Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemd9 Structured version   Unicode version

Theorem cdlemd9 30930
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 2-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
cdlemd9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( F `  R
)  =  ( G `
 R ) )

Proof of Theorem cdlemd9
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 simpl1 960 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `  P )  =  P )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
) )
2 simpl2 961 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `  P )  =  P )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
3 simpl3 962 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `  P )  =  P )  -> 
( F `  P
)  =  ( G `
 P ) )
4 simpr 448 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `  P )  =  P )  -> 
( F `  P
)  =  P )
5 cdlemd4.l . . . 4  |-  .<_  =  ( le `  K )
6 cdlemd4.j . . . 4  |-  .\/  =  ( join `  K )
7 cdlemd4.a . . . 4  |-  A  =  ( Atoms `  K )
8 cdlemd4.h . . . 4  |-  H  =  ( LHyp `  K
)
9 cdlemd4.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
105, 6, 7, 8, 9cdlemd8 30929 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  ( F `  R )  =  ( G `  R ) )
111, 2, 3, 4, 10syl112anc 1188 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `  P )  =  P )  -> 
( F `  R
)  =  ( G `
 R ) )
12 simpl11 1032 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `  P )  =/=  P )  -> 
( K  e.  HL  /\  W  e.  H ) )
13 simpl2 961 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `  P )  =/=  P )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
14 simp12l 1070 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  ->  F  e.  T )
1514adantr 452 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `  P )  =/=  P )  ->  F  e.  T )
165, 7, 8, 9ltrnel 30863 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
1712, 15, 13, 16syl3anc 1184 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `  P )  =/=  P )  -> 
( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
18 simpr 448 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `  P )  =/=  P )  -> 
( F `  P
)  =/=  P )
1918necomd 2681 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `  P )  =/=  P )  ->  P  =/=  ( F `  P ) )
205, 6, 7, 8cdlemb2 30765 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )  /\  P  =/=  ( F `  P
) )  ->  E. s  e.  A  ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  ( F `  P ) ) ) )
2112, 13, 17, 19, 20syl121anc 1189 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `  P )  =/=  P )  ->  E. s  e.  A  ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  ( F `
 P ) ) ) )
22 simp1l1 1050 . . . . 5  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `
 P )  =/= 
P )  /\  s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  ( F `  P ) ) ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
) )
23 simp1l2 1051 . . . . 5  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `
 P )  =/= 
P )  /\  s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  ( F `  P ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
24 simp2 958 . . . . . 6  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `
 P )  =/= 
P )  /\  s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  ( F `  P ) ) ) )  ->  s  e.  A )
25 simp3l 985 . . . . . 6  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `
 P )  =/= 
P )  /\  s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  ( F `  P ) ) ) )  ->  -.  s  .<_  W )
2624, 25jca 519 . . . . 5  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `
 P )  =/= 
P )  /\  s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  ( F `  P ) ) ) )  ->  ( s  e.  A  /\  -.  s  .<_  W ) )
27 simp1l3 1052 . . . . 5  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `
 P )  =/= 
P )  /\  s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  ( F `  P ) ) ) )  ->  ( F `  P )  =  ( G `  P ) )
28 simp3r 986 . . . . 5  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `
 P )  =/= 
P )  /\  s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  ( F `  P ) ) ) )  ->  -.  s  .<_  ( P  .\/  ( F `  P )
) )
295, 6, 7, 8, 9cdlemd7 30928 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( s  e.  A  /\  -.  s  .<_  W ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  -.  s  .<_  ( P  .\/  ( F `
 P ) ) ) )  ->  ( F `  R )  =  ( G `  R ) )
3022, 23, 26, 27, 28, 29syl122anc 1193 . . . 4  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T
)  /\  R  e.  A )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `
 P )  =/= 
P )  /\  s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  ( F `  P ) ) ) )  ->  ( F `  R )  =  ( G `  R ) )
3130rexlimdv3a 2824 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `  P )  =/=  P )  -> 
( E. s  e.  A  ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  ( F `  P )
) )  ->  ( F `  R )  =  ( G `  R ) ) )
3221, 31mpd 15 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  /\  ( F `  P )  =/=  P )  -> 
( F `  R
)  =  ( G `
 R ) )
3311, 32pm2.61dane 2676 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( F `  R
)  =  ( G `
 R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   Atomscatm 29988   HLchlt 30075   LHypclh 30708   LTrncltrn 30825
This theorem is referenced by:  cdlemd  30931
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-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-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-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