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Theorem cdlemg47a 31605
Description: TODO: fix comment. TODO: Use this above in place of  ( F `  P
)  =  P antecedents? (Contributed by NM, 5-Jun-2013.)
Hypotheses
Ref Expression
cdlemg46.b  |-  B  =  ( Base `  K
)
cdlemg46.h  |-  H  =  ( LHyp `  K
)
cdlemg46.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemg47a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( F  o.  G )  =  ( G  o.  F ) )

Proof of Theorem cdlemg47a
StepHypRef Expression
1 simp1 958 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2r 985 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  G  e.  T )
3 cdlemg46.b . . . . . 6  |-  B  =  ( Base `  K
)
4 cdlemg46.h . . . . . 6  |-  H  =  ( LHyp `  K
)
5 cdlemg46.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
63, 4, 5ltrn1o 30995 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G : B
-1-1-onto-> B )
71, 2, 6syl2anc 644 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  G : B
-1-1-onto-> B )
8 f1of 5677 . . . 4  |-  ( G : B -1-1-onto-> B  ->  G : B
--> B )
97, 8syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  G : B
--> B )
10 fcoi1 5620 . . 3  |-  ( G : B --> B  -> 
( G  o.  (  _I  |`  B ) )  =  G )
119, 10syl 16 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( G  o.  (  _I  |`  B ) )  =  G )
12 simp3 960 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  F  =  (  _I  |`  B ) )
1312coeq2d 5038 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( G  o.  F )  =  ( G  o.  (  _I  |`  B ) ) )
1412coeq1d 5037 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( F  o.  G )  =  ( (  _I  |`  B )  o.  G ) )
15 fcoi2 5621 . . . 4  |-  ( G : B --> B  -> 
( (  _I  |`  B )  o.  G )  =  G )
169, 15syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( (  _I  |`  B )  o.  G )  =  G )
1714, 16eqtrd 2470 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( F  o.  G )  =  G )
1811, 13, 173eqtr4rd 2481 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( F  o.  G )  =  ( G  o.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    _I cid 4496    |` cres 4883    o. ccom 4885   -->wf 5453   -1-1-onto->wf1o 5456   ` cfv 5457   Basecbs 13474   HLchlt 30222   LHypclh 30855   LTrncltrn 30972
This theorem is referenced by:  ltrncom  31609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-laut 30860  df-ldil 30975  df-ltrn 30976
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