Metamath Proof Explorer

Theorem dih1dimb2

Description: Isomorphism H at an atom under W . (Contributed by NM, 27-Apr-2014)

Ref Expression
Hypotheses dih1dimb2.b 
|- B = ( Base ` K )
dih1dimb2.l 
|- .<_ = ( le ` K )
dih1dimb2.a 
|- A = ( Atoms ` K )
dih1dimb2.h 
|- H = ( LHyp ` K )
dih1dimb2.t 
|- T = ( ( LTrn ` K ) ` W )
dih1dimb2.o 
|- O = ( h e. T |-> ( _I |` B ) )
dih1dimb2.u 
|- U = ( ( DVecH ` K ) ` W )
dih1dimb2.i 
|- I = ( ( DIsoH ` K ) ` W )
dih1dimb2.n 
|- N = ( LSpan ` U )
Assertion dih1dimb2 
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) -> E. f e. T ( f =/= ( _I |` B ) /\ ( I ` Q ) = ( N ` { <. f , O >. } ) ) )

Proof

Step Hyp Ref Expression
1 dih1dimb2.b 
 |-  B = ( Base ` K )
2 dih1dimb2.l 
 |-  .<_ = ( le ` K )
3 dih1dimb2.a 
 |-  A = ( Atoms ` K )
4 dih1dimb2.h 
 |-  H = ( LHyp ` K )
5 dih1dimb2.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 dih1dimb2.o 
 |-  O = ( h e. T |-> ( _I |` B ) )
7 dih1dimb2.u 
 |-  U = ( ( DVecH ` K ) ` W )
8 dih1dimb2.i 
 |-  I = ( ( DIsoH ` K ) ` W )
9 dih1dimb2.n 
 |-  N = ( LSpan ` U )
10 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
11 2 3 4 5 10 cdlemf 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) -> E. f e. T ( ( ( trL ` K ) ` W ) ` f ) = Q )
12 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) /\ f e. T /\ ( ( ( trL ` K ) ` W ) ` f ) = Q ) -> ( ( ( trL ` K ) ` W ) ` f ) = Q )
13 simp1rl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) /\ f e. T /\ ( ( ( trL ` K ) ` W ) ` f ) = Q ) -> Q e. A )
14 12 13 eqeltrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) /\ f e. T /\ ( ( ( trL ` K ) ` W ) ` f ) = Q ) -> ( ( ( trL ` K ) ` W ) ` f ) e. A )
15 simp1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) /\ f e. T /\ ( ( ( trL ` K ) ` W ) ` f ) = Q ) -> ( K e. HL /\ W e. H ) )
16 simp2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) /\ f e. T /\ ( ( ( trL ` K ) ` W ) ` f ) = Q ) -> f e. T )
17 1 3 4 5 10 trlnidatb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ f e. T ) -> ( f =/= ( _I |` B ) <-> ( ( ( trL ` K ) ` W ) ` f ) e. A ) )
18 15 16 17 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) /\ f e. T /\ ( ( ( trL ` K ) ` W ) ` f ) = Q ) -> ( f =/= ( _I |` B ) <-> ( ( ( trL ` K ) ` W ) ` f ) e. A ) )
19 14 18 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) /\ f e. T /\ ( ( ( trL ` K ) ` W ) ` f ) = Q ) -> f =/= ( _I |` B ) )
20 12 fveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) /\ f e. T /\ ( ( ( trL ` K ) ` W ) ` f ) = Q ) -> ( I ` ( ( ( trL ` K ) ` W ) ` f ) ) = ( I ` Q ) )
21 1 4 5 10 6 7 8 9 dih1dimb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ f e. T ) -> ( I ` ( ( ( trL ` K ) ` W ) ` f ) ) = ( N ` { <. f , O >. } ) )
22 15 16 21 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) /\ f e. T /\ ( ( ( trL ` K ) ` W ) ` f ) = Q ) -> ( I ` ( ( ( trL ` K ) ` W ) ` f ) ) = ( N ` { <. f , O >. } ) )
23 20 22 eqtr3d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) /\ f e. T /\ ( ( ( trL ` K ) ` W ) ` f ) = Q ) -> ( I ` Q ) = ( N ` { <. f , O >. } ) )
24 19 23 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) /\ f e. T /\ ( ( ( trL ` K ) ` W ) ` f ) = Q ) -> ( f =/= ( _I |` B ) /\ ( I ` Q ) = ( N ` { <. f , O >. } ) ) )
25 24 3expia 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) /\ f e. T ) -> ( ( ( ( trL ` K ) ` W ) ` f ) = Q -> ( f =/= ( _I |` B ) /\ ( I ` Q ) = ( N ` { <. f , O >. } ) ) ) )
26 25 reximdva 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) -> ( E. f e. T ( ( ( trL ` K ) ` W ) ` f ) = Q -> E. f e. T ( f =/= ( _I |` B ) /\ ( I ` Q ) = ( N ` { <. f , O >. } ) ) ) )
27 11 26 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ Q .<_ W ) ) -> E. f e. T ( f =/= ( _I |` B ) /\ ( I ` Q ) = ( N ` { <. f , O >. } ) ) )