# 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 ) )`