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Theorem tendoeq2 31585
Description: Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 31635, 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 31584 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
54adantrr 697 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
61, 2, 3tendoid 31584 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E
)  ->  ( V `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
76adantrl 696 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( V `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
85, 7eqtr4d 2331 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( U `  (  _I  |`  B ) )  =  ( V `  (  _I  |`  B ) ) )
9 fveq2 5541 . . . . . 6  |-  ( f  =  (  _I  |`  B )  ->  ( U `  f )  =  ( U `  (  _I  |`  B ) ) )
10 fveq2 5541 . . . . . 6  |-  ( f  =  (  _I  |`  B )  ->  ( V `  f )  =  ( V `  (  _I  |`  B ) ) )
119, 10eqeq12d 2310 . . . . 5  |-  ( f  =  (  _I  |`  B )  ->  ( ( U `
 f )  =  ( V `  f
)  <->  ( U `  (  _I  |`  B ) )  =  ( V `
 (  _I  |`  B ) ) ) )
128, 11syl5ibrcom 213 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( f  =  (  _I  |`  B )  ->  ( U `  f
)  =  ( V `
 f ) ) )
1312ralrimivw 2640 . . 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 2688 . . . . 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 758 . . . . . . 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 2465 . . . . . . . 8  |-  ( f  =  (  _I  |`  B )  \/  f  =/=  (  _I  |`  B ) )
17 pm5.5 326 . . . . . . . 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 242 . . . . . 6  |-  ( ( ( f  =  (  _I  |`  B )  ->  ( U `  f
)  =  ( V `
 f ) )  /\  ( f  =/=  (  _I  |`  B )  ->  ( U `  f )  =  ( V `  f ) ) )  <->  ( U `  f )  =  ( V `  f ) )
2019ralbii 2580 . . . . 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 242 . . . 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 31575 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  ->  U  =  V )
24233expia 1153 . . . 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 208 . . 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 656 . 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 1148 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    _I cid 4320    |` cres 4707   ` cfv 5271   Basecbs 13164   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   TEndoctendo 31563
This theorem is referenced by:  tendocan  31635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566
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