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Theorem 4atlem10a 33240
Description: Lemma for 4at 33249. 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 1060 . . 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 1063 . . 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 1064 . . 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 33000 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
51, 4syl 17 . . . 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 1030 . . . . 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 2471 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
8 4at.j . . . . . 6  |-  .\/  =  ( join `  K )
9 4at.a . . . . . 6  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 33003 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
116, 10syl 17 . . . 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 1065 . . . . 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 32926 . . . . 5  |-  ( W  e.  A  ->  W  e.  ( Base `  K
) )
1412, 13syl 17 . . . 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 16375 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  W )  e.  ( Base `  K ) )
165, 11, 14, 15syl3anc 1292 . . 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 1032 . . 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 33021 . . 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 1305 . 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 33237 . . . 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 1290 . . 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 4407 . 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 33237 . . . 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 1290 . . 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 2486 . 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 293 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 189    /\ w3a 1007    = wceq 1452    e. wcel 1904   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Basecbs 15199   lecple 15275   joincjn 16267   Latclat 16369   Atomscatm 32900   HLchlt 32987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-preset 16251  df-poset 16269  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-lat 16370  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988
This theorem is referenced by:  4atlem10b  33241
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