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Metamath Proof Explorer

Theorem cdlemn7

Description: Part of proof of Lemma N of Crawley p. 121 line 36. (Contributed by NM, 26-Feb-2014)

Ref Expression
Hypotheses cdlemn8.b 
|- B = ( Base ` K )
cdlemn8.l 
|- .<_ = ( le ` K )
cdlemn8.a 
|- A = ( Atoms ` K )
cdlemn8.h 
|- H = ( LHyp ` K )
cdlemn8.p 
|- P = ( ( oc ` K ) ` W )
cdlemn8.o 
|- O = ( h e. T |-> ( _I |` B ) )
cdlemn8.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemn8.e 
|- E = ( ( TEndo ` K ) ` W )
cdlemn8.u 
|- U = ( ( DVecH ` K ) ` W )
cdlemn8.s 
|- .+ = ( +g ` U )
cdlemn8.f 
|- F = ( iota_ h e. T ( h ` P ) = Q )
cdlemn8.g 
|- G = ( iota_ h e. T ( h ` P ) = R )
Assertion cdlemn7 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( G = ( ( s ` F ) o. g ) /\ ( _I |` T ) = s ) )

Proof

Step Hyp Ref Expression
1 cdlemn8.b 
 |-  B = ( Base ` K )
2 cdlemn8.l 
 |-  .<_ = ( le ` K )
3 cdlemn8.a 
 |-  A = ( Atoms ` K )
4 cdlemn8.h 
 |-  H = ( LHyp ` K )
5 cdlemn8.p 
 |-  P = ( ( oc ` K ) ` W )
6 cdlemn8.o 
 |-  O = ( h e. T |-> ( _I |` B ) )
7 cdlemn8.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemn8.e 
 |-  E = ( ( TEndo ` K ) ` W )
9 cdlemn8.u 
 |-  U = ( ( DVecH ` K ) ` W )
10 cdlemn8.s 
 |-  .+ = ( +g ` U )
11 cdlemn8.f 
 |-  F = ( iota_ h e. T ( h ` P ) = Q )
12 cdlemn8.g 
 |-  G = ( iota_ h e. T ( h ` P ) = R )
13 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) )
14 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( K e. HL /\ W e. H ) )
15 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
16 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( R e. A /\ -. R .<_ W ) )
17 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> s e. E )
18 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> g e. T )
19 1 2 3 4 5 6 7 8 9 10 11 cdlemn6 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T ) ) -> ( <. ( s ` F ) , s >. .+ <. g , O >. ) = <. ( ( s ` F ) o. g ) , s >. )
20 14 15 16 17 18 19 syl122anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( <. ( s ` F ) , s >. .+ <. g , O >. ) = <. ( ( s ` F ) o. g ) , s >. )
21 13 20 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> <. G , ( _I |` T ) >. = <. ( ( s ` F ) o. g ) , s >. )
22 fvex 
 |-  ( s ` F ) e. _V
23 vex 
 |-  g e. _V
24 22 23 coex 
 |-  ( ( s ` F ) o. g ) e. _V
25 vex 
 |-  s e. _V
26 24 25 opth2 
 |-  ( <. G , ( _I |` T ) >. = <. ( ( s ` F ) o. g ) , s >. <-> ( G = ( ( s ` F ) o. g ) /\ ( _I |` T ) = s ) )
27 21 26 sylib 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( s e. E /\ g e. T /\ <. G , ( _I |` T ) >. = ( <. ( s ` F ) , s >. .+ <. g , O >. ) ) ) -> ( G = ( ( s ` F ) o. g ) /\ ( _I |` T ) = s ) )