Theorem ltrncom 30206

Description: Composition is commutative for translations. Part of proof of Lemma G of [Crawley] p. 116 (Contributed by NM, 5-Jun-2013.)

Hypotheses

Ref	Expression
ltrncom.h	|- H = ( LHyp ` K )
ltrncom.t	|- T = ( ( LTrn ` K ) ` W )

Assertion

Ref	Expression
ltrncom	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( F o. G ) = ( G o. F ) )

Proof of Theorem ltrncom

Step	Hyp	Ref	Expression
1		simpl1 958	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> ( K e. HL /\ W e. H ) )
2		simpl2 959	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> F e. T )
3		simpl3 960	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> G e. T )
4		simpr 447	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> F = ( _I |` ( Base ` K ) ) )
5		eqid 2284	. . . 4 |- ( Base ` K ) = ( Base ` K )
6		ltrncom.h	. . . 4 |- H = ( LHyp ` K )
7		ltrncom.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
8	5, 6, 7	cdlemg47a 30202	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> ( F o. G ) = ( G o. F ) )
9	1, 2, 3, 4, 8	syl121anc 1187	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> ( F o. G ) = ( G o. F ) )
10		simpll1 994	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F =/= ( _I |` ( Base ` K ) ) ) /\ ( ( ( trL ` K ) ` W ) ` F ) = ( ( ( trL ` K ) ` W ) ` G ) ) -> ( K e. HL /\ W e. H ) )
11		simpll2 995	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F =/= ( _I |` ( Base ` K ) ) ) /\ ( ( ( trL ` K ) ` W ) ` F ) = ( ( ( trL ` K ) ` W ) ` G ) ) -> F e. T )
12		simpll3 996	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F =/= ( _I |` ( Base ` K ) ) ) /\ ( ( ( trL ` K ) ` W ) ` F ) = ( ( ( trL ` K ) ` W ) ` G ) ) -> G e. T )
13		simplr 731	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F =/= ( _I |` ( Base ` K ) ) ) /\ ( ( ( trL ` K ) ` W ) ` F ) = ( ( ( trL ` K ) ` W ) ` G ) ) -> F =/= ( _I |` ( Base ` K ) ) )
14		simpr 447	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F =/= ( _I |` ( Base ` K ) ) ) /\ ( ( ( trL ` K ) ` W ) ` F ) = ( ( ( trL ` K ) ` W ) ` G ) ) -> ( ( ( trL ` K ) ` W ) ` F ) = ( ( ( trL ` K ) ` W ) ` G ) )
15		eqid 2284	. . . . 5 |- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
16	5, 6, 7, 15	cdlemg48 30205	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= ( _I |` ( Base ` K ) ) /\ ( ( ( trL ` K ) ` W ) ` F ) = ( ( ( trL ` K ) ` W ) ` G ) ) ) -> ( F o. G ) = ( G o. F ) )
17	10, 11, 12, 13, 14, 16	syl122anc 1191	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F =/= ( _I |` ( Base ` K ) ) ) /\ ( ( ( trL ` K ) ` W ) ` F ) = ( ( ( trL ` K ) ` W ) ` G ) ) -> ( F o. G ) = ( G o. F ) )
18		simpll1 994	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F =/= ( _I |` ( Base ` K ) ) ) /\ ( ( ( trL ` K ) ` W ) ` F ) =/= ( ( ( trL ` K ) ` W ) ` G ) ) -> ( K e. HL /\ W e. H ) )
19		simpll2 995	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F =/= ( _I |` ( Base ` K ) ) ) /\ ( ( ( trL ` K ) ` W ) ` F ) =/= ( ( ( trL ` K ) ` W ) ` G ) ) -> F e. T )
20		simpll3 996	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F =/= ( _I |` ( Base ` K ) ) ) /\ ( ( ( trL ` K ) ` W ) ` F ) =/= ( ( ( trL ` K ) ` W ) ` G ) ) -> G e. T )
21		simpr 447	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F =/= ( _I |` ( Base ` K ) ) ) /\ ( ( ( trL ` K ) ` W ) ` F ) =/= ( ( ( trL ` K ) ` W ) ` G ) ) -> ( ( ( trL ` K ) ` W ) ` F ) =/= ( ( ( trL ` K ) ` W ) ` G ) )
22	6, 7, 15	cdlemg44 30201	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( ( trL ` K ) ` W ) ` F ) =/= ( ( ( trL ` K ) ` W ) ` G ) ) -> ( F o. G ) = ( G o. F ) )
23	18, 19, 20, 21, 22	syl121anc 1187	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F =/= ( _I |` ( Base ` K ) ) ) /\ ( ( ( trL ` K ) ` W ) ` F ) =/= ( ( ( trL ` K ) ` W ) ` G ) ) -> ( F o. G ) = ( G o. F ) )
24	17, 23	pm2.61dane 2525	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F =/= ( _I |` ( Base ` K ) ) ) -> ( F o. G ) = ( G o. F ) )
25	9, 24	pm2.61dane 2525	1 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( F o. G ) = ( G o. F ) )

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934   = wceq 1623   e. wcel 1685   =/= wne 2447   _I cid 4303   |` cres 4690   o. ccom 4692   ` cfv 5221   Basecbs 13144   HLchlt 28819   LHypclh 29452   LTrncltrn 29569   trLctrl 29626

This theorem is referenced by:  ltrnco4  30207  trljco2  30209  tgrpabl  30219  tendoplcom  30250  tendoicl  30264  cdlemk3  30301  cdlemk12  30318  cdlemk12u  30340  cdlemk46  30416  cdlemk49  30419  dvhvaddcomN  30565  cdlemn4  30667  cdlemn8  30673  dihopelvalcpre  30717

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-map 6770  df-poset 14076  df-plt 14088  df-lub 14104  df-glb 14105  df-join 14106  df-meet 14107  df-p0 14141  df-p1 14142  df-lat 14148  df-clat 14210  df-oposet 28645  df-ol 28647  df-oml 28648  df-covers 28735  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820  df-llines 28966  df-lplanes 28967  df-lvols 28968  df-lines 28969  df-psubsp 28971  df-pmap 28972  df-padd 29264  df-lhyp 29456  df-laut 29457  df-ldil 29572  df-ltrn 29573  df-trl 29627