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Theorem cdlemg47a 31370
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 957 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2r 984 . . . . 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 30760 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G : B
-1-1-onto-> B )
71, 2, 6syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  G : B
-1-1-onto-> B )
8 f1of 5665 . . . 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 5608 . . 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 959 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  F  =  (  _I  |`  B ) )
1312coeq2d 5026 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( G  o.  F )  =  ( G  o.  (  _I  |`  B ) ) )
1412coeq1d 5025 . . 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 5609 . . . 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 2467 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( F  o.  G )  =  G )
1811, 13, 173eqtr4rd 2478 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    _I cid 4485    |` cres 4871    o. ccom 4873   -->wf 5441   -1-1-onto->wf1o 5444   ` cfv 5445   Basecbs 13457   HLchlt 29987   LHypclh 30620   LTrncltrn 30737
This theorem is referenced by:  ltrncom  31374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-map 7011  df-laut 30625  df-ldil 30740  df-ltrn 30741
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