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Theorem trljco2 34481
Description: Trace joined with trace of composition. (Contributed by NM, 16-Jun-2013.)
Hypotheses
Ref Expression
trljco.j  |-  .\/  =  ( join `  K )
trljco.h  |-  H  =  ( LHyp `  K
)
trljco.t  |-  T  =  ( ( LTrn `  K
) `  W )
trljco.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trljco2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( ( R `  F )  .\/  ( R `  ( F  o.  G )
) )  =  ( ( R `  G
)  .\/  ( R `  ( F  o.  G
) ) ) )

Proof of Theorem trljco2
StepHypRef Expression
1 simp1l 1012 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  K  e.  HL )
2 hllat 33104 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  K  e.  Lat )
4 eqid 2443 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
5 trljco.h . . . . . 6  |-  H  =  ( LHyp `  K
)
6 trljco.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
7 trljco.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
84, 5, 6, 7trlcl 33904 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
983adant3 1008 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
104, 5, 6, 7trlcl 33904 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  e.  (
Base `  K )
)
11103adant2 1007 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( R `  G )  e.  (
Base `  K )
)
12 trljco.j . . . . 5  |-  .\/  =  ( join `  K )
134, 12latjcom 15250 . . . 4  |-  ( ( K  e.  Lat  /\  ( R `  F )  e.  ( Base `  K
)  /\  ( R `  G )  e.  (
Base `  K )
)  ->  ( ( R `  F )  .\/  ( R `  G
) )  =  ( ( R `  G
)  .\/  ( R `  F ) ) )
143, 9, 11, 13syl3anc 1218 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( ( R `  F )  .\/  ( R `  G
) )  =  ( ( R `  G
)  .\/  ( R `  F ) ) )
1512, 5, 6, 7trljco 34480 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  F  e.  T
)  ->  ( ( R `  G )  .\/  ( R `  ( G  o.  F )
) )  =  ( ( R `  G
)  .\/  ( R `  F ) ) )
16153com23 1193 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( ( R `  G )  .\/  ( R `  ( G  o.  F )
) )  =  ( ( R `  G
)  .\/  ( R `  F ) ) )
1714, 16eqtr4d 2478 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( ( R `  F )  .\/  ( R `  G
) )  =  ( ( R `  G
)  .\/  ( R `  ( G  o.  F
) ) ) )
1812, 5, 6, 7trljco 34480 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( ( R `  F )  .\/  ( R `  ( F  o.  G )
) )  =  ( ( R `  F
)  .\/  ( R `  G ) ) )
195, 6ltrncom 34478 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( F  o.  G )  =  ( G  o.  F ) )
2019fveq2d 5716 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( R `  ( F  o.  G
) )  =  ( R `  ( G  o.  F ) ) )
2120oveq2d 6128 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( ( R `  G )  .\/  ( R `  ( F  o.  G )
) )  =  ( ( R `  G
)  .\/  ( R `  ( G  o.  F
) ) ) )
2217, 18, 213eqtr4d 2485 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( ( R `  F )  .\/  ( R `  ( F  o.  G )
) )  =  ( ( R `  G
)  .\/  ( R `  ( F  o.  G
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    o. ccom 4865   ` cfv 5439  (class class class)co 6112   Basecbs 14195   joincjn 15135   Latclat 15236   HLchlt 33091   LHypclh 33724   LTrncltrn 33841   trLctrl 33898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-riotaBAD 32700
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-undef 6813  df-map 7237  df-poset 15137  df-plt 15149  df-lub 15165  df-glb 15166  df-join 15167  df-meet 15168  df-p0 15230  df-p1 15231  df-lat 15237  df-clat 15299  df-oposet 32917  df-ol 32919  df-oml 32920  df-covers 33007  df-ats 33008  df-atl 33039  df-cvlat 33063  df-hlat 33092  df-llines 33238  df-lplanes 33239  df-lvols 33240  df-lines 33241  df-psubsp 33243  df-pmap 33244  df-padd 33536  df-lhyp 33728  df-laut 33729  df-ldil 33844  df-ltrn 33845  df-trl 33899
This theorem is referenced by:  cdlemh1  34555
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