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Theorem tendoeq1 35560
Description: Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
tendof.h  |-  H  =  ( LHyp `  K
)
tendof.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendof.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
tendoeq1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  ->  U  =  V )
Distinct variable groups:    f, K    T, f    f, W    U, f    f, V
Allowed substitution hints:    E( f)    H( f)

Proof of Theorem tendoeq1
StepHypRef Expression
1 simp3 998 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  ->  A. f  e.  T  ( U `  f )  =  ( V `  f ) )
2 simp1 996 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
3 simp2l 1022 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  ->  U  e.  E )
4 tendof.h . . . . . 6  |-  H  =  ( LHyp `  K
)
5 tendof.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
6 tendof.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
74, 5, 6tendof 35559 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  U : T
--> T )
82, 3, 7syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  ->  U : T --> T )
9 ffn 5729 . . . 4  |-  ( U : T --> T  ->  U  Fn  T )
108, 9syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  ->  U  Fn  T )
11 simp2r 1023 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  ->  V  e.  E )
124, 5, 6tendof 35559 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E
)  ->  V : T
--> T )
132, 11, 12syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  ->  V : T --> T )
14 ffn 5729 . . . 4  |-  ( V : T --> T  ->  V  Fn  T )
1513, 14syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  ->  V  Fn  T )
16 eqfnfv 5973 . . 3  |-  ( ( U  Fn  T  /\  V  Fn  T )  ->  ( U  =  V  <->  A. f  e.  T  ( U `  f )  =  ( V `  f ) ) )
1710, 15, 16syl2anc 661 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  -> 
( U  =  V  <->  A. f  e.  T  ( U `  f )  =  ( V `  f ) ) )
181, 17mpbird 232 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E )  /\  A. f  e.  T  ( U `  f )  =  ( V `  f ) )  ->  U  =  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814    Fn wfn 5581   -->wf 5582   ` cfv 5586   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   TEndoctendo 35548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-tendo 35551
This theorem is referenced by:  tendoeq2  35570  tendoplcom  35578  tendoplass  35579  tendodi1  35580  tendodi2  35581  tendo0pl  35587  tendoipl  35593
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