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Theorem trlnle 31045
Description: The atom not under the fiducial co-atom  W is not less than the trace of a lattice translation. Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 26-May-2012.)
Hypotheses
Ref Expression
trlne.l  |-  .<_  =  ( le `  K )
trlne.a  |-  A  =  ( Atoms `  K )
trlne.h  |-  H  =  ( LHyp `  K
)
trlne.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlne.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlnle  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  ( R `  F
) )

Proof of Theorem trlnle
StepHypRef Expression
1 simpl1l 1009 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  K  e.  HL )
2 hlatl 30220 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
31, 2syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  K  e.  AtLat )
4 simpl3l 1013 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  P  e.  A )
5 trlne.l . . . . 5  |-  .<_  =  ( le `  K )
6 eqid 2438 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 trlne.a . . . . 5  |-  A  =  ( Atoms `  K )
85, 6, 7atnle0 30169 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  -.  P  .<_  ( 0. `  K ) )
93, 4, 8syl2anc 644 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  -.  P  .<_  ( 0. `  K ) )
10 simpl1 961 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( K  e.  HL  /\  W  e.  H ) )
11 simpl3 963 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
12 simpl2 962 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  F  e.  T )
13 simpr 449 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( F `  P )  =  P )
14 trlne.h . . . . . 6  |-  H  =  ( LHyp `  K
)
15 trlne.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
16 trlne.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
175, 6, 7, 14, 15, 16trl0 31029 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( R `  F )  =  ( 0. `  K ) )
1810, 11, 12, 13, 17syl112anc 1189 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( R `  F )  =  ( 0. `  K ) )
1918breq2d 4226 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( P  .<_  ( R `  F )  <->  P  .<_  ( 0. `  K ) ) )
209, 19mtbird 294 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  -.  P  .<_  ( R `  F ) )
215, 7, 14, 15, 16trlne 31044 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  =/=  ( R `  F ) )
2221adantr 453 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  P  =/=  ( R `  F
) )
23 simpl1l 1009 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  K  e.  HL )
2423, 2syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  K  e.  AtLat )
25 simpl3l 1013 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  P  e.  A )
26 simpl1 961 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( K  e.  HL  /\  W  e.  H ) )
27 simpl3 963 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
28 simpl2 962 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  F  e.  T )
29 simpr 449 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( F `  P )  =/=  P )
305, 7, 14, 15, 16trlat 31028 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
3126, 27, 28, 29, 30syl112anc 1189 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( R `  F )  e.  A )
325, 7atncmp 30172 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  ( R `  F )  e.  A )  ->  ( -.  P  .<_  ( R `
 F )  <->  P  =/=  ( R `  F ) ) )
3324, 25, 31, 32syl3anc 1185 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( -.  P  .<_  ( R `
 F )  <->  P  =/=  ( R `  F ) ) )
3422, 33mpbird 225 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  -.  P  .<_  ( R `  F ) )
3520, 34pm2.61dane 2684 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  ( R `  F
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456   lecple 13538   0.cp0 14468   Atomscatm 30123   AtLatcal 30124   HLchlt 30210   LHypclh 30843   LTrncltrn 30960   trLctrl 31017
This theorem is referenced by:  cdlemc3  31052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-lhyp 30847  df-laut 30848  df-ldil 30963  df-ltrn 30964  df-trl 31018
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