Metamath Proof Explorer

Theorem cdlemg7fvN

Description: Value of a translation composition in terms of an associated atom. (Contributed by NM, 28-Apr-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemg7fv.b 
|- B = ( Base ` K )
cdlemg7fv.l 
|- .<_ = ( le ` K )
cdlemg7fv.j 
|- .\/ = ( join ` K )
cdlemg7fv.m 
|- ./\ = ( meet ` K )
cdlemg7fv.a 
|- A = ( Atoms ` K )
cdlemg7fv.h 
|- H = ( LHyp ` K )
cdlemg7fv.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion cdlemg7fvN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( F ` ( G ` X ) ) = ( ( F ` ( G ` P ) ) .\/ ( X ./\ W ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg7fv.b 
 |-  B = ( Base ` K )
2 cdlemg7fv.l 
 |-  .<_ = ( le ` K )
3 cdlemg7fv.j 
 |-  .\/ = ( join ` K )
4 cdlemg7fv.m 
 |-  ./\ = ( meet ` K )
5 cdlemg7fv.a 
 |-  A = ( Atoms ` K )
6 cdlemg7fv.h 
 |-  H = ( LHyp ` K )
7 cdlemg7fv.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( K e. HL /\ W e. H ) )
9 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> G e. T )
10 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( P e. A /\ -. P .<_ W ) )
11 2 5 6 7 ltrnel 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
12 8 9 10 11 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
13 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( X e. B /\ -. X .<_ W ) )
14 2 5 6 7 1 cdlemg7fvbwN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ G e. T ) -> ( ( G ` X ) e. B /\ -. ( G ` X ) .<_ W ) )
15 8 13 9 14 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( G ` X ) e. B /\ -. ( G ` X ) .<_ W ) )
16 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> F e. T )
17 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( P .\/ ( X ./\ W ) ) = X )
18 6 7 2 3 5 4 1 cdlemg2fv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( G ` X ) = ( ( G ` P ) .\/ ( X ./\ W ) ) )
19 8 10 13 9 17 18 syl122anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( G ` X ) = ( ( G ` P ) .\/ ( X ./\ W ) ) )
20 19 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( G ` X ) ./\ W ) = ( ( ( G ` P ) .\/ ( X ./\ W ) ) ./\ W ) )
21 simp2rl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> X e. B )
22 1 2 3 4 5 6 lhpelim 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) /\ X e. B ) -> ( ( ( G ` P ) .\/ ( X ./\ W ) ) ./\ W ) = ( X ./\ W ) )
23 8 12 21 22 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( ( G ` P ) .\/ ( X ./\ W ) ) ./\ W ) = ( X ./\ W ) )
24 20 23 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( G ` X ) ./\ W ) = ( X ./\ W ) )
25 24 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( G ` P ) .\/ ( ( G ` X ) ./\ W ) ) = ( ( G ` P ) .\/ ( X ./\ W ) ) )
26 25 19 eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( G ` P ) .\/ ( ( G ` X ) ./\ W ) ) = ( G ` X ) )
27 6 7 2 3 5 4 1 cdlemg2fv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) /\ ( ( G ` X ) e. B /\ -. ( G ` X ) .<_ W ) ) /\ ( F e. T /\ ( ( G ` P ) .\/ ( ( G ` X ) ./\ W ) ) = ( G ` X ) ) ) -> ( F ` ( G ` X ) ) = ( ( F ` ( G ` P ) ) .\/ ( ( G ` X ) ./\ W ) ) )
28 8 12 15 16 26 27 syl122anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( F ` ( G ` X ) ) = ( ( F ` ( G ` P ) ) .\/ ( ( G ` X ) ./\ W ) ) )
29 24 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( F ` ( G ` P ) ) .\/ ( ( G ` X ) ./\ W ) ) = ( ( F ` ( G ` P ) ) .\/ ( X ./\ W ) ) )
30 28 29 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( F ` ( G ` X ) ) = ( ( F ` ( G ` P ) ) .\/ ( X ./\ W ) ) )