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Theorem cdlemg10bALTN 31447
Description: TODO: FIX COMMENT TODO: Can this be moved up as a stand-alone theorem in ltrn* area? TODO: Compare this proof to cdlemg2m 31415 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 985 . . . . 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 986 . . . . 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 518 . . . 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 954 . . . 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 987 . . . 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 2296 . . . . 5  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
136, 7, 8, 9, 10, 11, 12cdlemg2k 31412 . . . 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 1182 . . 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 5889 . 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 956 . . . . . 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 30950 . . . . . 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 1182 . . . . 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 2296 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
208, 11, 19, 10, 6lhpmat 30841 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F `
 P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )  -> 
( ( F `  P )  ./\  W
)  =  ( 0.
`  K ) )
213, 18, 20syl2anc 642 . . . 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 5889 . . 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 981 . . . . 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 30951 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
253, 5, 23, 24syl3anc 1182 . . . 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 30175 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
271, 26syl 15 . . . . 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 983 . . . . . 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 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
3029, 9, 10hlatjcl 30178 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
311, 23, 28, 30syl3anc 1182 . . . . 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 30809 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
332, 32syl 15 . . . . 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 14173 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
3527, 31, 33, 34syl3anc 1182 . . . 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 14199 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
3727, 31, 33, 36syl3anc 1182 . . . 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 30678 . . . 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 1195 . . 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 30173 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
411, 40syl 15 . . . 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 30038 . . . 4  |-  ( ( K  e.  OL  /\  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K
) )  ->  (
( 0. `  K
)  .\/  ( ( P  .\/  Q )  ./\  W ) )  =  ( ( P  .\/  Q
)  ./\  W )
)
4341, 35, 42syl2anc 642 . . 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 2336 . 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 2328 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   0.cp0 14159   Latclat 14167   OLcol 29986   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970
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