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Theorem cdlemg48 31534
Description: Elmininate  h from cdlemg47 31533. (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 )
cdlemg46.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg48  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  G ) ) )  ->  ( F  o.  G )  =  ( G  o.  F ) )

Proof of Theorem cdlemg48
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 cdlemg46.b . . . 4  |-  B  =  ( Base `  K
)
2 cdlemg46.h . . . 4  |-  H  =  ( LHyp `  K
)
3 cdlemg46.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
4 cdlemg46.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
51, 2, 3, 4cdlemftr1 31364 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. h  e.  T  ( h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )
653ad2ant1 978 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  G ) ) )  ->  E. h  e.  T  ( h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )
7 simp11 987 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  G ) ) )  /\  h  e.  T  /\  ( h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
8 simp12l 1070 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  G ) ) )  /\  h  e.  T  /\  ( h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  F  e.  T
)
9 simp12r 1071 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  G ) ) )  /\  h  e.  T  /\  ( h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  G  e.  T
)
10 simp2 958 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  G ) ) )  /\  h  e.  T  /\  ( h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  h  e.  T
)
11 simp13r 1073 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  G ) ) )  /\  h  e.  T  /\  ( h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( R `  F )  =  ( R `  G ) )
12 simp13l 1072 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  G ) ) )  /\  h  e.  T  /\  ( h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  F  =/=  (  _I  |`  B ) )
13 simp3l 985 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  G ) ) )  /\  h  e.  T  /\  ( h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  h  =/=  (  _I  |`  B ) )
14 simp3r 986 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  G ) ) )  /\  h  e.  T  /\  ( h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( R `  h )  =/=  ( R `  F )
)
151, 2, 3, 4cdlemg47 31533 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
h  e.  T  /\  ( R `  F )  =  ( R `  G ) )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  ->  ( F  o.  G )  =  ( G  o.  F ) )
167, 8, 9, 10, 11, 12, 13, 14, 15syl323anc 1214 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  G ) ) )  /\  h  e.  T  /\  ( h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( F  o.  G )  =  ( G  o.  F ) )
1716rexlimdv3a 2832 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  G ) ) )  ->  ( E. h  e.  T  ( h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) )  -> 
( F  o.  G
)  =  ( G  o.  F ) ) )
186, 17mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  G ) ) )  ->  ( 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    =/= wne 2599   E.wrex 2706    _I cid 4493    |` cres 4880    o. ccom 4882   ` cfv 5454   Basecbs 13469   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   trLctrl 30955
This theorem is referenced by:  ltrncom  31535
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956
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