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Theorem 4atlem12a 32608
Description: Lemma for 4at 32611. 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 1027 . . 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 1028 . . 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 1029 . . 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 32362 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
51, 4syl 17 . . . 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 1030 . . . . 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 1031 . . . . 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 2402 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 4at.j . . . . . 6  |-  .\/  =  ( join `  K )
10 4at.a . . . . . 6  |-  A  =  ( Atoms `  K )
118, 9, 10hlatjcl 32365 . . . . 5  |-  ( ( K  e.  HL  /\  U  e.  A  /\  V  e.  A )  ->  ( U  .\/  V
)  e.  ( Base `  K ) )
121, 6, 7, 11syl3anc 1230 . . . 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 1032 . . . . 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 32288 . . . . 5  |-  ( W  e.  A  ->  W  e.  ( Base `  K
) )
1513, 14syl 17 . . . 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 15897 . . . 4  |-  ( ( K  e.  Lat  /\  ( U  .\/  V )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( U  .\/  V )  .\/  W )  e.  ( Base `  K ) )
175, 12, 15, 16syl3anc 1230 . . 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 999 . . 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 32383 . . 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 1243 . 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 32597 . . . 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 1238 . . 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 4406 . 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 32597 . . . 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 1238 . . 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 2424 . 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 285 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   Basecbs 14733   lecple 14808   joincjn 15789   Latclat 15891   Atomscatm 32262   HLchlt 32349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-preset 15773  df-poset 15791  df-lub 15820  df-glb 15821  df-join 15822  df-meet 15823  df-lat 15892  df-ats 32266  df-atl 32297  df-cvlat 32321  df-hlat 32350
This theorem is referenced by:  4atlem12b  32609
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