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Theorem cdlemk5a 31471
Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-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
cdlemk5a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
( F `  P
)  .\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )  .<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' F
) ) ) )

Proof of Theorem cdlemk5a
StepHypRef Expression
1 simp1l 981 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  K  e.  HL )
2 simp1r 982 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  W  e.  H )
3 simp21 990 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  e.  T )
4 simp22 991 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  G  e.  T )
5 simp3 959 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( R `  G )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )
6 cdlemk.b . . . 4  |-  B  =  ( Base `  K
)
7 cdlemk.l . . . 4  |-  .<_  =  ( le `  K )
8 cdlemk.j . . . 4  |-  .\/  =  ( join `  K )
9 cdlemk.a . . . 4  |-  A  =  ( Atoms `  K )
10 cdlemk.h . . . 4  |-  H  =  ( LHyp `  K
)
11 cdlemk.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
12 cdlemk.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
13 cdlemk.m . . . 4  |-  ./\  =  ( meet `  K )
146, 7, 8, 9, 10, 11, 12, 13cdlemk3 31469 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
( F `  P
)  .\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )  =  ( F `
 P ) )
151, 2, 3, 4, 5, 14syl221anc 1195 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
( F `  P
)  .\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )  =  ( F `
 P ) )
16 simp23 992 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  X  e.  T )
17 simp33l 1084 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  P  e.  A )
18 simp33r 1085 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  -.  P  .<_  W )
196, 7, 8, 9, 10, 11, 12, 13cdlemk4 31470 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  .<_  ( ( X `  P ) 
.\/  ( R `  ( X  o.  `' F ) ) ) )
201, 2, 3, 16, 17, 18, 19syl222anc 1200 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( F `  P )  .<_  ( ( X `  P ) 
.\/  ( R `  ( X  o.  `' F ) ) ) )
2115, 20eqbrtrd 4224 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 F )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
( F `  P
)  .\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  ( R `  ( G  o.  `' F ) ) ) )  .<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' F
) ) ) )
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 4868    |` cres 4871    o. ccom 4873   ` cfv 5445  (class class class)co 6072   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   Atomscatm 29900   HLchlt 29987   LHypclh 30620   LTrncltrn 30737   trLctrl 30794
This theorem is referenced by:  cdlemk5  31472  cdlemk5auN  31496  cdlemk5u  31497
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 4692
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 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-map 7011  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741  df-trl 30795
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