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Metamath Proof Explorer

Theorem tendoeq1

Description: Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013)

Ref Expression
Hypotheses tendof.h 
|- H = ( LHyp ` K )
tendof.t 
|- T = ( ( LTrn ` K ) ` W )
tendof.e 
|- E = ( ( TEndo ` K ) ` W )
Assertion tendoeq1 
|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> U = V )

Proof

Step Hyp Ref Expression
1 tendof.h 
 |-  H = ( LHyp ` K )
2 tendof.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 tendof.e 
 |-  E = ( ( TEndo ` K ) ` W )
4 simp3 
 |-  ( ( ( 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 ) )
5 simp1 
 |-  ( ( ( 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 ) )
6 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> U e. E )
7 1 2 3 tendof 
 |-  ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> U : T --> T )
8 5 6 7 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> U : T --> T )
9 8 ffnd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> U Fn T )
10 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> V e. E )
11 1 2 3 tendof 
 |-  ( ( ( K e. HL /\ W e. H ) /\ V e. E ) -> V : T --> T )
12 5 10 11 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> V : T --> T )
13 12 ffnd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> V Fn T )
14 eqfnfv 
 |-  ( ( U Fn T /\ V Fn T ) -> ( U = V <-> A. f e. T ( U ` f ) = ( V ` f ) ) )
15 9 13 14 syl2anc 
 |-  ( ( ( 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 ) ) )
16 4 15 mpbird 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E ) /\ A. f e. T ( U ` f ) = ( V ` f ) ) -> U = V )