Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme26.b |
|- B = ( Base ` K ) |
2 |
|
cdleme26.l |
|- .<_ = ( le ` K ) |
3 |
|
cdleme26.j |
|- .\/ = ( join ` K ) |
4 |
|
cdleme26.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdleme26.a |
|- A = ( Atoms ` K ) |
6 |
|
cdleme26.h |
|- H = ( LHyp ` K ) |
7 |
|
cdleme27.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
8 |
|
cdleme27.f |
|- F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) |
9 |
|
cdleme27.z |
|- Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) ) |
10 |
|
cdleme27.n |
|- N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) ) |
11 |
|
cdleme27.d |
|- D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) ) |
12 |
|
cdleme27.c |
|- C = if ( s .<_ ( P .\/ Q ) , D , F ) |
13 |
|
cdleme27.g |
|- G = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) |
14 |
|
cdleme27.o |
|- O = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) |
15 |
|
cdleme27.e |
|- E = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) ) |
16 |
|
cdleme27.y |
|- Y = if ( t .<_ ( P .\/ Q ) , E , G ) |
17 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
|
cdleme27b |
|- ( s = t -> C = Y ) |
18 |
17
|
oveq1d |
|- ( s = t -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) ) |
19 |
18
|
adantl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s = t ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) ) |
20 |
|
simpl11 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( K e. HL /\ W e. H ) ) |
21 |
|
simpl12 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( P e. A /\ -. P .<_ W ) ) |
22 |
|
simpl13 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( Q e. A /\ -. Q .<_ W ) ) |
23 |
|
simpl21 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> P =/= Q ) |
24 |
|
simpl22 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( s e. A /\ -. s .<_ W ) ) |
25 |
|
simpl23 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( t e. A /\ -. t .<_ W ) ) |
26 |
|
simpr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> s =/= t ) |
27 |
|
simpl31 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( s .\/ ( X ./\ W ) ) = X ) |
28 |
|
simpl32 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( t .\/ ( X ./\ W ) ) = X ) |
29 |
27 28
|
jca |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) ) |
30 |
|
simpl33 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( X e. B /\ -. X .<_ W ) ) |
31 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
|
cdleme28b |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) ) |
32 |
20 21 22 23 24 25 26 29 30 31
|
syl333anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) /\ s =/= t ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) ) |
33 |
19 32
|
pm2.61dane |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) ) |