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

Theorem trlnid 30368
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 983 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  F  =/=  G
)
2 simp1 955 . . . . 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 981 . . . . 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 2283 . . . . . 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 30367 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =  (  _I  |`  B )  <-> 
( R `  F
)  =  ( 0.
`  K ) ) )
102, 3, 9syl2anc 642 . . . 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 471 . . . . . 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 984 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  ( R `  F )  =  ( R `  G ) )
1312eqeq1d 2291 . . . . . . . 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 470 . . . . . . 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 958 . . . . . . . 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 1009 . . . . . . . 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 30367 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( G  =  (  _I  |`  B )  <-> 
( R `  G
)  =  ( 0.
`  K ) ) )
1815, 16, 17syl2anc 642 . . . . . . 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 223 . . . . . 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 2318 . . . . 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 423 . . . 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 206 . . 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 2484 . 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 14 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    _I cid 4304    |` cres 4691   ` cfv 5255   Basecbs 13148   0.cp0 14143   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemk43N  31152  cdlemk35u  31153  cdlemk55u1  31154  cdlemk39u1  31156  cdlemk19u1  31158
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
  Copyright terms: Public domain W3C validator