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Theorem 4atlem10a 35725
Description: Lemma for 4at 35734. Substitute  V for  R. (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
4atlem10a  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  ( R  .<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  <-> 
( ( P  .\/  Q )  .\/  ( R 
.\/  W ) )  =  ( ( P 
.\/  Q )  .\/  ( V  .\/  W ) ) ) )

Proof of Theorem 4atlem10a
StepHypRef Expression
1 simp11 1024 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  K  e.  HL )
2 simp21 1027 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  R  e.  A )
3 simp22 1028 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  V  e.  A )
4 hllat 35485 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
51, 4syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  K  e.  Lat )
6 simp1 994 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) )
7 eqid 2454 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
8 4at.j . . . . . 6  |-  .\/  =  ( join `  K )
9 4at.a . . . . . 6  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 35488 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
116, 10syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
12 simp23 1029 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  W  e.  A )
137, 9atbase 35411 . . . . 5  |-  ( W  e.  A  ->  W  e.  ( Base `  K
) )
1412, 13syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  W  e.  ( Base `  K )
)
157, 8latjcl 15880 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  W )  e.  ( Base `  K ) )
165, 11, 14, 15syl3anc 1226 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  ( ( P  .\/  Q )  .\/  W )  e.  ( Base `  K ) )
17 simp3 996 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )
18 4at.l . . . 4  |-  .<_  =  ( le `  K )
197, 18, 8, 9hlexchb2 35506 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  V  e.  A  /\  ( ( P  .\/  Q )  .\/  W )  e.  ( Base `  K
) )  /\  -.  R  .<_  ( ( P 
.\/  Q )  .\/  W ) )  ->  ( R  .<_  ( V  .\/  ( ( P  .\/  Q )  .\/  W ) )  <->  ( R  .\/  ( ( P  .\/  Q )  .\/  W ) )  =  ( V 
.\/  ( ( P 
.\/  Q )  .\/  W ) ) ) )
201, 2, 3, 16, 17, 19syl131anc 1239 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  ( R  .<_  ( V  .\/  (
( P  .\/  Q
)  .\/  W )
)  <->  ( R  .\/  ( ( P  .\/  Q )  .\/  W ) )  =  ( V 
.\/  ( ( P 
.\/  Q )  .\/  W ) ) ) )
2118, 8, 94atlem4c 35722 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( V  e.  A  /\  W  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  ( V  .\/  W ) )  =  ( V  .\/  (
( P  .\/  Q
)  .\/  W )
) )
226, 3, 12, 21syl12anc 1224 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  ( ( P  .\/  Q )  .\/  ( V  .\/  W ) )  =  ( V 
.\/  ( ( P 
.\/  Q )  .\/  W ) ) )
2322breq2d 4451 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  ( R  .<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  <-> 
R  .<_  ( V  .\/  ( ( P  .\/  Q )  .\/  W ) ) ) )
2418, 8, 94atlem4c 35722 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  W  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  ( R  .\/  W ) )  =  ( R  .\/  (
( P  .\/  Q
)  .\/  W )
) )
256, 2, 12, 24syl12anc 1224 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  ( ( P  .\/  Q )  .\/  ( R  .\/  W ) )  =  ( R 
.\/  ( ( P 
.\/  Q )  .\/  W ) ) )
2625, 22eqeq12d 2476 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  ( (
( P  .\/  Q
)  .\/  ( R  .\/  W ) )  =  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  <-> 
( R  .\/  (
( P  .\/  Q
)  .\/  W )
)  =  ( V 
.\/  ( ( P 
.\/  Q )  .\/  W ) ) ) )
2720, 23, 263bitr4d 285 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  R  .<_  ( ( P  .\/  Q )  .\/  W ) )  ->  ( R  .<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  <-> 
( ( P  .\/  Q )  .\/  ( R 
.\/  W ) )  =  ( ( P 
.\/  Q )  .\/  ( V  .\/  W ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   Latclat 15874   Atomscatm 35385   HLchlt 35472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-lat 15875  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473
This theorem is referenced by:  4atlem10b  35726
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