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Metamath Proof Explorer

Theorem dihord10

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

Ref Expression
Hypotheses dihjust.b 
|- B = ( Base ` K )
dihjust.l 
|- .<_ = ( le ` K )
dihjust.j 
|- .\/ = ( join ` K )
dihjust.m 
|- ./\ = ( meet ` K )
dihjust.a 
|- A = ( Atoms ` K )
dihjust.h 
|- H = ( LHyp ` K )
dihjust.i 
|- I = ( ( DIsoB ` K ) ` W )
dihjust.J 
|- J = ( ( DIsoC ` K ) ` W )
dihjust.u 
|- U = ( ( DVecH ` K ) ` W )
dihjust.s 
|- .(+) = ( LSSum ` U )
dihord2c.t 
|- T = ( ( LTrn ` K ) ` W )
dihord2c.r 
|- R = ( ( trL ` K ) ` W )
dihord2c.o 
|- O = ( h e. T |-> ( _I |` B ) )
dihord2.p 
|- P = ( ( oc ` K ) ` W )
dihord2.e 
|- E = ( ( TEndo ` K ) ` W )
dihord2.d 
|- .+ = ( +g ` U )
dihord2.g 
|- G = ( iota_ h e. T ( h ` P ) = N )
Assertion dihord10 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ ( Y ./\ W ) /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( R ` f ) .<_ ( Y ./\ W ) )

Proof

Step Hyp Ref Expression
1 dihjust.b 
 |-  B = ( Base ` K )
2 dihjust.l 
 |-  .<_ = ( le ` K )
3 dihjust.j 
 |-  .\/ = ( join ` K )
4 dihjust.m 
 |-  ./\ = ( meet ` K )
5 dihjust.a 
 |-  A = ( Atoms ` K )
6 dihjust.h 
 |-  H = ( LHyp ` K )
7 dihjust.i 
 |-  I = ( ( DIsoB ` K ) ` W )
8 dihjust.J 
 |-  J = ( ( DIsoC ` K ) ` W )
9 dihjust.u 
 |-  U = ( ( DVecH ` K ) ` W )
10 dihjust.s 
 |-  .(+) = ( LSSum ` U )
11 dihord2c.t 
 |-  T = ( ( LTrn ` K ) ` W )
12 dihord2c.r 
 |-  R = ( ( trL ` K ) ` W )
13 dihord2c.o 
 |-  O = ( h e. T |-> ( _I |` B ) )
14 dihord2.p 
 |-  P = ( ( oc ` K ) ` W )
15 dihord2.e 
 |-  E = ( ( TEndo ` K ) ` W )
16 dihord2.d 
 |-  .+ = ( +g ` U )
17 dihord2.g 
 |-  G = ( iota_ h e. T ( h ` P ) = N )
18 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ ( Y ./\ W ) /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( K e. HL /\ W e. H ) )
19 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ ( Y ./\ W ) /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
20 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ ( Y ./\ W ) /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( N e. A /\ -. N .<_ W ) )
21 simp31l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ ( Y ./\ W ) /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> s e. E )
22 simp31r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ ( Y ./\ W ) /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> g e. T )
23 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ ( Y ./\ W ) /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) )
24 1 2 5 6 14 13 11 15 9 16 17 dihordlem7b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( f = g /\ O = s ) )
25 24 simpld 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> f = g )
26 18 19 20 21 22 23 25 syl123anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ ( Y ./\ W ) /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> f = g )
27 26 fveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ ( Y ./\ W ) /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( R ` f ) = ( R ` g ) )
28 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ ( Y ./\ W ) /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( R ` g ) .<_ ( Y ./\ W ) )
29 27 28 eqbrtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ ( ( s e. E /\ g e. T ) /\ ( R ` g ) .<_ ( Y ./\ W ) /\ <. f , O >. = ( <. ( s ` G ) , s >. .+ <. g , O >. ) ) ) -> ( R ` f ) .<_ ( Y ./\ W ) )