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Theorem 3atlem3 35622
Description: Lemma for 3at 35627. (Contributed by NM, 23-Jun-2012.)
Hypotheses
Ref Expression
3at.l  |-  .<_  =  ( le `  K )
3at.j  |-  .\/  =  ( join `  K )
3at.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
3atlem3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  U  /\  -.  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U ) )

Proof of Theorem 3atlem3
StepHypRef Expression
1 simpl1 997 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
) )
2 simpl21 1072 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  -.  R  .<_  ( P  .\/  Q
) )
3 simpl22 1073 . . . 4  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  P  =/=  U )
4 simpr 459 . . . 4  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  P  .<_  ( T  .\/  U ) )
53, 4jca 530 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  ( P  =/=  U  /\  P  .<_  ( T  .\/  U ) ) )
6 simpl23 1074 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  -.  Q  .<_  ( P  .\/  U
) )
7 simpl3 999 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )
8 3at.l . . . 4  |-  .<_  =  ( le `  K )
9 3at.j . . . 4  |-  .\/  =  ( join `  K )
10 3at.a . . . 4  |-  A  =  ( Atoms `  K )
118, 9, 103atlem2 35621 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T ) 
.\/  U ) )
121, 2, 5, 6, 7, 11syl131anc 1239 . 2  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  P  .<_  ( T  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T ) 
.\/  U ) )
13 simpl1 997 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  P  .<_  ( T  .\/  U
) )  ->  ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) ) )
14 simpl21 1072 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  P  .<_  ( T  .\/  U
) )  ->  -.  R  .<_  ( P  .\/  Q ) )
15 simpr 459 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  P  .<_  ( T  .\/  U
) )  ->  -.  P  .<_  ( T  .\/  U ) )
16 simpl23 1074 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  P  .<_  ( T  .\/  U
) )  ->  -.  Q  .<_  ( P  .\/  U ) )
17 simpl3 999 . . 3  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  P  .<_  ( T  .\/  U
) )  ->  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )
188, 9, 103atlem1 35620 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  P  .<_  ( T  .\/  U )  /\  -.  Q  .<_  ( P  .\/  U ) )  /\  ( ( P  .\/  Q ) 
.\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T ) 
.\/  U ) )
1913, 14, 15, 16, 17, 18syl131anc 1239 . 2  |-  ( ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/= 
U  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  /\  -.  P  .<_  ( T  .\/  U
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U ) )
2012, 19pm2.61dan 789 1  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  U  /\  -.  Q  .<_  ( P 
.\/  U ) )  /\  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   lecple 14709   joincjn 15690   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-plt 15705  df-lub 15721  df-glb 15722  df-join 15723  df-meet 15724  df-p0 15786  df-lat 15793  df-covers 35404  df-ats 35405  df-atl 35436  df-cvlat 35460  df-hlat 35489
This theorem is referenced by:  3atlem4  35623  3atlem5  35624
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