Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemefs32.b |
|- B = ( Base ` K ) |
2 |
|
cdlemefs32.l |
|- .<_ = ( le ` K ) |
3 |
|
cdlemefs32.j |
|- .\/ = ( join ` K ) |
4 |
|
cdlemefs32.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdlemefs32.a |
|- A = ( Atoms ` K ) |
6 |
|
cdlemefs32.h |
|- H = ( LHyp ` K ) |
7 |
|
cdlemefs32.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
8 |
|
cdlemefs32.d |
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) |
9 |
|
cdlemefs32.e |
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) ) |
10 |
|
cdlemefs32.i |
|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) |
11 |
|
cdlemefs32.n |
|- N = if ( s .<_ ( P .\/ Q ) , I , C ) |
12 |
|
cdleme29fs.o |
|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) ) |
13 |
|
cdleme29fs.f |
|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) ) |
14 |
|
breq1 |
|- ( s = R -> ( s .<_ ( P .\/ Q ) <-> R .<_ ( P .\/ Q ) ) ) |
15 |
|
simp1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) |
16 |
|
simp3l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> s e. A ) |
17 |
|
simp3rl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> -. s .<_ W ) |
18 |
16 17
|
jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> ( s e. A /\ -. s .<_ W ) ) |
19 |
|
simp3rr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> s .<_ ( P .\/ Q ) ) |
20 |
|
simp2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> P =/= Q ) |
21 |
1 2 3 4 5 6 7 8 9 10 11
|
cdlemefs27cl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B ) |
22 |
15 18 19 20 21
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> N e. B ) |
23 |
1 2 3 4 5 6 7 8 9 10 11
|
cdlemefs32snb |
|- ( ( ( ( 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 e. B ) |
24 |
1 2 3 4 5 6 14 22 23 12 13
|
cdlemefrs32fva1 |
|- ( ( ( ( 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 ) |