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Theorem trlco 31209
Description: The trace of a composition of translations is less than or equal to the join of their traces. Part of proof of Lemma G of [Crawley] p. 116, second paragraph on p. 117. (Contributed by NM, 2-Jun-2013.)
Hypotheses
Ref Expression
trlco.l  |-  .<_  =  ( le `  K )
trlco.j  |-  .\/  =  ( join `  K )
trlco.h  |-  H  =  ( LHyp `  K
)
trlco.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlco.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlco  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( R `  ( F  o.  G
) )  .<_  ( ( R `  F ) 
.\/  ( R `  G ) ) )

Proof of Theorem trlco
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 trlco.l . . . 4  |-  .<_  =  ( le `  K )
2 eqid 2404 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 trlco.h . . . 4  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexnle 30488 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  (
Atoms `  K )  -.  p  .<_  W )
543ad2ant1 978 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  E. p  e.  ( Atoms `  K )  -.  p  .<_  W )
6 simpl1 960 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 simpl2 961 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )  ->  F  e.  T
)
8 simpl3 962 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )  ->  G  e.  T
)
9 simpr 448 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )  ->  ( p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )
10 trlco.j . . . 4  |-  .\/  =  ( join `  K )
11 trlco.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
12 trlco.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
13 eqid 2404 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
141, 10, 3, 11, 12, 13, 2trlcolem 31208 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )  ->  ( R `  ( F  o.  G
) )  .<_  ( ( R `  F ) 
.\/  ( R `  G ) ) )
156, 7, 8, 9, 14syl121anc 1189 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )  ->  ( R `  ( F  o.  G
) )  .<_  ( ( R `  F ) 
.\/  ( R `  G ) ) )
165, 15rexlimddv 2794 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( R `  ( F  o.  G
) )  .<_  ( ( R `  F ) 
.\/  ( R `  G ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2667   class class class wbr 4172    o. ccom 4841   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640
This theorem is referenced by:  trlcone  31210  cdlemg46  31217  trljco  31222  tendopltp  31262  dialss  31529  diblss  31653
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641
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