Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemef50.b |
|- B = ( Base ` K ) |
2 |
|
cdlemef50.l |
|- .<_ = ( le ` K ) |
3 |
|
cdlemef50.j |
|- .\/ = ( join ` K ) |
4 |
|
cdlemef50.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdlemef50.a |
|- A = ( Atoms ` K ) |
6 |
|
cdlemef50.h |
|- H = ( LHyp ` K ) |
7 |
|
cdlemef50.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
8 |
|
cdlemef50.d |
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) |
9 |
|
cdlemefs50.e |
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) ) |
10 |
|
cdlemef50.f |
|- F = ( 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 ) ) |
11 |
|
eqid |
|- ( ( Q .\/ P ) ./\ W ) = ( ( Q .\/ P ) ./\ W ) |
12 |
|
eqid |
|- ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) = ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) |
13 |
|
eqid |
|- ( ( Q .\/ P ) ./\ ( ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) = ( ( Q .\/ P ) ./\ ( ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) |
14 |
|
eqid |
|- ( a e. B |-> if ( ( Q =/= P /\ -. a .<_ W ) , ( iota_ c e. B A. u e. A ( ( -. u .<_ W /\ ( u .\/ ( a ./\ W ) ) = a ) -> c = ( if ( u .<_ ( Q .\/ P ) , ( iota_ b e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( Q .\/ P ) ) -> b = ( ( Q .\/ P ) ./\ ( ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) ) ) , [_ u / v ]_ ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) ) .\/ ( a ./\ W ) ) ) ) , a ) ) = ( a e. B |-> if ( ( Q =/= P /\ -. a .<_ W ) , ( iota_ c e. B A. u e. A ( ( -. u .<_ W /\ ( u .\/ ( a ./\ W ) ) = a ) -> c = ( if ( u .<_ ( Q .\/ P ) , ( iota_ b e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( Q .\/ P ) ) -> b = ( ( Q .\/ P ) ./\ ( ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) .\/ ( ( u .\/ v ) ./\ W ) ) ) ) ) , [_ u / v ]_ ( ( v .\/ ( ( Q .\/ P ) ./\ W ) ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) ) .\/ ( a ./\ W ) ) ) ) , a ) ) |
15 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14
|
cdlemeg49lebilem |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ Y e. B ) ) -> ( X .<_ Y <-> ( F ` X ) .<_ ( F ` Y ) ) ) |