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Theorem cdlemd9 33850
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 991 . . 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 992 . . 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 993 . . 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 461 . . 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 33849 . . 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 1222 . 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 1063 . . . 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 992 . . . 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 1101 . . . . . 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 465 . . . . 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 33783 . . . . 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 1218 . . . 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 461 . . . . 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 2695 . . . 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 33685 . . . 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 1223 . . 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 1081 . . . . 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 1082 . . . . 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 989 . . . . . 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 1016 . . . . . 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 532 . . . . 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 1083 . . . . 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 1017 . . . . 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 33848 . . . . 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 1227 . . . 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 2843 . . 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 2689 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   lecple 14245   joincjn 15114   Atomscatm 32908   HLchlt 32995   LHypclh 33628   LTrncltrn 33745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-map 7216  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-p1 15210  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-llines 33142  df-psubsp 33147  df-pmap 33148  df-padd 33440  df-lhyp 33632  df-laut 33633  df-ldil 33748  df-ltrn 33749  df-trl 33803
This theorem is referenced by:  cdlemd  33851
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