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Metamath Proof Explorer

Theorem dihordlem7b

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 dihordlem7b 
|- ( ( ( 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 = 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 1 2 3 4 5 6 7 8 9 10 11 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 ) )
13 12 simpld 
 |-  ( ( ( 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 ) )
14 12 simprd 
 |-  ( ( ( 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 >. ) ) ) -> O = s )
15 14 fveq1d 
 |-  ( ( ( 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 >. ) ) ) -> ( O ` G ) = ( s ` G ) )
16 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 ) )
17 2 3 4 5 lhpocnel2 
 |-  ( ( K e. HL /\ W e. H ) -> ( P e. A /\ -. P .<_ W ) )
18 17 3ad2ant1 
 |-  ( ( ( 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 >. ) ) ) -> ( P e. A /\ -. P .<_ W ) )
19 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 ) )
20 2 3 4 7 11 ltrniotacl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) -> G e. T )
21 16 18 19 20 syl3anc 
 |-  ( ( ( 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 )
22 6 1 tendo02 
 |-  ( G e. T -> ( O ` G ) = ( _I |` B ) )
23 21 22 syl 
 |-  ( ( ( 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 >. ) ) ) -> ( O ` G ) = ( _I |` B ) )
24 15 23 eqtr3d 
 |-  ( ( ( 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 ) = ( _I |` B ) )
25 24 coeq1d 
 |-  ( ( ( 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 ) o. g ) = ( ( _I |` B ) o. g ) )
26 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 )
27 1 4 7 ltrn1o 
 |-  ( ( ( K e. HL /\ W e. H ) /\ g e. T ) -> g : B -1-1-onto-> B )
28 16 26 27 syl2anc 
 |-  ( ( ( 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 : B -1-1-onto-> B )
29 f1of 
 |-  ( g : B -1-1-onto-> B -> g : B --> B )
30 fcoi2 
 |-  ( g : B --> B -> ( ( _I |` B ) o. g ) = g )
31 28 29 30 3syl 
 |-  ( ( ( 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 >. ) ) ) -> ( ( _I |` B ) o. g ) = g )
32 13 25 31 3eqtrd 
 |-  ( ( ( 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 = g )
33 32 14 jca 
 |-  ( ( ( 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 = g /\ O = s ) )