Metamath Proof Explorer

Theorem dihordlem7

Description: Part of proof of Lemma N of Crawley p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014)

Ref Expression
Hypotheses dihordlem8.b 
|- B = ( Base ` K )
dihordlem8.l 
|- .<_ = ( le ` K )
dihordlem8.a 
|- A = ( Atoms ` K )
dihordlem8.h 
|- H = ( LHyp ` K )
dihordlem8.p 
|- P = ( ( oc ` K ) ` W )
dihordlem8.o 
|- O = ( h e. T |-> ( _I |` B ) )
dihordlem8.t 
|- T = ( ( LTrn ` K ) ` W )
dihordlem8.e 
|- E = ( ( TEndo ` K ) ` W )
dihordlem8.u 
|- U = ( ( DVecH ` K ) ` W )
dihordlem8.s 
|- .+ = ( +g ` U )
dihordlem8.g 
|- G = ( iota_ h e. T ( h ` P ) = R )
Assertion dihordlem7 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( f = ( ( s ` G ) o. g ) /\ O = s ) )

Proof

Step Hyp Ref Expression
1 dihordlem8.b 
 |-  B = ( Base ` K )
2 dihordlem8.l 
 |-  .<_ = ( le ` K )
3 dihordlem8.a 
 |-  A = ( Atoms ` K )
4 dihordlem8.h 
 |-  H = ( LHyp ` K )
5 dihordlem8.p 
 |-  P = ( ( oc ` K ) ` W )
6 dihordlem8.o 
 |-  O = ( h e. T |-> ( _I |` B ) )
7 dihordlem8.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 dihordlem8.e 
 |-  E = ( ( TEndo ` K ) ` W )
9 dihordlem8.u 
 |-  U = ( ( DVecH ` K ) ` W )
10 dihordlem8.s 
 |-  .+ = ( +g ` U )
11 dihordlem8.g 
 |-  G = ( iota_ h e. T ( h ` P ) = R )
12 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) )
13 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( K e. HL /\ W e. H ) )
14 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
15 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( R e. A /\ -. R .<_ W ) )
16 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> s e. E )
17 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> g e. T )
18 1 2 3 4 5 6 7 8 9 10 11 dihordlem6 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T ) ) -> ( <. ( s ` G ) , s >. .+ <. g , O >. ) = <. ( ( s ` G ) o. g ) , s >. )
19 13 14 15 16 17 18 syl122anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( <. ( s ` G ) , s >. .+ <. g , O >. ) = <. ( ( s ` G ) o. g ) , s >. )
20 12 19 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> <. f , O >. = <. ( ( s ` G ) o. g ) , s >. )
21 fvex 
 |-  ( s ` G ) e. _V
22 vex 
 |-  g e. _V
23 21 22 coex 
 |-  ( ( s ` G ) o. g ) e. _V
24 vex 
 |-  s e. _V
25 23 24 opth2 
 |-  ( <. f , O >. = <. ( ( s ` G ) o. g ) , s >. <-> ( f = ( ( s ` G ) o. g ) /\ O = s ) )
26 20 25 sylib 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( f = ( ( s ` G ) o. g ) /\ O = s ) )