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Theorem ltrncom 34705
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
StepHypRef Expression
1 simpl1 991 . . 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 992 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  ( Base `  K ) ) )  ->  F  e.  T )
3 simpl3 993 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  ( Base `  K ) ) )  ->  G  e.  T )
4 simpr 461 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  ( Base `  K ) ) )  ->  F  =  (  _I  |`  ( Base `  K ) ) )
5 eqid 2454 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
6 ltrncom.h . . . 4  |-  H  =  ( LHyp `  K
)
7 ltrncom.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
85, 6, 7cdlemg47a 34701 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  ( Base `  K ) ) )  ->  ( F  o.  G )  =  ( G  o.  F ) )
91, 2, 3, 4, 8syl121anc 1224 . 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 1027 . . . 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 1028 . . . 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 1029 . . . 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 754 . . . 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 461 . . . 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 2454 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
165, 6, 7, 15cdlemg48 34704 . . . 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 ) )
1710, 11, 12, 13, 14, 16syl122anc 1228 . . 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 1027 . . . 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 1028 . . . 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 1029 . . . 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 461 . . . 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 )
)
226, 7, 15cdlemg44 34700 . . . 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 ) )
2318, 19, 20, 21, 22syl121anc 1224 . . 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 ) )
2417, 23pm2.61dane 2769 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =/=  (  _I  |`  ( Base `  K ) ) )  ->  ( F  o.  G )  =  ( G  o.  F ) )
259, 24pm2.61dane 2769 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( F  o.  G )  =  ( G  o.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647    _I cid 4738    |` cres 4949    o. ccom 4951   ` cfv 5525   Basecbs 14291   HLchlt 33318   LHypclh 33951   LTrncltrn 34068   trLctrl 34125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-riotaBAD 32927
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-undef 6901  df-map 7325  df-poset 15234  df-plt 15246  df-lub 15262  df-glb 15263  df-join 15264  df-meet 15265  df-p0 15327  df-p1 15328  df-lat 15334  df-clat 15396  df-oposet 33144  df-ol 33146  df-oml 33147  df-covers 33234  df-ats 33235  df-atl 33266  df-cvlat 33290  df-hlat 33319  df-llines 33465  df-lplanes 33466  df-lvols 33467  df-lines 33468  df-psubsp 33470  df-pmap 33471  df-padd 33763  df-lhyp 33955  df-laut 33956  df-ldil 34071  df-ltrn 34072  df-trl 34126
This theorem is referenced by:  ltrnco4  34706  trljco2  34708  tgrpabl  34718  tendoplcom  34749  tendoicl  34763  cdlemk3  34800  cdlemk12  34817  cdlemk12u  34839  cdlemk46  34915  cdlemk49  34918  dvhvaddcomN  35064  cdlemn4  35166  cdlemn8  35172  dihopelvalcpre  35216
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