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Theorem tendoeq1 30326
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 957 . 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 955 . . . . 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 981 . . . . 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 30325 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  U : T
--> T )
82, 3, 7syl2anc 642 . . . 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 5389 . . . 4  |-  ( U : T --> T  ->  U  Fn  T )
108, 9syl 15 . . 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 982 . . . . 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 30325 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E
)  ->  V : T
--> T )
132, 11, 12syl2anc 642 . . . 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 5389 . . . 4  |-  ( V : T --> T  ->  V  Fn  T )
1513, 14syl 15 . . 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 5622 . . 3  |-  ( ( U  Fn  T  /\  V  Fn  T )  ->  ( U  =  V  <->  A. f  e.  T  ( U `  f )  =  ( V `  f ) ) )
1710, 15, 16syl2anc 642 . 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 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    Fn wfn 5250   -->wf 5251   ` cfv 5255   HLchlt 28913   LHypclh 29546   LTrncltrn 29663   TEndoctendo 30314
This theorem is referenced by:  tendoeq2  30336  tendoplcom  30344  tendoplass  30345  tendodi1  30346  tendodi2  30347  tendo0pl  30353  tendoipl  30359
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-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-map 6774  df-tendo 30317
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