Metamath Proof Explorer

Theorem cdlemg4b2

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 cdlemg4b2 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( ( G ` 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 ) -> ( ( G ` P ) .\/ V ) = ( ( G ` 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 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> P e. A )
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 3com23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
17 eqid 
 |-  ( ( P .\/ ( G ` P ) ) ( meet ` K ) W ) = ( ( P .\/ ( G ` P ) ) ( meet ` K ) W )
18 1 6 8 2 3 17 cdleme0cq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) ) ) -> ( ( G ` P ) .\/ ( ( P .\/ ( G ` P ) ) ( meet ` K ) W ) ) = ( P .\/ ( G ` P ) ) )
19 13 14 16 18 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( ( G ` P ) .\/ ( ( P .\/ ( G ` P ) ) ( meet ` K ) W ) ) = ( P .\/ ( G ` P ) ) )
20 12 19 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( ( G ` P ) .\/ V ) = ( P .\/ ( G ` P ) ) )