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Theorem tendoeq2 31571
Description: Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 31621, 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 31570 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
54adantrr 698 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
61, 2, 3tendoid 31570 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E
)  ->  ( V `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
76adantrl 697 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( V `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
85, 7eqtr4d 2471 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( U `  (  _I  |`  B ) )  =  ( V `  (  _I  |`  B ) ) )
9 fveq2 5728 . . . . . 6  |-  ( f  =  (  _I  |`  B )  ->  ( U `  f )  =  ( U `  (  _I  |`  B ) ) )
10 fveq2 5728 . . . . . 6  |-  ( f  =  (  _I  |`  B )  ->  ( V `  f )  =  ( V `  (  _I  |`  B ) ) )
119, 10eqeq12d 2450 . . . . 5  |-  ( f  =  (  _I  |`  B )  ->  ( ( U `
 f )  =  ( V `  f
)  <->  ( U `  (  _I  |`  B ) )  =  ( V `
 (  _I  |`  B ) ) ) )
128, 11syl5ibrcom 214 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( f  =  (  _I  |`  B )  ->  ( U `  f
)  =  ( V `
 f ) ) )
1312ralrimivw 2790 . . 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 2838 . . . . 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 759 . . . . . . 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 2607 . . . . . . . 8  |-  ( f  =  (  _I  |`  B )  \/  f  =/=  (  _I  |`  B ) )
17 pm5.5 327 . . . . . . . 8  |-  ( ( f  =  (  _I  |`  B )  \/  f  =/=  (  _I  |`  B ) )  ->  ( (
( f  =  (  _I  |`  B )  \/  f  =/=  (  _I  |`  B ) )  ->  ( U `  f )  =  ( V `  f ) )  <->  ( U `  f )  =  ( V `  f ) ) )
1816, 17ax-mp 8 . . . . . . 7  |-  ( ( ( f  =  (  _I  |`  B )  \/  f  =/=  (  _I  |`  B ) )  ->  ( U `  f )  =  ( V `  f ) )  <->  ( U `  f )  =  ( V `  f ) )
1915, 18bitr3i 243 . . . . . 6  |-  ( ( ( f  =  (  _I  |`  B )  ->  ( U `  f
)  =  ( V `
 f ) )  /\  ( f  =/=  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) ) )  <->  ( U `  f )  =  ( V `  f ) )
2019ralbii 2729 . . . . 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 243 . . . 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 31561 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  ->  U  =  V )
24233expia 1155 . . . 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 209 . . 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 657 . 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 1150 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 set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705    _I cid 4493    |` cres 4880   ` cfv 5454   Basecbs 13469   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   TEndoctendo 31549
This theorem is referenced by:  tendocan  31621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956  df-tendo 31552
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