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Theorem cdlemg9b 33999
Description: The triples  <. P , 
( F `  ( G `  P )
) ,  ( F `
 P ) >. and  <. Q , 
( F `  ( G `  Q )
) ,  ( F `
 Q ) >. are centrally perspective. TODO: FIX COMMENT (Contributed by NM, 1-May-2013.)
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
cdlemg9b  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  Q )  ./\  ( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) ) )  .<_  ( ( G `  P )  .\/  ( G `  Q
) ) )

Proof of Theorem cdlemg9b
StepHypRef Expression
1 cdlemg8.l . . 3  |-  .<_  =  ( le `  K )
2 cdlemg8.j . . 3  |-  .\/  =  ( join `  K )
3 cdlemg8.m . . 3  |-  ./\  =  ( meet `  K )
4 cdlemg8.a . . 3  |-  A  =  ( Atoms `  K )
5 cdlemg8.h . . 3  |-  H  =  ( LHyp `  K
)
6 cdlemg8.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
7 eqid 2441 . . 3  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
81, 2, 3, 4, 5, 6, 7cdlemg9a 33998 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  ( ( P 
.\/  Q )  ./\  W ) )  ./\  (
( F `  ( G `  P )
)  .\/  ( ( P  .\/  Q )  ./\  W ) ) )  .<_  ( ( G `  P )  .\/  (
( P  .\/  Q
)  ./\  W )
) )
9 simp1l 1007 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  K  e.  HL )
10 simp1r 1008 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  W  e.  H )
11 simp21 1016 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
12 simp22l 1102 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  Q  e.  A )
131, 2, 3, 4, 5, 7cdlemg3a 33963 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  ( P  .\/  Q )  =  ( P  .\/  (
( P  .\/  Q
)  ./\  W )
) )
149, 10, 11, 12, 13syl211anc 1219 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( P  .\/  Q )  =  ( P  .\/  ( ( P  .\/  Q ) 
./\  W ) ) )
15 simp1 983 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
16 simp22 1017 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
17 simp23 1018 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  F  e.  T )
18 simp31 1019 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  G  e.  T )
195, 6, 1, 2, 4, 3, 7cdlemg2l 33969 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) )  =  ( ( F `
 ( G `  P ) )  .\/  ( ( P  .\/  Q )  ./\  W )
) )
2015, 11, 16, 17, 18, 19syl122anc 1222 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =  ( ( F `  ( G `
 P ) ) 
.\/  ( ( P 
.\/  Q )  ./\  W ) ) )
2114, 20oveq12d 6108 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  Q )  ./\  ( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) ) )  =  ( ( P  .\/  ( ( P  .\/  Q ) 
./\  W ) ) 
./\  ( ( F `
 ( G `  P ) )  .\/  ( ( P  .\/  Q )  ./\  W )
) ) )
225, 6, 1, 2, 4, 3, 7cdlemg2k 33967 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  G  e.  T )  ->  (
( G `  P
)  .\/  ( G `  Q ) )  =  ( ( G `  P )  .\/  (
( P  .\/  Q
)  ./\  W )
) )
2315, 11, 16, 18, 22syl121anc 1218 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( G `  P )  .\/  ( G `  Q
) )  =  ( ( G `  P
)  .\/  ( ( P  .\/  Q )  ./\  W ) ) )
248, 21, 233brtr4d 4319 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  Q )  ./\  ( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) ) )  .<_  ( ( G `  P )  .\/  ( G `  Q
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   lecple 14241   joincjn 15110   meetcmee 15111   Atomscatm 32630   HLchlt 32717   LHypclh 33350   LTrncltrn 33467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-riotaBAD 32326
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-undef 6788  df-map 7212  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-p1 15206  df-lat 15212  df-clat 15274  df-oposet 32543  df-ol 32545  df-oml 32546  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718  df-llines 32864  df-lplanes 32865  df-lvols 32866  df-lines 32867  df-psubsp 32869  df-pmap 32870  df-padd 33162  df-lhyp 33354  df-laut 33355  df-ldil 33470  df-ltrn 33471  df-trl 33525
This theorem is referenced by:  cdlemg9  34000
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