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Theorem ltrncom 29616
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 2253 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
6 ltrncom.h . . . 4  |-  H  =  ( LHyp `  K
)
7 ltrncom.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
85, 6, 7cdlemg47a 29612 . . 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 2253 . . . . 5  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
165, 6, 7, 15cdlemg48 29615 . . . 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 29611 . . . 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 2490 . 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 2490 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 2412    _I cid 4197    |` cres 4582    o. ccom 4584   ` cfv 4592   Basecbs 13022   HLchlt 28229   LHypclh 28862   LTrncltrn 28979   trLctrl 29036
This theorem is referenced by:  ltrnco4  29617  trljco2  29619  tgrpabl  29629  tendoplcom  29660  tendoicl  29674  cdlemk3  29711  cdlemk12  29728  cdlemk12u  29750  cdlemk46  29826  cdlemk49  29829  dvhvaddcomN  29975  cdlemn4  30077  cdlemn8  30083  dihopelvalcpre  30127
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-llines 28376  df-lplanes 28377  df-lvols 28378  df-lines 28379  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866  df-laut 28867  df-ldil 28982  df-ltrn 28983  df-trl 29037
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