Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg9a Structured version   Unicode version

Theorem cdlemg9a 31356
Description: 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 )
cdlemg9.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdlemg9a  |-  ( ( ( 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  .\/  U )  ./\  ( ( F `  ( G `  P ) )  .\/  U ) )  .<_  ( ( G `  P )  .\/  U ) )

Proof of Theorem cdlemg9a
StepHypRef Expression
1 simp1l 981 . . 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 ) ) )  ->  K  e.  HL )
2 simp21l 1074 . . 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  e.  A )
3 simp1 957 . . . 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 ) )
4 simp23 992 . . . 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 )
5 simp31 993 . . . 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 )
6 cdlemg8.l . . . . 5  |-  .<_  =  ( le `  K )
7 cdlemg8.a . . . . 5  |-  A  =  ( Atoms `  K )
8 cdlemg8.h . . . . 5  |-  H  =  ( LHyp `  K
)
9 cdlemg8.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
106, 7, 8, 9ltrncoat 30868 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  ( F `  ( G `  P ) )  e.  A )
113, 4, 5, 2, 10syl121anc 1189 . . 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
) )  e.  A
)
12 simp1r 982 . . . 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 )
13 simp21 990 . . . 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 ) )
14 simp22l 1076 . . . 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 )
15 simp32 994 . . . 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  =/=  Q )
16 cdlemg8.j . . . . 5  |-  .\/  =  ( join `  K )
17 cdlemg8.m . . . . 5  |-  ./\  =  ( meet `  K )
18 cdlemg9.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
196, 16, 17, 7, 8, 18cdleme0a 30935 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
201, 12, 13, 14, 15, 19syl212anc 1194 . . 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 ) ) )  ->  U  e.  A )
21 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  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) )
22 simp22 991 . . . . . 6  |-  ( ( ( 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 ) )
238, 9, 6, 16, 7, 17, 18cdlemg2l 31327 . . . . . 6  |-  ( ( ( 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 ) )  .\/  U ) )
243, 13, 22, 4, 5, 23syl122anc 1193 . . . . 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  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =  ( ( F `  ( G `
 P ) ) 
.\/  U ) )
256, 16, 17, 7, 8, 18cdlemg3a 31321 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  ->  ( P  .\/  Q )  =  ( P  .\/  U
) )
261, 12, 13, 14, 25syl211anc 1190 . . . . 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  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( P  .\/  Q )  =  ( P  .\/  U ) )
2721, 24, 263netr3d 2624 . . . 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 `  ( G `  P ) )  .\/  U )  =/=  ( P 
.\/  U ) )
2827necomd 2681 . . 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  .\/  U )  =/=  (
( F `  ( G `  P )
)  .\/  U )
)
296, 16, 17, 72llnma3r 30512 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( F `  ( G `  P )
)  e.  A  /\  U  e.  A )  /\  ( P  .\/  U
)  =/=  ( ( F `  ( G `
 P ) ) 
.\/  U ) )  ->  ( ( P 
.\/  U )  ./\  ( ( F `  ( G `  P ) )  .\/  U ) )  =  U )
301, 2, 11, 20, 28, 29syl131anc 1197 . 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  .\/  U )  ./\  ( ( F `  ( G `  P ) )  .\/  U ) )  =  U )
316, 7, 8, 9ltrnat 30864 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  P  e.  A
)  ->  ( G `  P )  e.  A
)
323, 5, 2, 31syl3anc 1184 . . 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 ) ) )  ->  ( G `  P )  e.  A
)
336, 16, 7hlatlej2 30100 . . 3  |-  ( ( K  e.  HL  /\  ( G `  P )  e.  A  /\  U  e.  A )  ->  U  .<_  ( ( G `  P )  .\/  U
) )
341, 32, 20, 33syl3anc 1184 . 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 ) ) )  ->  U  .<_  ( ( G `  P
)  .\/  U )
)
3530, 34eqbrtrd 4224 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  .\/  U )  ./\  ( ( F `  ( G `  P ) )  .\/  U ) )  .<_  ( ( G `  P )  .\/  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   meetcmee 14394   Atomscatm 29988   HLchlt 30075   LHypclh 30708   LTrncltrn 30825
This theorem is referenced by:  cdlemg9b  31357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-llines 30222  df-lplanes 30223  df-lvols 30224  df-lines 30225  df-psubsp 30227  df-pmap 30228  df-padd 30520  df-lhyp 30712  df-laut 30713  df-ldil 30828  df-ltrn 30829  df-trl 30883
  Copyright terms: Public domain W3C validator