Theorem ltrncom 30177

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 963	. . 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 964	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> F e. T )
3		simpl3 965	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> G e. T )
4		simpr 449	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = ( _I |` ( Base ` K ) ) ) -> F = ( _I |` ( Base ` K ) ) )
5		eqid 2258	. . . 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 30173	. . 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 1192	. 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 999	. . . 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 1000	. . . 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 1001	. . . 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 734	. . . 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 449	. . . 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 2258	. . . . 5 |- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
16	5, 6, 7, 15	cdlemg48 30176	. . . 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 1196	. . 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 999	. . . 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 1000	. . . 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 1001	. . . 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 449	. . . 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 30172	. . . 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 1192	. . 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 2499	. 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 2499	1 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( F o. G ) = ( G o. F ) )

Syntax hints:   -> wi 6   /\ wa 360   /\ w3a 939   = wceq 1619   e. wcel 1621   =/= wne 2421   _I cid 4276   |` cres 4663   o. ccom 4665   ` cfv 4673   Basecbs 13111   HLchlt 28790   LHypclh 29423   LTrncltrn 29540   trLctrl 29597

This theorem is referenced by:  ltrnco4  30178  trljco2  30180  tgrpabl  30190  tendoplcom  30221  tendoicl  30235  cdlemk3  30272  cdlemk12  30289  cdlemk12u  30311  cdlemk46  30387  cdlemk49  30390  dvhvaddcomN  30536  cdlemn4  30638  cdlemn8  30644  dihopelvalcpre  30688

This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-map 6742  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-llines 28937  df-lplanes 28938  df-lvols 28939  df-lines 28940  df-psubsp 28942  df-pmap 28943  df-padd 29235  df-lhyp 29427  df-laut 29428  df-ldil 29543  df-ltrn 29544  df-trl 29598