Theorem ltrncom 31220

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 960	. . 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 961	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> F e. T )
3		simpl3 962	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> G e. T )
4		simpr 448	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> F = ( _I |` ( Base ` K ) ) )
5		eqid 2404	. . . 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 31216	. . 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 1189	. 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 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 ) ) -> ( K e. HL /\ W e. H ) )
11		simpll2 997	. . . 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 998	. . . 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 732	. . . 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 448	. . . 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 2404	. . . . 5 |- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
16	5, 6, 7, 15	cdlemg48 31219	. . . 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 1193	. . 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 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 ) ) -> ( K e. HL /\ W e. H ) )
19		simpll2 997	. . . 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 998	. . . 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 448	. . . 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 31215	. . . 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 1189	. . 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 2645	. 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 2645	1 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( F o. G ) = ( G o. F ) )

Syntax hints:   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   _I cid 4453   |` cres 4839   o. ccom 4841   ` cfv 5413   Basecbs 13424   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640

This theorem is referenced by:  ltrnco4  31221  trljco2  31223  tgrpabl  31233  tendoplcom  31264  tendoicl  31278  cdlemk3  31315  cdlemk12  31332  cdlemk12u  31354  cdlemk46  31430  cdlemk49  31433  dvhvaddcomN  31579  cdlemn4  31681  cdlemn8  31687  dihopelvalcpre  31731

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641