Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemd7 Unicode version

Theorem cdlemd7 30211
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)
Hypotheses
Ref Expression
cdlemd4.l  |-  .<_  =  ( le `  K )
cdlemd4.j  |-  .\/  =  ( join `  K )
cdlemd4.a  |-  A  =  ( Atoms `  K )
cdlemd4.h  |-  H  =  ( LHyp `  K
)
cdlemd4.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemd7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  ( F `  R )  =  ( G `  R ) )

Proof of Theorem cdlemd7
StepHypRef Expression
1 simp1 955 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A ) )
2 simp2l 981 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp2r 982 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
4 simp11l 1066 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  K  e.  HL )
5 hllat 29371 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
64, 5syl 15 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  K  e.  Lat )
7 simp2rl 1024 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  Q  e.  A )
8 eqid 2316 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
9 cdlemd4.a . . . . 5  |-  A  =  ( Atoms `  K )
108, 9atbase 29297 . . . 4  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
117, 10syl 15 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  Q  e.  ( Base `  K
) )
12 simp2ll 1022 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  P  e.  A )
138, 9atbase 29297 . . . 4  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1412, 13syl 15 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  P  e.  ( Base `  K
) )
15 simp11 985 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
16 simp12l 1068 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  F  e.  T )
17 cdlemd4.h . . . . 5  |-  H  =  ( LHyp `  K
)
18 cdlemd4.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
198, 17, 18ltrncl 30132 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( F `  P )  e.  (
Base `  K )
)
2015, 16, 14, 19syl3anc 1182 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  ( F `  P )  e.  ( Base `  K
) )
21 simp3r 984 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )
22 cdlemd4.l . . . . 5  |-  .<_  =  ( le `  K )
23 cdlemd4.j . . . . 5  |-  .\/  =  ( join `  K )
248, 22, 23latnlej1l 14224 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  ( F `  P )  e.  ( Base `  K
) )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  ->  Q  =/=  P )
2524necomd 2562 . . 3  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  ( F `  P )  e.  ( Base `  K
) )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  ->  P  =/=  Q )
266, 11, 14, 20, 21, 25syl131anc 1195 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  P  =/=  Q )
27 simp3l 983 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  ( F `  P )  =  ( G `  P ) )
28 simp12 986 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  ( F  e.  T  /\  G  e.  T )
)
2922, 23, 9, 17, 18cdlemd6 30210 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( F `  Q
)  =  ( G `
 Q ) )
3015, 28, 2, 3, 21, 27, 29syl231anc 1202 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  ( F `  Q )  =  ( G `  Q ) )
3122, 23, 9, 17, 18cdlemd5 30209 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( G `  R ) )
321, 2, 3, 26, 27, 30, 31syl132anc 1200 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  Q  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  ( F `  R )  =  ( G `  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   joincjn 14127   Latclat 14200   Atomscatm 29271   HLchlt 29358   LHypclh 29991   LTrncltrn 30108
This theorem is referenced by:  cdlemd9  30213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-map 6817  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-llines 29505  df-psubsp 29510  df-pmap 29511  df-padd 29803  df-lhyp 29995  df-laut 29996  df-ldil 30111  df-ltrn 30112  df-trl 30166
  Copyright terms: Public domain W3C validator