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Theorem cdlemg10bALTN 31433
Description: TODO: FIX COMMENT TODO: Can this be moved up as a stand-alone theorem in ltrn* area? TODO: Compare this proof to cdlemg2m 31401 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 987 . . . . 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 988 . . . . 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 519 . . . 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 956 . . . 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 989 . . . 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 2436 . . . . 5  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
136, 7, 8, 9, 10, 11, 12cdlemg2k 31398 . . . 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 1184 . . 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 6096 . 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 958 . . . . . 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 30936 . . . . . 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 1184 . . . . 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 2436 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
208, 11, 19, 10, 6lhpmat 30827 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F `
 P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )  -> 
( ( F `  P )  ./\  W
)  =  ( 0.
`  K ) )
213, 18, 20syl2anc 643 . . . 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 6096 . . 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 983 . . . . 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 30937 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
253, 5, 23, 24syl3anc 1184 . . . 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 30161 . . . . . 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 985 . . . . . 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 2436 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
3029, 9, 10hlatjcl 30164 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
311, 23, 28, 30syl3anc 1184 . . . . 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 30795 . . . . . 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 14480 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
3527, 31, 33, 34syl3anc 1184 . . . 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 14506 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
3727, 31, 33, 36syl3anc 1184 . . . 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 30664 . . . 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 1197 . . 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 30159 . . . . 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 30024 . . . 4  |-  ( ( K  e.  OL  /\  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K
) )  ->  (
( 0. `  K
)  .\/  ( ( P  .\/  Q )  ./\  W ) )  =  ( ( P  .\/  Q
)  ./\  W )
)
4341, 35, 42syl2anc 643 . . 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 2476 . 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 2468 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   0.cp0 14466   Latclat 14474   OLcol 29972   Atomscatm 30061   HLchlt 30148   LHypclh 30781   LTrncltrn 30898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956
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