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Theorem 4atlem10b 30402
Description: Lemma for 4at 30410. Substitute  V for  R (cont.). (Contributed by NM, 10-Jul-2012.)
Hypotheses
Ref Expression
4at.l  |-  .<_  =  ( le `  K )
4at.j  |-  .\/  =  ( join `  K )
4at.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
4atlem10b  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  ( ( P 
.\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) )

Proof of Theorem 4atlem10b
StepHypRef Expression
1 simprr 734 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) )
2 simprl 733 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  R  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) )
3 simpl1 960 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) )
4 simpl21 1035 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  R  e.  A
)
5 simpl23 1037 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  V  e.  A
)
6 simpl31 1038 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  W  e.  A
)
7 simpl32 1039 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  -.  R  .<_  ( ( P  .\/  Q
)  .\/  W )
)
8 4at.l . . . . . . 7  |-  .<_  =  ( le `  K )
9 4at.j . . . . . . 7  |-  .\/  =  ( join `  K )
10 4at.a . . . . . . 7  |-  A  =  ( Atoms `  K )
118, 9, 104atlem10a 30401 . . . . . 6  |-  ( ( ( 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 ) ) ) )
123, 4, 5, 6, 7, 11syl131anc 1197 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  ( R  .<_  ( ( P  .\/  Q
)  .\/  ( V  .\/  W ) )  <->  ( ( P  .\/  Q )  .\/  ( R  .\/  W ) )  =  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )
132, 12mpbid 202 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  ( ( P 
.\/  Q )  .\/  ( R  .\/  W ) )  =  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) )
141, 13breqtrrd 4238 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  S  .<_  ( ( P  .\/  Q ) 
.\/  ( R  .\/  W ) ) )
15 simpl22 1036 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  S  e.  A
)
16 simpl33 1040 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  -.  S  .<_  ( ( P  .\/  Q
)  .\/  R )
)
178, 9, 104atlem9 30400 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( S  .<_  ( ( P  .\/  Q )  .\/  ( R 
.\/  W ) )  <-> 
( ( P  .\/  Q )  .\/  ( R 
.\/  S ) )  =  ( ( P 
.\/  Q )  .\/  ( R  .\/  W ) ) ) )
183, 4, 15, 6, 16, 17syl131anc 1197 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  ( S  .<_  ( ( P  .\/  Q
)  .\/  ( R  .\/  W ) )  <->  ( ( P  .\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  Q ) 
.\/  ( R  .\/  W ) ) ) )
1914, 18mpbid 202 . 2  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  ( ( P 
.\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  Q ) 
.\/  ( R  .\/  W ) ) )
2019, 13eqtrd 2468 1  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  V  e.  A )  /\  ( W  e.  A  /\  -.  R  .<_  ( ( P  .\/  Q ) 
.\/  W )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) )  /\  ( R 
.<_  ( ( P  .\/  Q )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) ) )  ->  ( ( P 
.\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  Q ) 
.\/  ( V  .\/  W ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   lecple 13536   joincjn 14401   Atomscatm 30061   HLchlt 30148
This theorem is referenced by:  4atlem10  30403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-lub 14431  df-join 14433  df-lat 14475  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149
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