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Metamath Proof Explorer

Theorem cdlemd8

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 1-Jun-2012)

Ref Expression
Hypotheses cdlemd4.l 
|- .<_ = ( le ` K )
cdlemd4.j 
|- .\/ = ( join ` K )
cdlemd4.a 
|- A = ( Atoms ` K )
cdlemd4.h 
|- H = ( LHyp ` K )
cdlemd4.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion cdlemd8 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> ( F ` R ) = ( G ` R ) )

Proof

Step Hyp Ref Expression
1 cdlemd4.l 
 |-  .<_ = ( le ` K )
2 cdlemd4.j 
 |-  .\/ = ( join ` K )
3 cdlemd4.a 
 |-  A = ( Atoms ` K )
4 cdlemd4.h 
 |-  H = ( LHyp ` K )
5 cdlemd4.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 simp3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> ( F ` P ) = P )
7 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> ( K e. HL /\ W e. H ) )
8 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> F e. T )
9 simp2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> ( P e. A /\ -. P .<_ W ) )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 1 3 4 5 ltrnideq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` P ) = P ) )
12 7 8 9 11 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` P ) = P ) )
13 6 12 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> F = ( _I |` ( Base ` K ) ) )
14 13 fveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> ( F ` R ) = ( ( _I |` ( Base ` K ) ) ` R ) )
15 simp3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> ( F ` P ) = ( G ` P ) )
16 15 6 eqtr3d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> ( G ` P ) = P )
17 simp12r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> G e. T )
18 10 1 3 4 5 ltrnideq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( G = ( _I |` ( Base ` K ) ) <-> ( G ` P ) = P ) )
19 7 17 9 18 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> ( G = ( _I |` ( Base ` K ) ) <-> ( G ` P ) = P ) )
20 16 19 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> G = ( _I |` ( Base ` K ) ) )
21 20 fveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> ( G ` R ) = ( ( _I |` ( Base ` K ) ) ` R ) )
22 14 21 eqtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ R e. A ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) = ( G ` P ) /\ ( F ` P ) = P ) ) -> ( F ` R ) = ( G ` R ) )