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Theorem tendoeq2 30963
Description: Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 31013, 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 30962 . . . . . . 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 30962 . . . . . . 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 2318 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E ) )  -> 
( U `  (  _I  |`  B ) )  =  ( V `  (  _I  |`  B ) ) )
9 fveq2 5525 . . . . . 6  |-  ( f  =  (  _I  |`  B )  ->  ( U `  f )  =  ( U `  (  _I  |`  B ) ) )
10 fveq2 5525 . . . . . 6  |-  ( f  =  (  _I  |`  B )  ->  ( V `  f )  =  ( V `  (  _I  |`  B ) ) )
119, 10eqeq12d 2297 . . . . 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 2627 . . 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 2675 . . . . 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 2452 . . . . . . . 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 2567 . . . . 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 30953 . . . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    _I cid 4304    |` cres 4691   ` cfv 5255   Basecbs 13148   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941
This theorem is referenced by:  tendocan  31013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944
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