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Theorem ltrnatlw 30669
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 986 . 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 960 . . . . 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 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 )  ->  F  e.  T
)
4 simpl22 1036 . . . . 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 1037 . . . . . 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 448 . . . . . 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 519 . . . . 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 1012 . . . . 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 30668 . . . . 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 1197 . . . 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 424 . . 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 2633 . 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 15 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   class class class wbr 4176   ` cfv 5417   lecple 13495   Atomscatm 29750   HLchlt 29837   LHypclh 30470   LTrncltrn 30587
This theorem is referenced by:  cdlemg18  31168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-map 6983  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591  df-trl 30645
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