Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemk5.b |
|- B = ( Base ` K ) |
2 |
|
cdlemk5.l |
|- .<_ = ( le ` K ) |
3 |
|
cdlemk5.j |
|- .\/ = ( join ` K ) |
4 |
|
cdlemk5.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdlemk5.a |
|- A = ( Atoms ` K ) |
6 |
|
cdlemk5.h |
|- H = ( LHyp ` K ) |
7 |
|
cdlemk5.t |
|- T = ( ( LTrn ` K ) ` W ) |
8 |
|
cdlemk5.r |
|- R = ( ( trL ` K ) ` W ) |
9 |
|
cdlemk5.z |
|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) |
10 |
|
cdlemk5.y |
|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) ) |
11 |
|
cdlemk5.x |
|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) ) |
12 |
|
cdlemk5.u |
|- U = ( g e. T |-> if ( F = N , g , X ) ) |
13 |
|
simp22 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> F =/= N ) |
14 |
|
simp21 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> F e. T ) |
15 |
11 12
|
cdlemk40f |
|- ( ( F =/= N /\ F e. T ) -> ( U ` F ) = [_ F / g ]_ X ) |
16 |
13 14 15
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( U ` F ) = [_ F / g ]_ X ) |
17 |
16
|
fveq1d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( U ` F ) ` P ) = ( [_ F / g ]_ X ` P ) ) |
18 |
|
simp1l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( K e. HL /\ W e. H ) ) |
19 |
|
simp23 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> N e. T ) |
20 |
|
simp1r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( R ` N ) ) |
21 |
1 6 7 8
|
trlnid |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( F =/= N /\ ( R ` F ) = ( R ` N ) ) ) -> F =/= ( _I |` B ) ) |
22 |
18 14 19 13 20 21
|
syl122anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> F =/= ( _I |` B ) ) |
23 |
14 22 19
|
3jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) ) |
24 |
1 2 3 4 5 6 7 8 9 10 11
|
cdlemk19x |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( [_ F / g ]_ X ` P ) = ( N ` P ) ) |
25 |
23 24
|
syld3an2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( [_ F / g ]_ X ` P ) = ( N ` P ) ) |
26 |
17 25
|
eqtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ F =/= N /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( U ` F ) ` P ) = ( N ` P ) ) |