Step |
Hyp |
Ref |
Expression |
1 |
|
cdleml6.b |
|- B = ( Base ` K ) |
2 |
|
cdleml6.j |
|- .\/ = ( join ` K ) |
3 |
|
cdleml6.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdleml6.h |
|- H = ( LHyp ` K ) |
5 |
|
cdleml6.t |
|- T = ( ( LTrn ` K ) ` W ) |
6 |
|
cdleml6.r |
|- R = ( ( trL ` K ) ` W ) |
7 |
|
cdleml6.p |
|- Q = ( ( oc ` K ) ` W ) |
8 |
|
cdleml6.z |
|- Z = ( ( Q .\/ ( R ` b ) ) ./\ ( ( h ` Q ) .\/ ( R ` ( b o. `' ( s ` h ) ) ) ) ) |
9 |
|
cdleml6.y |
|- Y = ( ( Q .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) ) |
10 |
|
cdleml6.x |
|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` ( s ` h ) ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` Q ) = Y ) ) |
11 |
|
cdleml6.u |
|- U = ( g e. T |-> if ( ( s ` h ) = h , g , X ) ) |
12 |
|
cdleml6.e |
|- E = ( ( TEndo ` K ) ` W ) |
13 |
|
cdleml6.o |
|- .0. = ( f e. T |-> ( _I |` B ) ) |
14 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( K e. HL /\ W e. H ) ) |
15 |
|
simp3l |
|- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> s e. E ) |
16 |
|
simp2 |
|- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> h e. T ) |
17 |
4 5 12
|
tendocl |
|- ( ( ( K e. HL /\ W e. H ) /\ s e. E /\ h e. T ) -> ( s ` h ) e. T ) |
18 |
14 15 16 17
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( s ` h ) e. T ) |
19 |
1 4 5 6 12 13
|
tendotr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( s e. E /\ s =/= .0. ) /\ h e. T ) -> ( R ` ( s ` h ) ) = ( R ` h ) ) |
20 |
19
|
3com23 |
|- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( R ` ( s ` h ) ) = ( R ` h ) ) |
21 |
|
eqid |
|- ( oc ` K ) = ( oc ` K ) |
22 |
|
eqid |
|- ( Atoms ` K ) = ( Atoms ` K ) |
23 |
1 2 3 21 22 4 5 6 7 8 9 10 11 12
|
cdlemk56w |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( s ` h ) e. T /\ h e. T ) /\ ( R ` ( s ` h ) ) = ( R ` h ) ) -> ( U e. E /\ ( U ` ( s ` h ) ) = h ) ) |
24 |
14 18 16 20 23
|
syl121anc |
|- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( U e. E /\ ( U ` ( s ` h ) ) = h ) ) |