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 |
|
simp22 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G e. T ) |
12 |
10
|
cdlemk41 |
|- ( G e. T -> [_ G / g ]_ Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) ) |
13 |
11 12
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> [_ G / g ]_ Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) ) |
14 |
|
simp1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F = N ) ) |
15 |
|
simp21l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T ) |
16 |
|
simp21r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F =/= ( _I |` B ) ) |
17 |
|
simp23l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> b e. T ) |
18 |
|
simp31 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` b ) =/= ( R ` F ) ) |
19 |
|
simp33 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
20 |
1 2 3 4 5 6 7 8 9
|
cdlemkfid2N |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ b e. T ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> Z = ( b ` P ) ) |
21 |
14 15 16 17 18 19 20
|
syl132anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> Z = ( b ` P ) ) |
22 |
21
|
oveq1d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( Z .\/ ( R ` ( G o. `' b ) ) ) = ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) |
23 |
22
|
oveq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) ) |
24 |
|
simp1l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( K e. HL /\ W e. H ) ) |
25 |
|
simp23r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> b =/= ( _I |` B ) ) |
26 |
|
simp32 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` b ) =/= ( R ` G ) ) |
27 |
26
|
necomd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` G ) =/= ( R ` b ) ) |
28 |
1 2 3 4 5 6 7 8
|
cdlemkfid1N |
|- ( ( ( K e. HL /\ W e. H ) /\ ( b e. T /\ b =/= ( _I |` B ) /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` b ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) = ( G ` P ) ) |
29 |
24 17 25 11 27 19 28
|
syl132anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) = ( G ` P ) ) |
30 |
13 23 29
|
3eqtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> [_ G / g ]_ Y = ( G ` P ) ) |