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Theorem 4atlem11a 33590
Description: Lemma for 4at 33596. 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 1018 . . 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 1020 . . 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 1021 . . 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 33347 . . . . 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 1019 . . . . 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 1022 . . . . 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 2454 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 4at.j . . . . . 6  |-  .\/  =  ( join `  K )
10 4at.a . . . . . 6  |-  A  =  ( Atoms `  K )
118, 9, 10hlatjcl 33350 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  V  e.  A )  ->  ( P  .\/  V
)  e.  ( Base `  K ) )
121, 6, 7, 11syl3anc 1219 . . . 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 1023 . . . . 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 33273 . . . . 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 15341 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  V )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  V )  .\/  W )  e.  ( Base `  K ) )
175, 12, 15, 16syl3anc 1219 . . 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 990 . . 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 33368 . . 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 1232 . 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 33583 . . . 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 1227 . . 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 4413 . 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 33583 . . . 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 1227 . . 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 2476 . 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 965    = wceq 1370    e. wcel 1758   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   Latclat 15335   Atomscatm 33247   HLchlt 33334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-poset 15236  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-lat 15336  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335
This theorem is referenced by:  4atlem11b  33591
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