Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg47a Structured version   Unicode version

Theorem cdlemg47a 34039
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 1005 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2r 1032 . . . . 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 33427 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G : B
-1-1-onto-> B )
71, 2, 6syl2anc 665 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  G : B
-1-1-onto-> B )
8 f1of 5822 . . . 4  |-  ( G : B -1-1-onto-> B  ->  G : B
--> B )
97, 8syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  G : B
--> B )
10 fcoi1 5765 . . 3  |-  ( G : B --> B  -> 
( G  o.  (  _I  |`  B ) )  =  G )
119, 10syl 17 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( G  o.  (  _I  |`  B ) )  =  G )
12 simp3 1007 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  F  =  (  _I  |`  B ) )
1312coeq2d 5008 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( G  o.  F )  =  ( G  o.  (  _I  |`  B ) ) )
1412coeq1d 5007 . . 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 5766 . . . 4  |-  ( G : B --> B  -> 
( (  _I  |`  B )  o.  G )  =  G )
169, 15syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( (  _I  |`  B )  o.  G )  =  G )
1714, 16eqtrd 2461 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( F  o.  G )  =  G )
1811, 13, 173eqtr4rd 2472 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    _I cid 4755    |` cres 4847    o. ccom 4849   -->wf 5588   -1-1-onto->wf1o 5591   ` cfv 5592   Basecbs 15073   HLchlt 32654   LHypclh 33287   LTrncltrn 33404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7473  df-laut 33292  df-ldil 33407  df-ltrn 33408
This theorem is referenced by:  ltrncom  34043
  Copyright terms: Public domain W3C validator