Metamath Proof Explorer

Theorem cdlemg2fvlem

Description: Lemma for cdlemg2fv . (Contributed by NM, 23-Apr-2013)

Ref Expression
Hypotheses cdlemg2.b 
|- B = ( Base ` K )
cdlemg2.l 
|- .<_ = ( le ` K )
cdlemg2.j 
|- .\/ = ( join ` K )
cdlemg2.m 
|- ./\ = ( meet ` K )
cdlemg2.a 
|- A = ( Atoms ` K )
cdlemg2.h 
|- H = ( LHyp ` K )
cdlemg2.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg2ex.u 
|- U = ( ( p .\/ q ) ./\ W )
cdlemg2ex.d 
|- D = ( ( t .\/ U ) ./\ ( q .\/ ( ( p .\/ t ) ./\ W ) ) )
cdlemg2ex.e 
|- E = ( ( p .\/ q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemg2ex.g 
|- G = ( x e. B |-> if ( ( p =/= q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( p .\/ q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( p .\/ q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
Assertion cdlemg2fvlem 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( F ` X ) = ( ( F ` P ) .\/ ( X ./\ W ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg2.b 
 |-  B = ( Base ` K )
2 cdlemg2.l 
 |-  .<_ = ( le ` K )
3 cdlemg2.j 
 |-  .\/ = ( join ` K )
4 cdlemg2.m 
 |-  ./\ = ( meet ` K )
5 cdlemg2.a 
 |-  A = ( Atoms ` K )
6 cdlemg2.h 
 |-  H = ( LHyp ` K )
7 cdlemg2.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemg2ex.u 
 |-  U = ( ( p .\/ q ) ./\ W )
9 cdlemg2ex.d 
 |-  D = ( ( t .\/ U ) ./\ ( q .\/ ( ( p .\/ t ) ./\ W ) ) )
10 cdlemg2ex.e 
 |-  E = ( ( p .\/ q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
11 cdlemg2ex.g 
 |-  G = ( x e. B |-> if ( ( p =/= q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( p .\/ q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( p .\/ q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
12 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( K e. HL /\ W e. H ) )
13 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> F e. T )
14 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( X e. B /\ -. X .<_ W ) )
15 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( P e. A /\ -. P .<_ W ) )
16 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( P .\/ ( X ./\ W ) ) = X )
17 15 16 jca 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( ( P e. A /\ -. P .<_ W ) /\ ( P .\/ ( X ./\ W ) ) = X ) )
18 fveq1 
 |-  ( F = G -> ( F ` X ) = ( G ` X ) )
19 fveq1 
 |-  ( F = G -> ( F ` P ) = ( G ` P ) )
20 19 oveq1d 
 |-  ( F = G -> ( ( F ` P ) .\/ ( X ./\ W ) ) = ( ( G ` P ) .\/ ( X ./\ W ) ) )
21 18 20 eqeq12d 
 |-  ( F = G -> ( ( F ` X ) = ( ( F ` P ) .\/ ( X ./\ W ) ) <-> ( G ` X ) = ( ( G ` P ) .\/ ( X ./\ W ) ) ) )
22 1 2 3 4 5 6 8 9 10 11 cdleme48fvg 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ -. p .<_ W ) /\ ( q e. A /\ -. q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( G ` X ) = ( ( G ` P ) .\/ ( X ./\ W ) ) )
23 22 3expb 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ -. p .<_ W ) /\ ( q e. A /\ -. q .<_ W ) ) /\ ( ( X e. B /\ -. X .<_ W ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( P .\/ ( X ./\ W ) ) = X ) ) ) -> ( G ` X ) = ( ( G ` P ) .\/ ( X ./\ W ) ) )
24 1 2 3 4 5 6 7 8 9 10 11 21 23 cdlemg2ce 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( X e. B /\ -. X .<_ W ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( P .\/ ( X ./\ W ) ) = X ) ) ) -> ( F ` X ) = ( ( F ` P ) .\/ ( X ./\ W ) ) )
25 12 13 14 17 24 syl112anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ ( P .\/ ( X ./\ W ) ) = X ) ) -> ( F ` X ) = ( ( F ` P ) .\/ ( X ./\ W ) ) )