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Theorem cdlemg47a 30923
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 955 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2r 982 . . . . 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 30313 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G : B
-1-1-onto-> B )
71, 2, 6syl2anc 642 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  G : B
-1-1-onto-> B )
8 f1of 5472 . . . 4  |-  ( G : B -1-1-onto-> B  ->  G : B
--> B )
97, 8syl 15 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  G : B
--> B )
10 fcoi1 5415 . . 3  |-  ( G : B --> B  -> 
( G  o.  (  _I  |`  B ) )  =  G )
119, 10syl 15 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( G  o.  (  _I  |`  B ) )  =  G )
12 simp3 957 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  F  =  (  _I  |`  B ) )
1312coeq2d 4846 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( G  o.  F )  =  ( G  o.  (  _I  |`  B ) ) )
1412coeq1d 4845 . . 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 5416 . . . 4  |-  ( G : B --> B  -> 
( (  _I  |`  B )  o.  G )  =  G )
169, 15syl 15 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( (  _I  |`  B )  o.  G )  =  G )
1714, 16eqtrd 2315 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( F  o.  G )  =  G )
1811, 13, 173eqtr4rd 2326 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    _I cid 4304    |` cres 4691    o. ccom 4693   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255   Basecbs 13148   HLchlt 29540   LHypclh 30173   LTrncltrn 30290
This theorem is referenced by:  ltrncom  30927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-laut 30178  df-ldil 30293  df-ltrn 30294
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