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Theorem 4atlem12a 30408
Description: Lemma for 4at 30411. 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 988 . . 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 989 . . 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 990 . . 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 30162 . . . . 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 991 . . . . 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 992 . . . . 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 2437 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 4at.j . . . . . 6  |-  .\/  =  ( join `  K )
10 4at.a . . . . . 6  |-  A  =  ( Atoms `  K )
118, 9, 10hlatjcl 30165 . . . . 5  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
121, 6, 7, 11syl3anc 1185 . . . 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 993 . . . . 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 30088 . . . . 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 14480 . . . 4  |-  ( ( K  e.  Lat  /\  ( U  .\/  V )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( U  .\/  V )  .\/  W )  e.  ( Base `  K ) )
175, 12, 15, 16syl3anc 1185 . . 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 960 . . 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 30183 . . 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 1198 . 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 30397 . . . 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 1193 . . 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 4225 . 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 30397 . . . 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 1193 . . 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 2451 . 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 278 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 178    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   joincjn 14402   Latclat 14475   Atomscatm 30062   HLchlt 30149
This theorem is referenced by:  4atlem12b  30409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-lub 14432  df-join 14434  df-lat 14476  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150
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