Theorem ltrncom 31473

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 2436	. . . 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 31469	. . 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 2436	. . . . 5 |- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
16	5, 6, 7, 15	cdlemg48 31472	. . . 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 31468	. . . 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 2677	. 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 2677	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 1652   e. wcel 1725   =/= wne 2599   _I cid 4486   |` cres 4873   o. ccom 4875   ` cfv 5447   Basecbs 13462   HLchlt 30086   LHypclh 30719   LTrncltrn 30836   trLctrl 30893

This theorem is referenced by:  ltrnco4  31474  trljco2  31476  tgrpabl  31486  tendoplcom  31517  tendoicl  31531  cdlemk3  31568  cdlemk12  31585  cdlemk12u  31607  cdlemk46  31683  cdlemk49  31686  dvhvaddcomN  31832  cdlemn4  31934  cdlemn8  31940  dihopelvalcpre  31984

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 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-iin 4089  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-undef 6536  df-riota 6542  df-map 7013  df-poset 14396  df-plt 14408  df-lub 14424  df-glb 14425  df-join 14426  df-meet 14427  df-p0 14461  df-p1 14462  df-lat 14468  df-clat 14530  df-oposet 29912  df-ol 29914  df-oml 29915  df-covers 30002  df-ats 30003  df-atl 30034  df-cvlat 30058  df-hlat 30087  df-llines 30233  df-lplanes 30234  df-lvols 30235  df-lines 30236  df-psubsp 30238  df-pmap 30239  df-padd 30531  df-lhyp 30723  df-laut 30724  df-ldil 30839  df-ltrn 30840  df-trl 30894