Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemftr.b |
|- B = ( Base ` K ) |
2 |
|
cdlemftr.h |
|- H = ( LHyp ` K ) |
3 |
|
cdlemftr.t |
|- T = ( ( LTrn ` K ) ` W ) |
4 |
|
cdlemftr.r |
|- R = ( ( trL ` K ) ` W ) |
5 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
6 |
|
eqid |
|- ( Atoms ` K ) = ( Atoms ` K ) |
7 |
5 6 2
|
lhpexle3 |
|- ( ( K e. HL /\ W e. H ) -> E. u e. ( Atoms ` K ) ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) |
8 |
|
df-rex |
|- ( E. u e. ( Atoms ` K ) ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> E. u ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) ) |
9 |
7 8
|
sylib |
|- ( ( K e. HL /\ W e. H ) -> E. u ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) ) |
10 |
1 5 6 2 3 4
|
cdlemfnid |
|- ( ( ( K e. HL /\ W e. H ) /\ ( u e. ( Atoms ` K ) /\ u ( le ` K ) W ) ) -> E. f e. T ( ( R ` f ) = u /\ f =/= ( _I |` B ) ) ) |
11 |
10
|
adantrrr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) ) -> E. f e. T ( ( R ` f ) = u /\ f =/= ( _I |` B ) ) ) |
12 |
|
eqcom |
|- ( ( R ` f ) = u <-> u = ( R ` f ) ) |
13 |
12
|
anbi1i |
|- ( ( ( R ` f ) = u /\ f =/= ( _I |` B ) ) <-> ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) ) |
14 |
13
|
rexbii |
|- ( E. f e. T ( ( R ` f ) = u /\ f =/= ( _I |` B ) ) <-> E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) ) |
15 |
11 14
|
sylib |
|- ( ( ( K e. HL /\ W e. H ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) ) -> E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) ) |
16 |
|
simprrr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) ) -> ( u =/= X /\ u =/= Y /\ u =/= Z ) ) |
17 |
15 16
|
jca |
|- ( ( ( K e. HL /\ W e. H ) /\ ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) ) -> ( E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) |
18 |
17
|
ex |
|- ( ( K e. HL /\ W e. H ) -> ( ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) -> ( E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) ) |
19 |
18
|
eximdv |
|- ( ( K e. HL /\ W e. H ) -> ( E. u ( u e. ( Atoms ` K ) /\ ( u ( le ` K ) W /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) -> E. u ( E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) ) |
20 |
9 19
|
mpd |
|- ( ( K e. HL /\ W e. H ) -> E. u ( E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) |
21 |
|
rexcom4 |
|- ( E. f e. T E. u ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> E. u E. f e. T ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) |
22 |
|
anass |
|- ( ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> ( u = ( R ` f ) /\ ( f =/= ( _I |` B ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) ) |
23 |
22
|
exbii |
|- ( E. u ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> E. u ( u = ( R ` f ) /\ ( f =/= ( _I |` B ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) ) |
24 |
|
fvex |
|- ( R ` f ) e. _V |
25 |
|
neeq1 |
|- ( u = ( R ` f ) -> ( u =/= X <-> ( R ` f ) =/= X ) ) |
26 |
|
neeq1 |
|- ( u = ( R ` f ) -> ( u =/= Y <-> ( R ` f ) =/= Y ) ) |
27 |
|
neeq1 |
|- ( u = ( R ` f ) -> ( u =/= Z <-> ( R ` f ) =/= Z ) ) |
28 |
25 26 27
|
3anbi123d |
|- ( u = ( R ` f ) -> ( ( u =/= X /\ u =/= Y /\ u =/= Z ) <-> ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) ) |
29 |
28
|
anbi2d |
|- ( u = ( R ` f ) -> ( ( f =/= ( _I |` B ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) ) ) |
30 |
24 29
|
ceqsexv |
|- ( E. u ( u = ( R ` f ) /\ ( f =/= ( _I |` B ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) <-> ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) ) |
31 |
23 30
|
bitri |
|- ( E. u ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) ) |
32 |
31
|
rexbii |
|- ( E. f e. T E. u ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> E. f e. T ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) ) |
33 |
|
r19.41v |
|- ( E. f e. T ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> ( E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) |
34 |
33
|
exbii |
|- ( E. u E. f e. T ( ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> E. u ( E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) ) |
35 |
21 32 34
|
3bitr3ri |
|- ( E. u ( E. f e. T ( u = ( R ` f ) /\ f =/= ( _I |` B ) ) /\ ( u =/= X /\ u =/= Y /\ u =/= Z ) ) <-> E. f e. T ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) ) |
36 |
20 35
|
sylib |
|- ( ( K e. HL /\ W e. H ) -> E. f e. T ( f =/= ( _I |` B ) /\ ( ( R ` f ) =/= X /\ ( R ` f ) =/= Y /\ ( R ` f ) =/= Z ) ) ) |