Theorem trljco2 36029

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

Step	Hyp	Ref	Expression
1		simp1l 1085	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> K e. HL )
2		hllat 34650	. . . . 5 |- ( K e. HL -> K e. Lat )
3	1, 2	syl 17	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> K e. Lat )
4		eqid 2622	. . . . . 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 )
8	4, 5, 6, 7	trlcl 35451	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) )
9	8	3adant3 1081	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` F ) e. ( Base ` K ) )
10	4, 5, 6, 7	trlcl 35451	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) e. ( Base ` K ) )
11	10	3adant2 1080	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` G ) e. ( Base ` K ) )
12		trljco.j	. . . . 5 |- .\/ = ( join ` K )
13	4, 12	latjcom 17059	. . . 4 |- ( ( K e. Lat /\ ( R ` F ) e. ( Base ` K ) /\ ( R ` G ) e. ( Base ` K ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) = ( ( R ` G ) .\/ ( R ` F ) ) )
14	3, 9, 11, 13	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` G ) ) = ( ( R ` G ) .\/ ( R ` F ) ) )
15	12, 5, 6, 7	trljco 36028	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ F e. T ) -> ( ( R ` G ) .\/ ( R ` ( G o. F ) ) ) = ( ( R ` G ) .\/ ( R ` F ) ) )
16	15	3com23 1271	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` G ) .\/ ( R ` ( G o. F ) ) ) = ( ( R ` G ) .\/ ( R ` F ) ) )
17	14, 16	eqtr4d 2659	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` G ) ) = ( ( R ` G ) .\/ ( R ` ( G o. F ) ) ) )
18	12, 5, 6, 7	trljco 36028	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( R ` F ) .\/ ( R ` ( F o. G ) ) ) = ( ( R ` F ) .\/ ( R ` G ) ) )
19	5, 6	ltrncom 36026	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( F o. G ) = ( G o. F ) )
20	19	fveq2d 6195	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( R ` ( F o. G ) ) = ( R ` ( G o. F ) ) )
21	20	oveq2d 6666	. 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 ) ) ) )
22	17, 18, 21	3eqtr4d 2666	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 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   o. ccom 5118   ` cfv 5888  (class class class)co 6650   Basecbs 15857   joincjn 16944   Latclat 17045   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446

This theorem is referenced by:  cdlemh1  36103