Metamath Proof Explorer

Theorem cdleml2N

Description: Part of proof of Lemma L of Crawley p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdleml1.b 
|- B = ( Base ` K )
cdleml1.h 
|- H = ( LHyp ` K )
cdleml1.t 
|- T = ( ( LTrn ` K ) ` W )
cdleml1.r 
|- R = ( ( trL ` K ) ` W )
cdleml1.e 
|- E = ( ( TEndo ` K ) ` W )
Assertion cdleml2N 
|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> E. s e. E ( s ` ( U ` f ) ) = ( V ` f ) )

Proof

Step Hyp Ref Expression
1 cdleml1.b 
 |-  B = ( Base ` K )
2 cdleml1.h 
 |-  H = ( LHyp ` K )
3 cdleml1.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 cdleml1.r 
 |-  R = ( ( trL ` K ) ` W )
5 cdleml1.e 
 |-  E = ( ( TEndo ` K ) ` W )
6 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( K e. HL /\ W e. H ) )
7 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> U e. E )
8 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> f e. T )
9 2 3 5 tendocl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ f e. T ) -> ( U ` f ) e. T )
10 6 7 8 9 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( U ` f ) e. T )
11 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> V e. E )
12 2 3 5 tendocl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ V e. E /\ f e. T ) -> ( V ` f ) e. T )
13 6 11 8 12 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( V ` f ) e. T )
14 1 2 3 4 5 cdleml1N 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> ( R ` ( U ` f ) ) = ( R ` ( V ` f ) ) )
15 2 3 4 5 cdlemk 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( U ` f ) e. T /\ ( V ` f ) e. T ) /\ ( R ` ( U ` f ) ) = ( R ` ( V ` f ) ) ) -> E. s e. E ( s ` ( U ` f ) ) = ( V ` f ) )
16 6 10 13 14 15 syl121anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ V e. E /\ f e. T ) /\ ( f =/= ( _I |` B ) /\ ( U ` f ) =/= ( _I |` B ) /\ ( V ` f ) =/= ( _I |` B ) ) ) -> E. s e. E ( s ` ( U ` f ) ) = ( V ` f ) )