Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemef44.b |
|- B = ( Base ` K ) |
2 |
|
cdlemef44.l |
|- .<_ = ( le ` K ) |
3 |
|
cdlemef44.j |
|- .\/ = ( join ` K ) |
4 |
|
cdlemef44.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdlemef44.a |
|- A = ( Atoms ` K ) |
6 |
|
cdlemef44.h |
|- H = ( LHyp ` K ) |
7 |
|
cdlemef44.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
8 |
|
cdlemef44.d |
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) |
9 |
|
cdlemef44.o |
|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) |
10 |
|
cdlemef44.f |
|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) ) |
11 |
|
cdlemefs44.e |
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) ) |
12 |
|
cdlemefs44.i |
|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) |
13 |
|
eqid |
|- if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D ) = if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D ) |
14 |
|
eqid |
|- ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) |
15 |
|
eqid |
|- ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) |
16 |
1 2 3 4 5 6 7 8 11 12 13 9 10 14 15
|
cdlemefs31fv1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F ` R ) = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) ) |
17 |
|
simp22l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> R e. A ) |
18 |
|
simp23l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. A ) |
19 |
8 11 14 15
|
cdleme31sde |
|- ( ( R e. A /\ S e. A ) -> [_ R / s ]_ [_ S / t ]_ E = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) ) |
20 |
17 18 19
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> [_ R / s ]_ [_ S / t ]_ E = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) ) |
21 |
16 20
|
eqtr4d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F ` R ) = [_ R / s ]_ [_ S / t ]_ E ) |