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Theorem ltrncom 35552
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 999 . . 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 1000 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  ( Base `  K ) ) )  ->  F  e.  T )
3 simpl3 1001 . . 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 2467 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
6 ltrncom.h . . . 4  |-  H  =  ( LHyp `  K
)
7 ltrncom.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
85, 6, 7cdlemg47a 35548 . . 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 1233 . 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 1035 . . . 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 1036 . . . 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 1037 . . . 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 2467 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
165, 6, 7, 15cdlemg48 35551 . . . 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 1237 . . 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 1035 . . . 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 1036 . . . 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 1037 . . . 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 35547 . . . 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 1233 . . 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 2785 . 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 2785 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662    _I cid 4790    |` cres 5001    o. ccom 5003   ` cfv 5588   Basecbs 14490   HLchlt 34165   LHypclh 34798   LTrncltrn 34915   trLctrl 34972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-riotaBAD 33774
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-undef 7002  df-map 7422  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-p1 15527  df-lat 15533  df-clat 15595  df-oposet 33991  df-ol 33993  df-oml 33994  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166  df-llines 34312  df-lplanes 34313  df-lvols 34314  df-lines 34315  df-psubsp 34317  df-pmap 34318  df-padd 34610  df-lhyp 34802  df-laut 34803  df-ldil 34918  df-ltrn 34919  df-trl 34973
This theorem is referenced by:  ltrnco4  35553  trljco2  35555  tgrpabl  35565  tendoplcom  35596  tendoicl  35610  cdlemk3  35647  cdlemk12  35664  cdlemk12u  35686  cdlemk46  35762  cdlemk49  35765  dvhvaddcomN  35911  cdlemn4  36013  cdlemn8  36019  dihopelvalcpre  36063
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