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Theorem 4atlem10b 30476
Description: Lemma for 4at 30484. 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 735 . . . 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 734 . . . . 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 961 . . . . . 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 1036 . . . . . 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 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 ) ) ) )  ->  V  e.  A
)
6 simpl31 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 ) ) ) )  ->  W  e.  A
)
7 simpl32 1040 . . . . . 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 30475 . . . . . 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 1198 . . . . 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 203 . . . 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 4241 . . 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 1037 . . . 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 1041 . . . 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 30474 . . . 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 1198 . . 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 203 . 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 2470 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   lecple 13541   joincjn 14406   Atomscatm 30135   HLchlt 30222
This theorem is referenced by:  4atlem10  30477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-lub 14436  df-join 14438  df-lat 14480  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223
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