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Theorem tendoeq2 34050
Description: Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 34100, we show that we only need to consider a single non-identity translation. (Contributed by NM, 21-Jun-2013.)
Hypotheses
Ref Expression
tendoeq2.b  |-  B  =  ( Base `  K
)
tendoeq2.h  |-  H  =  ( LHyp `  K
)
tendoeq2.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendoeq2.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
tendoeq2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  (
f  =/=  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) ) )  ->  U  =  V )
Distinct variable groups:    f, E    f, H    f, K    T, f    f, W    U, f    f, V
Allowed substitution hint:    B( f)

Proof of Theorem tendoeq2
StepHypRef Expression
1 tendoeq2.b . . . . . . . 8  |-  B  =  ( Base `  K
)
2 tendoeq2.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
3 tendoeq2.e . . . . . . . 8  |-  E  =  ( ( TEndo `  K
) `  W )
41, 2, 3tendoid 34049 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
54adantrr 721 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
61, 2, 3tendoid 34049 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E
)  ->  ( V `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
76adantrl 720 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( V `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
85, 7eqtr4d 2473 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( U `  (  _I  |`  B ) )  =  ( V `  (  _I  |`  B ) ) )
9 fveq2 5881 . . . . . 6  |-  ( f  =  (  _I  |`  B )  ->  ( U `  f )  =  ( U `  (  _I  |`  B ) ) )
10 fveq2 5881 . . . . . 6  |-  ( f  =  (  _I  |`  B )  ->  ( V `  f )  =  ( V `  (  _I  |`  B ) ) )
119, 10eqeq12d 2451 . . . . 5  |-  ( f  =  (  _I  |`  B )  ->  ( ( U `
 f )  =  ( V `  f
)  <->  ( U `  (  _I  |`  B ) )  =  ( V `
 (  _I  |`  B ) ) ) )
128, 11syl5ibrcom 225 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( f  =  (  _I  |`  B )  ->  ( U `  f
)  =  ( V `
 f ) ) )
1312ralrimivw 2847 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  ->  A. f  e.  T  ( f  =  (  _I  |`  B )  ->  ( U `  f
)  =  ( V `
 f ) ) )
14 r19.26 2962 . . . . 5  |-  ( A. f  e.  T  (
( f  =  (  _I  |`  B )  ->  ( U `  f
)  =  ( V `
 f ) )  /\  ( f  =/=  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) ) )  <->  ( A. f  e.  T  (
f  =  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) )  /\  A. f  e.  T  ( f  =/=  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) ) ) )
15 jaob 790 . . . . . . 7  |-  ( ( ( f  =  (  _I  |`  B )  \/  f  =/=  (  _I  |`  B ) )  ->  ( U `  f )  =  ( V `  f ) )  <->  ( ( f  =  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) )  /\  ( f  =/=  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) ) ) )
16 exmidne 2636 . . . . . . . 8  |-  ( f  =  (  _I  |`  B )  \/  f  =/=  (  _I  |`  B ) )
17 pm5.5 337 . . . . . . . 8  |-  ( ( f  =  (  _I  |`  B )  \/  f  =/=  (  _I  |`  B ) )  ->  ( (
( f  =  (  _I  |`  B )  \/  f  =/=  (  _I  |`  B ) )  ->  ( U `  f )  =  ( V `  f ) )  <->  ( U `  f )  =  ( V `  f ) ) )
1816, 17ax-mp 5 . . . . . . 7  |-  ( ( ( f  =  (  _I  |`  B )  \/  f  =/=  (  _I  |`  B ) )  ->  ( U `  f )  =  ( V `  f ) )  <->  ( U `  f )  =  ( V `  f ) )
1915, 18bitr3i 254 . . . . . 6  |-  ( ( ( f  =  (  _I  |`  B )  ->  ( U `  f
)  =  ( V `
 f ) )  /\  ( f  =/=  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) ) )  <->  ( U `  f )  =  ( V `  f ) )
2019ralbii 2863 . . . . 5  |-  ( A. f  e.  T  (
( f  =  (  _I  |`  B )  ->  ( U `  f
)  =  ( V `
 f ) )  /\  ( f  =/=  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) ) )  <->  A. f  e.  T  ( U `  f )  =  ( V `  f ) )
2114, 20bitr3i 254 . . . 4  |-  ( ( A. f  e.  T  ( f  =  (  _I  |`  B )  ->  ( U `  f
)  =  ( V `
 f ) )  /\  A. f  e.  T  ( f  =/=  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) ) )  <->  A. f  e.  T  ( U `  f )  =  ( V `  f ) )
22 tendoeq2.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
232, 22, 3tendoeq1 34040 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  ->  U  =  V )
24233expia 1207 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( A. f  e.  T  ( U `  f )  =  ( V `  f )  ->  U  =  V ) )
2521, 24syl5bi 220 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( ( A. f  e.  T  ( f  =  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) )  /\  A. f  e.  T  ( f  =/=  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) ) )  ->  U  =  V ) )
2613, 25mpand 679 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( A. f  e.  T  ( f  =/=  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) )  ->  U  =  V ) )
27263impia 1202 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  (
f  =/=  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) ) )  ->  U  =  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782    _I cid 4764    |` cres 4856   ` cfv 5601   Basecbs 15084   HLchlt 32625   LHypclh 33258   LTrncltrn 33375   TEndoctendo 34028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-lhyp 33262  df-laut 33263  df-ldil 33378  df-ltrn 33379  df-trl 33434  df-tendo 34031
This theorem is referenced by:  tendocan  34100
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