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Theorem trlco 36554
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 2457 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 trlco.h . . . 4  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexnle 35831 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  (
Atoms `  K )  -.  p  .<_  W )
543ad2ant1 1017 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  E. p  e.  ( Atoms `  K )  -.  p  .<_  W )
6 simpl1 999 . . 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 1000 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )  ->  F  e.  T
)
8 simpl3 1001 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )  ->  G  e.  T
)
9 simpr 461 . . 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 2457 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
141, 10, 3, 11, 12, 13, 2trlcolem 36553 . . 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 1233 . 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 2953 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   E.wrex 2808   class class class wbr 4456    o. ccom 5012   ` cfv 5594  (class class class)co 6296   lecple 14718   joincjn 15699   meetcmee 15700   Atomscatm 35089   HLchlt 35176   LHypclh 35809   LTrncltrn 35926   trLctrl 35984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34785
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-map 7440  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-llines 35323  df-lplanes 35324  df-lvols 35325  df-lines 35326  df-psubsp 35328  df-pmap 35329  df-padd 35621  df-lhyp 35813  df-laut 35814  df-ldil 35929  df-ltrn 35930  df-trl 35985
This theorem is referenced by:  trlcone  36555  cdlemg46  36562  trljco  36567  tendopltp  36607  dialss  36874  diblss  36998
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