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Theorem ltrncom 34222
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 2438 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
6 ltrncom.h . . . 4  |-  H  =  ( LHyp `  K
)
7 ltrncom.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
85, 6, 7cdlemg47a 34218 . . 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 1223 . 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 2438 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
165, 6, 7, 15cdlemg48 34221 . . . 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 1227 . . 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 34217 . . . 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 1223 . . 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 2684 . 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 2684 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 1369    e. wcel 1756    =/= wne 2601    _I cid 4626    |` cres 4837    o. ccom 4839   ` cfv 5413   Basecbs 14166   HLchlt 32835   LHypclh 33468   LTrncltrn 33585   trLctrl 33642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-riotaBAD 32444
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-undef 6784  df-map 7208  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-llines 32982  df-lplanes 32983  df-lvols 32984  df-lines 32985  df-psubsp 32987  df-pmap 32988  df-padd 33280  df-lhyp 33472  df-laut 33473  df-ldil 33588  df-ltrn 33589  df-trl 33643
This theorem is referenced by:  ltrnco4  34223  trljco2  34225  tgrpabl  34235  tendoplcom  34266  tendoicl  34280  cdlemk3  34317  cdlemk12  34334  cdlemk12u  34356  cdlemk46  34432  cdlemk49  34435  dvhvaddcomN  34581  cdlemn4  34683  cdlemn8  34689  dihopelvalcpre  34733
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