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Theorem 4atlem11a 35744
Description: Lemma for 4at 35750. Substitute  U for  Q. (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
4atlem11a  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  ( Q  .<_  ( ( P  .\/  U )  .\/  ( V 
.\/  W ) )  <-> 
( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  =  ( ( P 
.\/  U )  .\/  ( V  .\/  W ) ) ) )

Proof of Theorem 4atlem11a
StepHypRef Expression
1 simp11 1024 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  K  e.  HL )
2 simp13 1026 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  Q  e.  A )
3 simp21 1027 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  U  e.  A )
4 hllat 35501 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
51, 4syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  K  e.  Lat )
6 simp12 1025 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  P  e.  A )
7 simp22 1028 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  V  e.  A )
8 eqid 2382 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 4at.j . . . . . 6  |-  .\/  =  ( join `  K )
10 4at.a . . . . . 6  |-  A  =  ( Atoms `  K )
118, 9, 10hlatjcl 35504 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  V  e.  A )  ->  ( P  .\/  V
)  e.  ( Base `  K ) )
121, 6, 7, 11syl3anc 1226 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  ( P  .\/  V )  e.  (
Base `  K )
)
13 simp23 1029 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  W  e.  A )
148, 10atbase 35427 . . . . 5  |-  ( W  e.  A  ->  W  e.  ( Base `  K
) )
1513, 14syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  W  e.  ( Base `  K )
)
168, 9latjcl 15798 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  V )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  V )  .\/  W )  e.  ( Base `  K ) )
175, 12, 15, 16syl3anc 1226 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  ( ( P  .\/  V )  .\/  W )  e.  ( Base `  K ) )
18 simp3 996 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )
19 4at.l . . . 4  |-  .<_  =  ( le `  K )
208, 19, 9, 10hlexchb2 35522 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  U  e.  A  /\  ( ( P  .\/  V )  .\/  W )  e.  ( Base `  K
) )  /\  -.  Q  .<_  ( ( P 
.\/  V )  .\/  W ) )  ->  ( Q  .<_  ( U  .\/  ( ( P  .\/  V )  .\/  W ) )  <->  ( Q  .\/  ( ( P  .\/  V )  .\/  W ) )  =  ( U 
.\/  ( ( P 
.\/  V )  .\/  W ) ) ) )
211, 2, 3, 17, 18, 20syl131anc 1239 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  ( Q  .<_  ( U  .\/  (
( P  .\/  V
)  .\/  W )
)  <->  ( Q  .\/  ( ( P  .\/  V )  .\/  W ) )  =  ( U 
.\/  ( ( P 
.\/  V )  .\/  W ) ) ) )
2219, 9, 104atlem4b 35737 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  /\  ( V  e.  A  /\  W  e.  A
) )  ->  (
( P  .\/  U
)  .\/  ( V  .\/  W ) )  =  ( U  .\/  (
( P  .\/  V
)  .\/  W )
) )
231, 6, 3, 7, 13, 22syl32anc 1234 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  ( ( P  .\/  U )  .\/  ( V  .\/  W ) )  =  ( U 
.\/  ( ( P 
.\/  V )  .\/  W ) ) )
2423breq2d 4379 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  ( Q  .<_  ( ( P  .\/  U )  .\/  ( V 
.\/  W ) )  <-> 
Q  .<_  ( U  .\/  ( ( P  .\/  V )  .\/  W ) ) ) )
2519, 9, 104atlem4b 35737 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( V  e.  A  /\  W  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  ( V  .\/  W ) )  =  ( Q  .\/  (
( P  .\/  V
)  .\/  W )
) )
261, 6, 2, 7, 13, 25syl32anc 1234 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  ( ( P  .\/  Q )  .\/  ( V  .\/  W ) )  =  ( Q 
.\/  ( ( P 
.\/  V )  .\/  W ) ) )
2726, 23eqeq12d 2404 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  ( (
( P  .\/  Q
)  .\/  ( V  .\/  W ) )  =  ( ( P  .\/  U )  .\/  ( V 
.\/  W ) )  <-> 
( Q  .\/  (
( P  .\/  V
)  .\/  W )
)  =  ( U 
.\/  ( ( P 
.\/  V )  .\/  W ) ) ) )
2821, 24, 273bitr4d 285 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  Q  .<_  ( ( P  .\/  V )  .\/  W ) )  ->  ( Q  .<_  ( ( P  .\/  U )  .\/  ( V 
.\/  W ) )  <-> 
( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  =  ( ( P 
.\/  U )  .\/  ( V  .\/  W ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1399    e. wcel 1826   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   Basecbs 14634   lecple 14709   joincjn 15690   Latclat 15792   Atomscatm 35401   HLchlt 35488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-preset 15674  df-poset 15692  df-lub 15721  df-glb 15722  df-join 15723  df-meet 15724  df-lat 15793  df-ats 35405  df-atl 35436  df-cvlat 35460  df-hlat 35489
This theorem is referenced by:  4atlem11b  35745
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