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 |
|
cdlemk5c.s |
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) |
12 |
|
cdlemk5a.u2 |
|- C = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( e o. `' b ) ) ) ) ) ) |
13 |
|
simp11 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( K e. HL /\ W e. H ) ) |
14 |
|
simp12l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> F e. T ) |
15 |
|
simp31 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> b e. T ) |
16 |
|
simp21 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> N e. T ) |
17 |
|
simp13l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> G e. T ) |
18 |
|
simp331 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> I e. T ) |
19 |
16 17 18
|
3jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( N e. T /\ G e. T /\ I e. T ) ) |
20 |
|
simp22 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
21 |
|
simp23 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( R ` F ) = ( R ` N ) ) |
22 |
|
simp12r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> F =/= ( _I |` B ) ) |
23 |
|
simp321 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> b =/= ( _I |` B ) ) |
24 |
|
simp13r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> G =/= ( _I |` B ) ) |
25 |
22 23 24
|
3jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( F =/= ( _I |` B ) /\ b =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) ) |
26 |
|
simp332 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> I =/= ( _I |` B ) ) |
27 |
|
simp322 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( R ` b ) =/= ( R ` F ) ) |
28 |
|
simp323 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( R ` b ) =/= ( R ` G ) ) |
29 |
28
|
necomd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( R ` G ) =/= ( R ` b ) ) |
30 |
|
simp333 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( R ` b ) =/= ( R ` I ) ) |
31 |
30
|
necomd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( R ` I ) =/= ( R ` b ) ) |
32 |
27 29 31
|
3jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( ( R ` b ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` b ) /\ ( R ` I ) =/= ( R ` b ) ) ) |
33 |
|
eqid |
|- ( S ` b ) = ( S ` b ) |
34 |
|
eqid |
|- ( ( ( G ` P ) .\/ ( I ` P ) ) ./\ ( ( R ` ( G o. `' b ) ) .\/ ( R ` ( I o. `' b ) ) ) ) = ( ( ( G ` P ) .\/ ( I ` P ) ) ./\ ( ( R ` ( G o. `' b ) ) .\/ ( R ` ( I o. `' b ) ) ) ) |
35 |
1 2 3 4 5 6 7 8 11 33 12 34
|
cdlemk11u |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ b e. T ) /\ ( ( N e. T /\ G e. T /\ I e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ b =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ I =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` b ) /\ ( R ` I ) =/= ( R ` b ) ) ) ) -> ( ( C ` G ) ` P ) .<_ ( ( ( C ` I ) ` P ) .\/ ( R ` ( I o. `' G ) ) ) ) |
36 |
13 14 15 19 20 21 25 26 32 35
|
syl333anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( ( C ` G ) ` P ) .<_ ( ( ( C ` I ) ` P ) .\/ ( R ` ( I o. `' G ) ) ) ) |
37 |
|
simp32 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) |
38 |
15 37
|
jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) |
39 |
1 2 3 4 5 6 7 8 9 10 11 12
|
cdlemkyuu |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> [_ G / g ]_ Y = ( ( C ` G ) ` P ) ) |
40 |
38 39
|
syld3an3 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> [_ G / g ]_ Y = ( ( C ` G ) ` P ) ) |
41 |
|
simp12 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( F e. T /\ F =/= ( _I |` B ) ) ) |
42 |
18 26
|
jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( I e. T /\ I =/= ( _I |` B ) ) ) |
43 |
|
simp2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) |
44 |
23 27 30
|
3jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` I ) ) ) |
45 |
1 2 3 4 5 6 7 8 9 10 11 12
|
cdlemkyuu |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> [_ I / g ]_ Y = ( ( C ` I ) ` P ) ) |
46 |
13 41 42 43 15 44 45
|
syl312anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> [_ I / g ]_ Y = ( ( C ` I ) ` P ) ) |
47 |
46
|
oveq1d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( [_ I / g ]_ Y .\/ ( R ` ( I o. `' G ) ) ) = ( ( ( C ` I ) ` P ) .\/ ( R ` ( I o. `' G ) ) ) ) |
48 |
36 40 47
|
3brtr4d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> [_ G / g ]_ Y .<_ ( [_ I / g ]_ Y .\/ ( R ` ( I o. `' G ) ) ) ) |