Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg12b Structured version   Unicode version

Theorem cdlemg12b 34607
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 988 . . 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 989 . . 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 1024 . . 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 1025 . . 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 1021 . . . . 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 1107 . . . . 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 1026 . . . . 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 34605 . . . . 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 1236 . . . 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 1012 . . . . 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 1013 . . . . 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 2452 . . . . . 6  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
208, 9, 10, 11, 12, 19cdlemg3a 34560 . . . . 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 1225 . . . 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 1022 . . . . 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 34564 . . . . 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 1224 . . . 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 2752 . . 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 34606 . . 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 1231 . 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 6213 . 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 1023 . . 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 34566 . . 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 1228 . 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 4425 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   lecple 14359   joincjn 15228   meetcmee 15229   Atomscatm 33227   HLchlt 33314   LHypclh 33947   LTrncltrn 34064   trLctrl 34121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-riotaBAD 32923
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-undef 6897  df-map 7321  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-llines 33461  df-lplanes 33462  df-lvols 33463  df-lines 33464  df-psubsp 33466  df-pmap 33467  df-padd 33759  df-lhyp 33951  df-laut 33952  df-ldil 34067  df-ltrn 34068  df-trl 34122
This theorem is referenced by:  cdlemg12c  34608
  Copyright terms: Public domain W3C validator