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Theorem ltrnatlw 30431
Description: If the value of an atom equals the atom in a non-identity translation, the atom is under the fiducial hyperplane. (Contributed by NM, 15-May-2013.)
Hypotheses
Ref Expression
ltrn2eq.l  |-  .<_  =  ( le `  K )
ltrn2eq.a  |-  A  =  ( Atoms `  K )
ltrn2eq.h  |-  H  =  ( LHyp `  K
)
ltrn2eq.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrnatlw  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( ( F `  P )  =/=  P  /\  ( F `  Q
)  =  Q ) )  ->  Q  .<_  W )

Proof of Theorem ltrnatlw
StepHypRef Expression
1 simp3r 985 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( ( F `  P )  =/=  P  /\  ( F `  Q
)  =  Q ) )  ->  ( F `  Q )  =  Q )
2 simpl1 959 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( F `  P )  =/=  P  /\  ( F `
 Q )  =  Q ) )  /\  -.  Q  .<_  W )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simpl21 1034 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( F `  P )  =/=  P  /\  ( F `
 Q )  =  Q ) )  /\  -.  Q  .<_  W )  ->  F  e.  T
)
4 simpl22 1035 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( F `  P )  =/=  P  /\  ( F `
 Q )  =  Q ) )  /\  -.  Q  .<_  W )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5 simpl23 1036 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( F `  P )  =/=  P  /\  ( F `
 Q )  =  Q ) )  /\  -.  Q  .<_  W )  ->  Q  e.  A
)
6 simpr 447 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( F `  P )  =/=  P  /\  ( F `
 Q )  =  Q ) )  /\  -.  Q  .<_  W )  ->  -.  Q  .<_  W )
75, 6jca 518 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( F `  P )  =/=  P  /\  ( F `
 Q )  =  Q ) )  /\  -.  Q  .<_  W )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
8 simpl3l 1011 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( F `  P )  =/=  P  /\  ( F `
 Q )  =  Q ) )  /\  -.  Q  .<_  W )  ->  ( F `  P )  =/=  P
)
9 ltrn2eq.l . . . . . 6  |-  .<_  =  ( le `  K )
10 ltrn2eq.a . . . . . 6  |-  A  =  ( Atoms `  K )
11 ltrn2eq.h . . . . . 6  |-  H  =  ( LHyp `  K
)
12 ltrn2eq.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
139, 10, 11, 12ltrnatneq 30430 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( F `  Q )  =/=  Q )
142, 3, 4, 7, 8, 13syl131anc 1196 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( F `  P )  =/=  P  /\  ( F `
 Q )  =  Q ) )  /\  -.  Q  .<_  W )  ->  ( F `  Q )  =/=  Q
)
1514ex 423 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( ( F `  P )  =/=  P  /\  ( F `  Q
)  =  Q ) )  ->  ( -.  Q  .<_  W  ->  ( F `  Q )  =/=  Q ) )
1615necon4bd 2591 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( ( F `  P )  =/=  P  /\  ( F `  Q
)  =  Q ) )  ->  ( ( F `  Q )  =  Q  ->  Q  .<_  W ) )
171, 16mpd 14 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( ( F `  P )  =/=  P  /\  ( F `  Q
)  =  Q ) )  ->  Q  .<_  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   ` cfv 5358   lecple 13423   Atomscatm 29512   HLchlt 29599   LHypclh 30232   LTrncltrn 30349
This theorem is referenced by:  cdlemg18  30930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-map 6917  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29425  df-ol 29427  df-oml 29428  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-lhyp 30236  df-laut 30237  df-ldil 30352  df-ltrn 30353  df-trl 30407
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