Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemefr27.b |
|- B = ( Base ` K ) |
2 |
|
cdlemefr27.l |
|- .<_ = ( le ` K ) |
3 |
|
cdlemefr27.j |
|- .\/ = ( join ` K ) |
4 |
|
cdlemefr27.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdlemefr27.a |
|- A = ( Atoms ` K ) |
6 |
|
cdlemefr27.h |
|- H = ( LHyp ` K ) |
7 |
|
cdlemefr27.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
8 |
|
cdlemefr27.c |
|- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) |
9 |
|
cdlemefr27.n |
|- N = if ( s .<_ ( P .\/ Q ) , I , C ) |
10 |
|
cdleme29fr.o |
|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) ) |
11 |
|
cdleme29fr.f |
|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) ) |
12 |
|
cdleme43frv.x |
|- X = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) |
13 |
1 2 3 4 5 6 7 8 9 10 11
|
cdlemefr32fva1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( F ` R ) = [_ R / s ]_ N ) |
14 |
|
simp2rl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R e. A ) |
15 |
|
simp3 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> -. R .<_ ( P .\/ Q ) ) |
16 |
8 9 12
|
cdleme31sn2 |
|- ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = X ) |
17 |
14 15 16
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = X ) |
18 |
13 17
|
eqtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( F ` R ) = X ) |