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Theorem 4atlem9 33566
Description: Lemma for 4at 33576. Substitute  W for  S. (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
4atlem9  |-  ( ( ( 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 ) ) ) )

Proof of Theorem 4atlem9
StepHypRef Expression
1 simp11 1018 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  K  e.  HL )
2 simp22 1022 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  S  e.  A )
3 simp23 1023 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  W  e.  A )
4 hllat 33327 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
51, 4syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  K  e.  Lat )
6 simp1 988 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) )
7 eqid 2452 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
8 4at.j . . . . . 6  |-  .\/  =  ( join `  K )
9 4at.a . . . . . 6  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 33330 . . . . 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  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
12 simp21 1021 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  R  e.  A )
137, 9atbase 33253 . . . . 5  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1412, 13syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  R  e.  ( Base `  K )
)
157, 8latjcl 15335 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
165, 11, 14, 15syl3anc 1219 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
17 simp3 990 . . 3  |-  ( ( ( 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 ) )
18 4at.l . . . 4  |-  .<_  =  ( le `  K )
197, 18, 8, 9hlexchb2 33348 . . 3  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  W  e.  A  /\  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K
) )  /\  -.  S  .<_  ( ( P 
.\/  Q )  .\/  R ) )  ->  ( S  .<_  ( W  .\/  ( ( P  .\/  Q )  .\/  R ) )  <->  ( S  .\/  ( ( P  .\/  Q )  .\/  R ) )  =  ( W 
.\/  ( ( P 
.\/  Q )  .\/  R ) ) ) )
201, 2, 3, 16, 17, 19syl131anc 1232 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( S  .<_  ( W  .\/  (
( P  .\/  Q
)  .\/  R )
)  <->  ( S  .\/  ( ( P  .\/  Q )  .\/  R ) )  =  ( W 
.\/  ( ( P 
.\/  Q )  .\/  R ) ) ) )
2118, 8, 94atlem4d 33565 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  W  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  ( R  .\/  W ) )  =  ( W  .\/  (
( P  .\/  Q
)  .\/  R )
) )
226, 12, 3, 21syl12anc 1217 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( ( P  .\/  Q )  .\/  ( R  .\/  W ) )  =  ( W 
.\/  ( ( P 
.\/  Q )  .\/  R ) ) )
2322breq2d 4407 . 2  |-  ( ( ( 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 ) )  <-> 
S  .<_  ( W  .\/  ( ( P  .\/  Q )  .\/  R ) ) ) )
2418, 8, 94atlem4d 33565 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  ( R  .\/  S ) )  =  ( S  .\/  (
( P  .\/  Q
)  .\/  R )
) )
256, 12, 2, 24syl12anc 1217 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( ( P  .\/  Q )  .\/  ( R  .\/  S ) )  =  ( S 
.\/  ( ( P 
.\/  Q )  .\/  R ) ) )
2625, 22eqeq12d 2474 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  W  e.  A
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )  ->  ( (
( P  .\/  Q
)  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  Q )  .\/  ( R 
.\/  W ) )  <-> 
( S  .\/  (
( P  .\/  Q
)  .\/  R )
)  =  ( W 
.\/  ( ( P 
.\/  Q )  .\/  R ) ) ) )
2720, 23, 263bitr4d 285 1  |-  ( ( ( 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 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   Latclat 15329   Atomscatm 33227   HLchlt 33314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-poset 15230  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-lat 15330  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315
This theorem is referenced by:  4atlem10b  33568
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