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Theorem trlnid 33197
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 1025 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  F  =/=  G
)
2 simp1 997 . . . . 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 1023 . . . . 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 2402 . . . . . 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 33196 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =  (  _I  |`  B )  <-> 
( R `  F
)  =  ( 0.
`  K ) ) )
102, 3, 9syl2anc 659 . . . 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 483 . . . . . 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 1026 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  ( F  =/=  G  /\  ( R `  F )  =  ( R `  G ) ) )  ->  ( R `  F )  =  ( R `  G ) )
1312eqeq1d 2404 . . . . . . . 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 482 . . . . . . 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 1000 . . . . . . . 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 1051 . . . . . . . 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 33196 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( G  =  (  _I  |`  B )  <-> 
( R `  G
)  =  ( 0.
`  K ) ) )
1815, 16, 17syl2anc 659 . . . . . . 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 2446 . . . . 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 432 . . . 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 2627 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598    _I cid 4733    |` cres 4825   ` cfv 5569   Basecbs 14841   0.cp0 15991   HLchlt 32368   LHypclh 33001   LTrncltrn 33118   trLctrl 33176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-map 7459  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-lhyp 33005  df-laut 33006  df-ldil 33121  df-ltrn 33122  df-trl 33177
This theorem is referenced by:  cdlemk43N  33982  cdlemk35u  33983  cdlemk55u1  33984  cdlemk39u1  33986  cdlemk19u1  33988
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