Metamath Proof Explorer

Theorem cdlemg4b1

Description: TODO: FIX COMMENT. (Contributed by NM, 24-Apr-2013)

Ref Expression
Hypotheses cdlemg4.l 
|- .<_ = ( le ` K )
cdlemg4.a 
|- A = ( Atoms ` K )
cdlemg4.h 
|- H = ( LHyp ` K )
cdlemg4.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg4.r 
|- R = ( ( trL ` K ) ` W )
cdlemg4.j 
|- .\/ = ( join ` K )
cdlemg4b.v 
|- V = ( R ` G )
Assertion cdlemg4b1 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( P .\/ V ) = ( P .\/ ( G ` P ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg4.l 
 |-  .<_ = ( le ` K )
2 cdlemg4.a 
 |-  A = ( Atoms ` K )
3 cdlemg4.h 
 |-  H = ( LHyp ` K )
4 cdlemg4.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 cdlemg4.r 
 |-  R = ( ( trL ` K ) ` W )
6 cdlemg4.j 
 |-  .\/ = ( join ` K )
7 cdlemg4b.v 
 |-  V = ( R ` G )
8 eqid 
 |-  ( meet ` K ) = ( meet ` K )
9 1 6 8 2 3 4 5 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` G ) = ( ( P .\/ ( G ` P ) ) ( meet ` K ) W ) )
10 9 3com23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( R ` G ) = ( ( P .\/ ( G ` P ) ) ( meet ` K ) W ) )
11 7 10 syl5eq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> V = ( ( P .\/ ( G ` P ) ) ( meet ` K ) W ) )
12 11 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( P .\/ V ) = ( P .\/ ( ( P .\/ ( G ` P ) ) ( meet ` K ) W ) ) )
13 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( K e. HL /\ W e. H ) )
14 simp2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( P e. A /\ -. P .<_ W ) )
15 1 2 3 4 ltrnel 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
16 15 simpld 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( G ` P ) e. A )
17 16 3com23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( G ` P ) e. A )
18 eqid 
 |-  ( ( P .\/ ( G ` P ) ) ( meet ` K ) W ) = ( ( P .\/ ( G ` P ) ) ( meet ` K ) W )
19 1 6 8 2 3 18 cdleme0cp 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( G ` P ) e. A ) ) -> ( P .\/ ( ( P .\/ ( G ` P ) ) ( meet ` K ) W ) ) = ( P .\/ ( G ` P ) ) )
20 13 14 17 19 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( P .\/ ( ( P .\/ ( G ` P ) ) ( meet ` K ) W ) ) = ( P .\/ ( G ` P ) ) )
21 12 20 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( P .\/ V ) = ( P .\/ ( G ` P ) ) )