Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemef46.b |
|- B = ( Base ` K ) |
2 |
|
cdlemef46.l |
|- .<_ = ( le ` K ) |
3 |
|
cdlemef46.j |
|- .\/ = ( join ` K ) |
4 |
|
cdlemef46.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdlemef46.a |
|- A = ( Atoms ` K ) |
6 |
|
cdlemef46.h |
|- H = ( LHyp ` K ) |
7 |
|
cdlemef46.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
8 |
|
cdlemef46.d |
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) |
9 |
|
cdlemefs46.e |
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) ) |
10 |
|
cdlemef46.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 |
|
simpl3r |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P = Q ) -> ( S .\/ ( X ./\ W ) ) = X ) |
12 |
|
simp3ll |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> S e. A ) |
13 |
12
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P = Q ) -> S e. A ) |
14 |
1 5
|
atbase |
|- ( S e. A -> S e. B ) |
15 |
13 14
|
syl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P = Q ) -> S e. B ) |
16 |
10
|
cdleme31id |
|- ( ( S e. B /\ P = Q ) -> ( F ` S ) = S ) |
17 |
15 16
|
sylancom |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P = Q ) -> ( F ` S ) = S ) |
18 |
17
|
oveq1d |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P = Q ) -> ( ( F ` S ) .\/ ( X ./\ W ) ) = ( S .\/ ( X ./\ W ) ) ) |
19 |
|
simp2l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> X e. B ) |
20 |
10
|
cdleme31id |
|- ( ( X e. B /\ P = Q ) -> ( F ` X ) = X ) |
21 |
19 20
|
sylan |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P = Q ) -> ( F ` X ) = X ) |
22 |
11 18 21
|
3eqtr4rd |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P = Q ) -> ( F ` X ) = ( ( F ` S ) .\/ ( X ./\ W ) ) ) |
23 |
|
simpl1 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P =/= Q ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) |
24 |
|
simpr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P =/= Q ) -> P =/= Q ) |
25 |
|
simpl2 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P =/= Q ) -> ( X e. B /\ -. X .<_ W ) ) |
26 |
|
simpl3 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P =/= Q ) -> ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) |
27 |
1 2 3 4 5 6 7 8 9 10
|
cdleme48fv |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> ( F ` X ) = ( ( F ` S ) .\/ ( X ./\ W ) ) ) |
28 |
23 24 25 26 27
|
syl121anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) /\ P =/= Q ) -> ( F ` X ) = ( ( F ` S ) .\/ ( X ./\ W ) ) ) |
29 |
22 28
|
pm2.61dane |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> ( F ` X ) = ( ( F ` S ) .\/ ( X ./\ W ) ) ) |