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Theorem trlnid 33663
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 2438 . . . . . 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 33662 . . . . 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 2446 . . . . . . . 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 33662 . . . . . . . 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 2473 . . . . 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 2641 . 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 1369    e. wcel 1756    =/= wne 2601    _I cid 4626    |` cres 4837   ` cfv 5413   Basecbs 14166   0.cp0 15199   HLchlt 32835   LHypclh 33468   LTrncltrn 33585   trLctrl 33642
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-map 7208  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-lhyp 33472  df-laut 33473  df-ldil 33588  df-ltrn 33589  df-trl 33643
This theorem is referenced by:  cdlemk43N  34447  cdlemk35u  34448  cdlemk55u1  34449  cdlemk39u1  34451  cdlemk19u1  34453
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