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

Theorem cdlemg47a 34383
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 988 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2r 1015 . . . . 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 33773 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G : B
-1-1-onto-> B )
71, 2, 6syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  G : B
-1-1-onto-> B )
8 f1of 5646 . . . 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 5590 . . 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 990 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  F  =  (  _I  |`  B ) )
1312coeq2d 5007 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( G  o.  F )  =  ( G  o.  (  _I  |`  B ) ) )
1412coeq1d 5006 . . 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 5591 . . . 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 2475 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( F  o.  G )  =  G )
1811, 13, 173eqtr4rd 2486 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    _I cid 4636    |` cres 4847    o. ccom 4849   -->wf 5419   -1-1-onto->wf1o 5422   ` cfv 5423   Basecbs 14179   HLchlt 33000   LHypclh 33633   LTrncltrn 33750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-map 7221  df-laut 33638  df-ldil 33753  df-ltrn 33754
This theorem is referenced by:  ltrncom  34387
  Copyright terms: Public domain W3C validator