This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem tendo0mul

Description: Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 1-Aug-2013)

Ref Expression
Hypotheses tendoid0.b 
|- B = ( Base ` K )
tendoid0.h 
|- H = ( LHyp ` K )
tendoid0.t 
|- T = ( ( LTrn ` K ) ` W )
tendoid0.e 
|- E = ( ( TEndo ` K ) ` W )
tendoid0.o 
|- O = ( f e. T |-> ( _I |` B ) )
Assertion tendo0mul 
|- ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( O o. U ) = O )

Proof

Step Hyp Ref Expression
1 tendoid0.b 
 |-  B = ( Base ` K )
2 tendoid0.h 
 |-  H = ( LHyp ` K )
3 tendoid0.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 tendoid0.e 
 |-  E = ( ( TEndo ` K ) ` W )
5 tendoid0.o 
 |-  O = ( f e. T |-> ( _I |` B ) )
6 1 2 3 cdlemftr0 
 |-  ( ( K e. HL /\ W e. H ) -> E. g e. T g =/= ( _I |` B ) )
7 6 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> E. g e. T g =/= ( _I |` B ) )
8 simpll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( K e. HL /\ W e. H ) )
9 1 2 3 4 5 tendo0cl 
 |-  ( ( K e. HL /\ W e. H ) -> O e. E )
10 9 ad2antrr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> O e. E )
11 simplr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> U e. E )
12 2 4 tendococl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ O e. E /\ U e. E ) -> ( O o. U ) e. E )
13 8 10 11 12 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( O o. U ) e. E )
14 simprl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> g e. T )
15 2 3 4 tendocl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ g e. T ) -> ( U ` g ) e. T )
16 8 11 14 15 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( U ` g ) e. T )
17 5 1 tendo02 
 |-  ( ( U ` g ) e. T -> ( O ` ( U ` g ) ) = ( _I |` B ) )
18 16 17 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( O ` ( U ` g ) ) = ( _I |` B ) )
19 2 3 4 tendocoval 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( O e. E /\ U e. E ) /\ g e. T ) -> ( ( O o. U ) ` g ) = ( O ` ( U ` g ) ) )
20 8 10 11 14 19 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( ( O o. U ) ` g ) = ( O ` ( U ` g ) ) )
21 5 1 tendo02 
 |-  ( g e. T -> ( O ` g ) = ( _I |` B ) )
22 21 ad2antrl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( O ` g ) = ( _I |` B ) )
23 18 20 22 3eqtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( ( O o. U ) ` g ) = ( O ` g ) )
24 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( g e. T /\ g =/= ( _I |` B ) ) )
25 1 2 3 4 tendocan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( O o. U ) e. E /\ O e. E /\ ( ( O o. U ) ` g ) = ( O ` g ) ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( O o. U ) = O )
26 8 13 10 23 24 25 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ U e. E ) /\ ( g e. T /\ g =/= ( _I |` B ) ) ) -> ( O o. U ) = O )
27 7 26 rexlimddv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ U e. E ) -> ( O o. U ) = O )