Metamath Proof Explorer

Theorem cdlemfnid

Description: cdlemf with additional constraint of non-identity. (Contributed by NM, 20-Jun-2013)

Ref Expression
Hypotheses cdlemfnid.b 
|- B = ( Base ` K )
cdlemfnid.l 
|- .<_ = ( le ` K )
cdlemfnid.a 
|- A = ( Atoms ` K )
cdlemfnid.h 
|- H = ( LHyp ` K )
cdlemfnid.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemfnid.r 
|- R = ( ( trL ` K ) ` W )
Assertion cdlemfnid 
|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) -> E. f e. T ( ( R ` f ) = U /\ f =/= ( _I |` B ) ) )

Proof

Step Hyp Ref Expression
1 cdlemfnid.b 
 |-  B = ( Base ` K )
2 cdlemfnid.l 
 |-  .<_ = ( le ` K )
3 cdlemfnid.a 
 |-  A = ( Atoms ` K )
4 cdlemfnid.h 
 |-  H = ( LHyp ` K )
5 cdlemfnid.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 cdlemfnid.r 
 |-  R = ( ( trL ` K ) ` W )
7 2 3 4 5 6 cdlemf 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) -> E. f e. T ( R ` f ) = U )
8 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ f e. T /\ ( R ` f ) = U ) -> ( R ` f ) = U )
9 simp1rl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ f e. T /\ ( R ` f ) = U ) -> U e. A )
10 8 9 eqeltrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ f e. T /\ ( R ` f ) = U ) -> ( R ` f ) e. A )
11 simp1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ f e. T /\ ( R ` f ) = U ) -> ( K e. HL /\ W e. H ) )
12 simp2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ f e. T /\ ( R ` f ) = U ) -> f e. T )
13 1 3 4 5 6 trlnidatb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ f e. T ) -> ( f =/= ( _I |` B ) <-> ( R ` f ) e. A ) )
14 11 12 13 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ f e. T /\ ( R ` f ) = U ) -> ( f =/= ( _I |` B ) <-> ( R ` f ) e. A ) )
15 10 14 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ f e. T /\ ( R ` f ) = U ) -> f =/= ( _I |` B ) )
16 8 15 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ f e. T /\ ( R ` f ) = U ) -> ( ( R ` f ) = U /\ f =/= ( _I |` B ) ) )
17 16 3expia 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) /\ f e. T ) -> ( ( R ` f ) = U -> ( ( R ` f ) = U /\ f =/= ( _I |` B ) ) ) )
18 17 reximdva 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) -> ( E. f e. T ( R ` f ) = U -> E. f e. T ( ( R ` f ) = U /\ f =/= ( _I |` B ) ) ) )
19 7 18 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) ) -> E. f e. T ( ( R ` f ) = U /\ f =/= ( _I |` B ) ) )