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Theorem trlco 30046
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
StepHypRef Expression
1 trlco.l . . . 4  |-  .<_  =  ( le `  K )
2 eqid 2256 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 trlco.h . . . 4  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexnle 29325 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  (
Atoms `  K )  -.  p  .<_  W )
543ad2ant1 981 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  E. p  e.  ( Atoms `  K )  -.  p  .<_  W )
6 simpl1 963 . . . . 5  |-  ( ( ( ( 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 964 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )  ->  F  e.  T
)
8 simpl3 965 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p  .<_  W ) )  ->  G  e.  T
)
9 simpr 449 . . . . 5  |-  ( ( ( ( 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 . . . . . 6  |-  .\/  =  ( join `  K )
11 trlco.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
12 trlco.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
13 eqid 2256 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
141, 10, 3, 11, 12, 13, 2trlcolem 30045 . . . . 5  |-  ( ( ( 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 1192 . . . 4  |-  ( ( ( ( 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 ) ) )
1615exp32 591 . . 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 ) ) ) ) )
1716rexlimdv 2637 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( E. p  e.  ( Atoms `  K )  -.  p  .<_  W  ->  ( R `  ( F  o.  G
) )  .<_  ( ( R `  F ) 
.\/  ( R `  G ) ) ) )
185, 17mpd 16 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   E.wrex 2517   class class class wbr 3963    o. ccom 4630   ` cfv 4638  (class class class)co 5757   lecple 13142   joincjn 14005   meetcmee 14006   Atomscatm 28583   HLchlt 28670   LHypclh 29303   LTrncltrn 29420   trLctrl 29477
This theorem is referenced by:  trlcone  30047  cdlemg46  30054  trljco  30059  tendopltp  30099  dialss  30366  diblss  30490
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-map 6707  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478
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