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Theorem cdlemg12b 30760
Description: The triples  <. P , 
( F `  P
) ,  ( F `
 ( G `  P ) ) >. and  <. Q , 
( F `  Q
) ,  ( F `
 ( G `  Q ) ) >. are centrally perspective. TODO: FIX COMMENT (Contributed by NM, 5-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg12b  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  Q )  ./\  ( ( G `  P )  .\/  ( G `  Q
) ) )  .<_  ( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) ) )

Proof of Theorem cdlemg12b
StepHypRef Expression
1 simp1 957 . . 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  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp2 958 . . 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  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
) )
3 simp31 993 . . 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  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  G  e.  T )
4 simp32 994 . . 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  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  P  =/=  Q )
5 simp21 990 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
6 simp22l 1076 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  Q  e.  A )
7 simp33 995 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  -.  ( R `  G
)  .<_  ( P  .\/  Q ) )
8 cdlemg12.l . . . . . 6  |-  .<_  =  ( le `  K )
9 cdlemg12.j . . . . . 6  |-  .\/  =  ( join `  K )
10 cdlemg12.m . . . . . 6  |-  ./\  =  ( meet `  K )
11 cdlemg12.a . . . . . 6  |-  A  =  ( Atoms `  K )
12 cdlemg12.h . . . . . 6  |-  H  =  ( LHyp `  K
)
13 cdlemg12.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
14 cdlemg12b.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
158, 9, 10, 11, 12, 13, 14cdlemg11b 30758 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( G  e.  T  /\  P  =/=  Q  /\  -.  ( R `  G )  .<_  ( P 
.\/  Q ) ) )  ->  ( P  .\/  Q )  =/=  (
( G `  P
)  .\/  ( G `  Q ) ) )
161, 5, 6, 3, 4, 7, 15syl123anc 1201 . . . 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  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( P  .\/  Q
)  =/=  ( ( G `  P ) 
.\/  ( G `  Q ) ) )
17 simp1l 981 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  K  e.  HL )
18 simp1r 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  W  e.  H )
19 eqid 2389 . . . . . 6  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
208, 9, 10, 11, 12, 19cdlemg3a 30713 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  ( P  .\/  Q )  =  ( P  .\/  (
( P  .\/  Q
)  ./\  W )
) )
2117, 18, 5, 6, 20syl211anc 1190 . . . 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  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( P  .\/  Q
)  =  ( P 
.\/  ( ( P 
.\/  Q )  ./\  W ) ) )
22 simp22 991 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
2312, 13, 8, 9, 11, 10, 19cdlemg2k 30717 . . . . 5  |-  ( ( ( 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 )
) )
241, 5, 22, 3, 23syl121anc 1189 . . . 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  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( G `  P )  .\/  ( G `  Q )
)  =  ( ( G `  P ) 
.\/  ( ( P 
.\/  Q )  ./\  W ) ) )
2516, 21, 243netr3d 2578 . . 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  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( P  .\/  (
( P  .\/  Q
)  ./\  W )
)  =/=  ( ( G `  P ) 
.\/  ( ( P 
.\/  Q )  ./\  W ) ) )
268, 9, 10, 11, 12, 13, 19cdlemg12a 30759 . . 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  /\  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
)  =/=  ( ( G `  P ) 
.\/  ( ( P 
.\/  Q )  ./\  W ) ) ) )  ->  ( ( P 
.\/  ( ( P 
.\/  Q )  ./\  W ) )  ./\  (
( G `  P
)  .\/  ( ( P  .\/  Q )  ./\  W ) ) )  .<_  ( ( F `  ( G `  P ) )  .\/  ( ( P  .\/  Q ) 
./\  W ) ) )
271, 2, 3, 4, 25, 26syl113anc 1196 . 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  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  ( ( P  .\/  Q )  ./\  W )
)  ./\  ( ( G `  P )  .\/  ( ( P  .\/  Q )  ./\  W )
) )  .<_  ( ( F `  ( G `
 P ) ) 
.\/  ( ( P 
.\/  Q )  ./\  W ) ) )
2821, 24oveq12d 6040 . 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  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  Q )  ./\  ( ( G `  P )  .\/  ( G `  Q
) ) )  =  ( ( P  .\/  ( ( P  .\/  Q )  ./\  W )
)  ./\  ( ( G `  P )  .\/  ( ( P  .\/  Q )  ./\  W )
) ) )
29 simp23 992 . . 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  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  F  e.  T )
3012, 13, 8, 9, 11, 10, 19cdlemg2l 30719 . . 3  |-  ( ( ( 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 )
) )
311, 5, 22, 29, 3, 30syl122anc 1193 . 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  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) )  =  ( ( F `
 ( G `  P ) )  .\/  ( ( P  .\/  Q )  ./\  W )
) )
3227, 28, 313brtr4d 4185 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  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  Q )  ./\  ( ( G `  P )  .\/  ( G `  Q
) ) )  .<_  ( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   lecple 13465   joincjn 14330   meetcmee 14331   Atomscatm 29380   HLchlt 29467   LHypclh 30100   LTrncltrn 30217   trLctrl 30274
This theorem is referenced by:  cdlemg12c  30761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-map 6958  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-llines 29614  df-lplanes 29615  df-lvols 29616  df-lines 29617  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104  df-laut 30105  df-ldil 30220  df-ltrn 30221  df-trl 30275
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