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Theorem ltrncom 30057
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 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 2256 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
6 ltrncom.h . . . 4  |-  H  =  ( LHyp `  K
)
7 ltrncom.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
85, 6, 7cdlemg47a 30053 . . 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 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 2256 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
165, 6, 7, 15cdlemg48 30056 . . . 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 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 )
)
226, 7, 15cdlemg44 30052 . . . 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 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 ) )
2417, 23pm2.61dane 2497 . 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 2497 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
)  ->  ( F  o.  G )  =  ( G  o.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419    _I cid 4241    |` cres 4628    o. ccom 4630   ` cfv 4638   Basecbs 13075   HLchlt 28670   LHypclh 29303   LTrncltrn 29420   trLctrl 29477
This theorem is referenced by:  ltrnco4  30058  trljco2  30060  tgrpabl  30070  tendoplcom  30101  tendoicl  30115  cdlemk3  30152  cdlemk12  30169  cdlemk12u  30191  cdlemk46  30267  cdlemk49  30270  dvhvaddcomN  30416  cdlemn4  30518  cdlemn8  30524  dihopelvalcpre  30568
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 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-map 6707  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478
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