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Theorem 4atlem10a 30463
Description: Lemma for 4at 30472. 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 988 . . 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 991 . . 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 992 . . 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 30223 . . . . 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 958 . . . . 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 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
8 4at.j . . . . . 6  |-  .\/  =  ( join `  K )
9 4at.a . . . . . 6  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 30226 . . . . 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 993 . . . . 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 30149 . . . . 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 14481 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  W )  e.  ( Base `  K ) )
165, 11, 14, 15syl3anc 1185 . . 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 960 . . 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 30244 . . 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 1198 . 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 30460 . . . 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 1183 . . 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 4226 . 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 30460 . . . 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 1183 . . 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 2452 . 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 278 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   Latclat 14476   Atomscatm 30123   HLchlt 30210
This theorem is referenced by:  4atlem10b  30464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-lub 14433  df-join 14435  df-lat 14477  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211
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