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Theorem cdlemd9 33471
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 1008 . . 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 1009 . . 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 1010 . . 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 462 . . 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 33470 . . 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 1268 . 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 1080 . . . 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 1009 . . . 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 1118 . . . . . 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 466 . . . . 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 33403 . . . . 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 1264 . . . 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 462 . . . . 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 2702 . . . 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 33305 . . . 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 1269 . . 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 1098 . . . . 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 1099 . . . . 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 1006 . . . . . 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 1033 . . . . . 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 534 . . . . 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 1100 . . . . 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 1034 . . . . 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 33469 . . . . 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 1273 . . . 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 2926 . . 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 2749 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   lecple 15150   joincjn 16131   Atomscatm 32528   HLchlt 32615   LHypclh 33248   LTrncltrn 33365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7482  df-preset 16115  df-poset 16133  df-plt 16146  df-lub 16162  df-glb 16163  df-join 16164  df-meet 16165  df-p0 16227  df-p1 16228  df-lat 16234  df-clat 16296  df-oposet 32441  df-ol 32443  df-oml 32444  df-covers 32531  df-ats 32532  df-atl 32563  df-cvlat 32587  df-hlat 32616  df-llines 32762  df-psubsp 32767  df-pmap 32768  df-padd 33060  df-lhyp 33252  df-laut 33253  df-ldil 33368  df-ltrn 33369  df-trl 33424
This theorem is referenced by:  cdlemd  33472
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