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

Theorem trlnid 34131
Description: Different translations with the same trace cannot be the identity. (Contributed by NM, 26-Jul-2013.)
Hypotheses
Ref Expression
trlnid.b  |-  B  =  ( Base `  K
)
trlnid.h  |-  H  =  ( LHyp `  K
)
trlnid.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlnid.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlnid  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  F  =/=  (  _I  |`  B ) )

Proof of Theorem trlnid
StepHypRef Expression
1 simp3l 1016 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  F  =/=  G
)
2 simp1 988 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simp2l 1014 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  F  e.  T
)
4 trlnid.b . . . . . 6  |-  B  =  ( Base `  K
)
5 eqid 2451 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 trlnid.h . . . . . 6  |-  H  =  ( LHyp `  K
)
7 trlnid.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
8 trlnid.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
94, 5, 6, 7, 8trlid0b 34130 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =  (  _I  |`  B )  <-> 
( R `  F
)  =  ( 0.
`  K ) ) )
102, 3, 9syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  ( F  =  (  _I  |`  B )  <-> 
( R `  F
)  =  ( 0.
`  K ) ) )
1110biimpar 485 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F
)  =  ( R `
 G ) ) )  /\  ( R `
 F )  =  ( 0. `  K
) )  ->  F  =  (  _I  |`  B ) )
12 simp3r 1017 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  ( R `  F )  =  ( R `  G ) )
1312eqeq1d 2453 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  ( ( R `
 F )  =  ( 0. `  K
)  <->  ( R `  G )  =  ( 0. `  K ) ) )
1413biimpa 484 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F
)  =  ( R `
 G ) ) )  /\  ( R `
 F )  =  ( 0. `  K
) )  ->  ( R `  G )  =  ( 0. `  K ) )
15 simpl1 991 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F
)  =  ( R `
 G ) ) )  /\  ( R `
 F )  =  ( 0. `  K
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
16 simpl2r 1042 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F
)  =  ( R `
 G ) ) )  /\  ( R `
 F )  =  ( 0. `  K
) )  ->  G  e.  T )
174, 5, 6, 7, 8trlid0b 34130 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( G  =  (  _I  |`  B )  <-> 
( R `  G
)  =  ( 0.
`  K ) ) )
1815, 16, 17syl2anc 661 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F
)  =  ( R `
 G ) ) )  /\  ( R `
 F )  =  ( 0. `  K
) )  ->  ( G  =  (  _I  |`  B )  <->  ( R `  G )  =  ( 0. `  K ) ) )
1914, 18mpbird 232 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F
)  =  ( R `
 G ) ) )  /\  ( R `
 F )  =  ( 0. `  K
) )  ->  G  =  (  _I  |`  B ) )
2011, 19eqtr4d 2495 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F
)  =  ( R `
 G ) ) )  /\  ( R `
 F )  =  ( 0. `  K
) )  ->  F  =  G )
2120ex 434 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  ( ( R `
 F )  =  ( 0. `  K
)  ->  F  =  G ) )
2210, 21sylbid 215 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  ( F  =  (  _I  |`  B )  ->  F  =  G ) )
2322necon3d 2672 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  ( F  =/= 
G  ->  F  =/=  (  _I  |`  B ) ) )
241, 23mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  F  =/=  (  _I  |`  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644    _I cid 4731    |` cres 4942   ` cfv 5518   Basecbs 14278   0.cp0 15311   HLchlt 33303   LHypclh 33936   LTrncltrn 34053   trLctrl 34110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-map 7318  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-p1 15314  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-lhyp 33940  df-laut 33941  df-ldil 34056  df-ltrn 34057  df-trl 34111
This theorem is referenced by:  cdlemk43N  34915  cdlemk35u  34916  cdlemk55u1  34917  cdlemk39u1  34919  cdlemk19u1  34921
  Copyright terms: Public domain W3C validator