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Theorem cdlemg10bALTN 34285
Description: TODO: FIX COMMENT TODO: Can this be moved up as a stand-alone theorem in ltrn* area? TODO: Compare this proof to cdlemg2m 34253 and pick best, if moved to ltrn* area. (Contributed by NM, 4-May-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemg8.l  |-  .<_  =  ( le `  K )
cdlemg8.j  |-  .\/  =  ( join `  K )
cdlemg8.m  |-  ./\  =  ( meet `  K )
cdlemg8.a  |-  A  =  ( Atoms `  K )
cdlemg8.h  |-  H  =  ( LHyp `  K
)
cdlemg8.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemg10bALTN  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( F `
 P )  .\/  ( F `  Q ) )  ./\  W )  =  ( ( P 
.\/  Q )  ./\  W ) )

Proof of Theorem cdlemg10bALTN
StepHypRef Expression
1 simp11 1018 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  K  e.  HL )
2 simp12 1019 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  W  e.  H )
31, 2jca 532 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
4 3simpc 987 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
5 simp13 1020 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
6 cdlemg8.h . . . . 5  |-  H  =  ( LHyp `  K
)
7 cdlemg8.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemg8.l . . . . 5  |-  .<_  =  ( le `  K )
9 cdlemg8.j . . . . 5  |-  .\/  =  ( join `  K )
10 cdlemg8.a . . . . 5  |-  A  =  ( Atoms `  K )
11 cdlemg8.m . . . . 5  |-  ./\  =  ( meet `  K )
12 eqid 2443 . . . . 5  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
136, 7, 8, 9, 10, 11, 12cdlemg2k 34250 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  (
( F `  P
)  .\/  ( F `  Q ) )  =  ( ( F `  P )  .\/  (
( P  .\/  Q
)  ./\  W )
) )
143, 4, 5, 13syl3anc 1218 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( F `  P )  .\/  ( F `  Q )
)  =  ( ( F `  P ) 
.\/  ( ( P 
.\/  Q )  ./\  W ) ) )
1514oveq1d 6111 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( F `
 P )  .\/  ( F `  Q ) )  ./\  W )  =  ( ( ( F `  P ) 
.\/  ( ( P 
.\/  Q )  ./\  W ) )  ./\  W
) )
16 simp2 989 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
178, 10, 6, 7ltrnel 33788 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
183, 5, 16, 17syl3anc 1218 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
19 eqid 2443 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
208, 11, 19, 10, 6lhpmat 33679 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F `
 P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )  -> 
( ( F `  P )  ./\  W
)  =  ( 0.
`  K ) )
213, 18, 20syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( F `  P )  ./\  W
)  =  ( 0.
`  K ) )
2221oveq1d 6111 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( F `
 P )  ./\  W )  .\/  ( ( P  .\/  Q ) 
./\  W ) )  =  ( ( 0.
`  K )  .\/  ( ( P  .\/  Q )  ./\  W )
) )
23 simp2l 1014 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  P  e.  A )
248, 10, 6, 7ltrnat 33789 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
253, 5, 23, 24syl3anc 1218 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( F `  P
)  e.  A )
26 hllat 33013 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
271, 26syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  K  e.  Lat )
28 simp3l 1016 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  Q  e.  A )
29 eqid 2443 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
3029, 9, 10hlatjcl 33016 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
311, 23, 28, 30syl3anc 1218 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
3229, 6lhpbase 33647 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
332, 32syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  W  e.  ( Base `  K ) )
3429, 11latmcl 15227 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
3527, 31, 33, 34syl3anc 1218 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K
) )
3629, 8, 11latmle2 15252 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
3727, 31, 33, 36syl3anc 1218 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( P  .\/  Q )  ./\  W )  .<_  W )
3829, 8, 9, 11, 10atmod4i2 33516 . . . 4  |-  ( ( K  e.  HL  /\  ( ( F `  P )  e.  A  /\  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  /\  ( ( P  .\/  Q )  ./\  W )  .<_  W )  ->  ( ( ( F `
 P )  ./\  W )  .\/  ( ( P  .\/  Q ) 
./\  W ) )  =  ( ( ( F `  P ) 
.\/  ( ( P 
.\/  Q )  ./\  W ) )  ./\  W
) )
391, 25, 35, 33, 37, 38syl131anc 1231 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( F `
 P )  ./\  W )  .\/  ( ( P  .\/  Q ) 
./\  W ) )  =  ( ( ( F `  P ) 
.\/  ( ( P 
.\/  Q )  ./\  W ) )  ./\  W
) )
40 hlol 33011 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
411, 40syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  K  e.  OL )
4229, 9, 19olj02 32876 . . . 4  |-  ( ( K  e.  OL  /\  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K
) )  ->  (
( 0. `  K
)  .\/  ( ( P  .\/  Q )  ./\  W ) )  =  ( ( P  .\/  Q
)  ./\  W )
)
4341, 35, 42syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( 0. `  K )  .\/  (
( P  .\/  Q
)  ./\  W )
)  =  ( ( P  .\/  Q ) 
./\  W ) )
4422, 39, 433eqtr3d 2483 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( F `
 P )  .\/  ( ( P  .\/  Q )  ./\  W )
)  ./\  W )  =  ( ( P 
.\/  Q )  ./\  W ) )
4515, 44eqtrd 2475 1  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( F `
 P )  .\/  ( F `  Q ) )  ./\  W )  =  ( ( P 
.\/  Q )  ./\  W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Basecbs 14179   lecple 14250   joincjn 15119   meetcmee 15120   0.cp0 15212   Latclat 15220   OLcol 32824   Atomscatm 32913   HLchlt 33000   LHypclh 33633   LTrncltrn 33750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-riotaBAD 32609
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-undef 6797  df-map 7221  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-p1 15215  df-lat 15221  df-clat 15283  df-oposet 32826  df-ol 32828  df-oml 32829  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001  df-llines 33147  df-lplanes 33148  df-lvols 33149  df-lines 33150  df-psubsp 33152  df-pmap 33153  df-padd 33445  df-lhyp 33637  df-laut 33638  df-ldil 33753  df-ltrn 33754  df-trl 33808
This theorem is referenced by: (None)
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