P  .<_  ( T  .\/  ( ( U  .\/  V )  .\/  W ) )  <->  ( P  .\/  ( ( U  .\/  V )  .\/  W ) )  =  ( T 
.\/  ( ( U 
.\/  V )  .\/  W ) ) ) )
211, 2, 3, 17, 18, 20syl131anc 1197 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  ( P  .<_  ( T  .\/  (
( U  .\/  V
)  .\/  W )
)  <->  ( P  .\/  ( ( U  .\/  V )  .\/  W ) )  =  ( T 
.\/  ( ( U 
.\/  V )  .\/  W ) ) ) )
2219, 9, 104atlem4a 29715 . . . 4  |-  ( ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  /\  ( V  e.  A  /\  W  e.  A
) )  ->  (
( T  .\/  U
)  .\/  ( V  .\/  W ) )  =  ( T  .\/  (
( U  .\/  V
)  .\/  W )
) )
231, 3, 6, 7, 13, 22syl32anc 1192 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  ( ( T  .\/  U )  .\/  ( V  .\/  W ) )  =  ( T 
.\/  ( ( U 
.\/  V )  .\/  W ) ) )
2423breq2d 4167 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  ( P  .<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  <-> 
P  .<_  ( T  .\/  ( ( U  .\/  V )  .\/  W ) ) ) )
2519, 9, 104atlem4a 29715 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  /\  ( V  e.  A  /\  W  e.  A
) )  ->  (
( P  .\/  U
)  .\/  ( V  .\/  W ) )  =  ( P  .\/  (
( U  .\/  V
)  .\/  W )
) )
261, 2, 6, 7, 13, 25syl32anc 1192 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  ( ( P  .\/  U )  .\/  ( V  .\/  W ) )  =  ( P 
.\/  ( ( U 
.\/  V )  .\/  W ) ) )
2726, 23eqeq12d 2403 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  ( (
( P  .\/  U
)  .\/  ( V  .\/  W ) )  =  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  <-> 
( P  .\/  (
( U  .\/  V
)  .\/  W )
)  =  ( T 
.\/  ( ( U 
.\/  V )  .\/  W ) ) ) )
2821, 24, 273bitr4d 277 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  ( P  .<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  <-> 
( ( P  .\/  U )  .\/  ( V 
.\/  W ) )  =  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   Basecbs 13398   lecple 13465   joincjn 14330   Latclat 14403   Atomscatm 29380   HLchlt 29467
This theorem is referenced by:  4atlem12b  29727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-lub 14360  df-join 14362  df-lat 14404  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468
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