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Theorem trljat3 30892
Description: The value of a translation of an atom  P not under the fiducial co-atom  W, joined with trace. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 22-May-2012.)
Hypotheses
Ref Expression
trljat.l  |-  .<_  =  ( le `  K )
trljat.j  |-  .\/  =  ( join `  K )
trljat.a  |-  A  =  ( Atoms `  K )
trljat.h  |-  H  =  ( LHyp `  K
)
trljat.t  |-  T  =  ( ( LTrn `  K
) `  W )
trljat.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trljat3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F
) )  =  ( ( F `  P
)  .\/  ( R `  F ) ) )

Proof of Theorem trljat3
StepHypRef Expression
1 trljat.l . . 3  |-  .<_  =  ( le `  K )
2 trljat.j . . 3  |-  .\/  =  ( join `  K )
3 trljat.a . . 3  |-  A  =  ( Atoms `  K )
4 trljat.h . . 3  |-  H  =  ( LHyp `  K
)
5 trljat.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
6 trljat.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
71, 2, 3, 4, 5, 6trljat1 30890 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F
) )  =  ( P  .\/  ( F `
 P ) ) )
81, 2, 3, 4, 5, 6trljat2 30891 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  .\/  ( R `  F
) )  =  ( P  .\/  ( F `
 P ) ) )
97, 8eqtr4d 2470 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F
) )  =  ( ( F `  P
)  .\/  ( R `  F ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   Atomscatm 29988   HLchlt 30075   LHypclh 30708   LTrncltrn 30825   trLctrl 30882
This theorem is referenced by:  trlcoabs  31445  cdlemk1  31555  cdlemk2  31556  cdlemk1u  31583  cdlemkfid1N  31645  cdlemkid1  31646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-psubsp 30227  df-pmap 30228  df-padd 30520  df-lhyp 30712  df-laut 30713  df-ldil 30828  df-ltrn 30829  df-trl 30883
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