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Theorem cdlemg48 35408
Description: Elmininate  h from cdlemg47 35407. (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 35238 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. h  e.  T  ( h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )
653ad2ant1 1012 . 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 1021 . . . 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 1104 . . . 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 1105 . . . 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 992 . . . 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 1107 . . . 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 1106 . . . 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 1019 . . . 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 1020 . . . 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 35407 . . . 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 1253 . . 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 2950 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808    _I cid 4783    |` cres 4994    o. ccom 4996   ` cfv 5579   Basecbs 14479   HLchlt 34022   LHypclh 34655   LTrncltrn 34772   trLctrl 34829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-riotaBAD 33631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-undef 6992  df-map 7412  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170  df-lvols 34171  df-lines 34172  df-psubsp 34174  df-pmap 34175  df-padd 34467  df-lhyp 34659  df-laut 34660  df-ldil 34775  df-ltrn 34776  df-trl 34830
This theorem is referenced by:  ltrncom  35409
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