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Theorem tendoeq1 31561
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 959 . 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 957 . . . . 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 983 . . . . 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 31560 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  U : T
--> T )
82, 3, 7syl2anc 643 . . . 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 5591 . . . 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 984 . . . . 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 31560 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E
)  ->  V : T
--> T )
132, 11, 12syl2anc 643 . . . 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 5591 . . . 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 5827 . . 3  |-  ( ( U  Fn  T  /\  V  Fn  T )  ->  ( U  =  V  <->  A. f  e.  T  ( U `  f )  =  ( V `  f ) ) )
1710, 15, 16syl2anc 643 . 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 224 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    Fn wfn 5449   -->wf 5450   ` cfv 5454   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   TEndoctendo 31549
This theorem is referenced by:  tendoeq2  31571  tendoplcom  31579  tendoplass  31580  tendodi1  31581  tendodi2  31582  tendo0pl  31588  tendoipl  31594
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-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-map 7020  df-tendo 31552
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