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Theorem cdlemk6 31561
Description: Part of proof of Lemma K of [Crawley] p. 118. Apply dalaw 30610. (Contributed by NM, 25-Jun-2013.)
Hypotheses
Ref Expression
cdlemk.b  |-  B  =  ( Base `  K
)
cdlemk.l  |-  .<_  =  ( le `  K )
cdlemk.j  |-  .\/  =  ( join `  K )
cdlemk.a  |-  A  =  ( Atoms `  K )
cdlemk.h  |-  H  =  ( LHyp `  K
)
cdlemk.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
cdlemk6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  -> 
( ( P  .\/  ( G `  P ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) 
.<_  ( ( ( ( G `  P ) 
.\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )  .\/  ( ( ( X `
 P )  .\/  P )  ./\  ( ( R `  ( X  o.  `' F ) )  .\/  ( N `  P ) ) ) ) )

Proof of Theorem cdlemk6
StepHypRef Expression
1 simp31 993 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  ->  F  =/=  (  _I  |`  B ) )
2 simp32 994 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  ->  G  =/=  (  _I  |`  B ) )
3 simp33l 1084 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  -> 
( R `  G
)  =/=  ( R `
 F ) )
41, 2, 33jca 1134 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  -> 
( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  F )
) )
5 cdlemk.b . . . 4  |-  B  =  ( Base `  K
)
6 cdlemk.l . . . 4  |-  .<_  =  ( le `  K )
7 cdlemk.j . . . 4  |-  .\/  =  ( join `  K )
8 cdlemk.a . . . 4  |-  A  =  ( Atoms `  K )
9 cdlemk.h . . . 4  |-  H  =  ( LHyp `  K
)
10 cdlemk.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
11 cdlemk.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
12 cdlemk.m . . . 4  |-  ./\  =  ( meet `  K )
135, 6, 7, 8, 9, 10, 11, 12cdlemk5 31560 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  F )
) )  ->  (
( P  .\/  ( N `  P )
)  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) 
.<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' F ) ) ) )
144, 13syld3an3 1229 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  -> 
( ( P  .\/  ( N `  P ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) 
.<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' F ) ) ) )
15 simp11l 1068 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  ->  K  e.  HL )
16 simp22l 1076 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  ->  P  e.  A )
17 simp11 987 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
18 simp13 989 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  ->  G  e.  T )
196, 8, 9, 10ltrnat 30864 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  P  e.  A
)  ->  ( G `  P )  e.  A
)
2017, 18, 16, 19syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  -> 
( G `  P
)  e.  A )
21 simp21r 1075 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  ->  X  e.  T )
226, 8, 9, 10ltrnat 30864 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  T  /\  P  e.  A
)  ->  ( X `  P )  e.  A
)
2317, 21, 16, 22syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  -> 
( X `  P
)  e.  A )
24 simp21l 1074 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  ->  N  e.  T )
256, 8, 9, 10ltrnat 30864 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  N  e.  T  /\  P  e.  A
)  ->  ( N `  P )  e.  A
)
2617, 24, 16, 25syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  -> 
( N `  P
)  e.  A )
27 simp12 988 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  ->  F  e.  T )
288, 9, 10, 11trlcocnvat 31448 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  F  e.  T )  /\  ( R `  G )  =/=  ( R `  F
) )  ->  ( R `  ( G  o.  `' F ) )  e.  A )
2917, 18, 27, 3, 28syl121anc 1189 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  -> 
( R `  ( G  o.  `' F
) )  e.  A
)
30 simp33r 1085 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  -> 
( R `  X
)  =/=  ( R `
 F ) )
318, 9, 10, 11trlcocnvat 31448 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  T  /\  F  e.  T )  /\  ( R `  X )  =/=  ( R `  F
) )  ->  ( R `  ( X  o.  `' F ) )  e.  A )
3217, 21, 27, 30, 31syl121anc 1189 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  -> 
( R `  ( X  o.  `' F
) )  e.  A
)
336, 7, 12, 8dalaw 30610 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( G `  P
)  e.  A  /\  ( X `  P )  e.  A )  /\  ( ( N `  P )  e.  A  /\  ( R `  ( G  o.  `' F
) )  e.  A  /\  ( R `  ( X  o.  `' F
) )  e.  A
) )  ->  (
( ( P  .\/  ( N `  P ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) 
.<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' F ) ) )  ->  ( ( P 
.\/  ( G `  P ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )  .<_  ( (
( ( G `  P )  .\/  ( X `  P )
)  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )  .\/  ( ( ( X `
 P )  .\/  P )  ./\  ( ( R `  ( X  o.  `' F ) )  .\/  ( N `  P ) ) ) ) ) )
3415, 16, 20, 23, 26, 29, 32, 33syl133anc 1207 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  -> 
( ( ( P 
.\/  ( N `  P ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )  .<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' F
) ) )  -> 
( ( P  .\/  ( G `  P ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) 
.<_  ( ( ( ( G `  P ) 
.\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )  .\/  ( ( ( X `
 P )  .\/  P )  ./\  ( ( R `  ( X  o.  `' F ) )  .\/  ( N `  P ) ) ) ) ) )
3514, 34mpd 15 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) ) )  -> 
( ( P  .\/  ( G `  P ) )  ./\  ( ( N `  P )  .\/  ( R `  ( G  o.  `' F
) ) ) ) 
.<_  ( ( ( ( G `  P ) 
.\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )  .\/  ( ( ( X `
 P )  .\/  P )  ./\  ( ( R `  ( X  o.  `' F ) )  .\/  ( N `  P ) ) ) ) )
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    _I cid 4485   `'ccnv 4869    |` cres 4872    o. ccom 4874   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Atomscatm 29988   HLchlt 30075   LHypclh 30708   LTrncltrn 30825   trLctrl 30882
This theorem is referenced by:  cdlemk7  31572
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
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