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 |
1 4 5 12 13
|
tendo1ne0 |
|- ( ( K e. HL /\ W e. H ) -> ( _I |` T ) =/= .0. ) |
15 |
14
|
3ad2ant1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) /\ ( s e. E /\ s =/= .0. ) ) -> ( _I |` T ) =/= .0. ) |
16 |
1 2 3 4 5 6 7 8 9 10 11 12 13
|
cdleml8 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) /\ ( s e. E /\ s =/= .0. ) ) -> ( U o. s ) = ( _I |` T ) ) |
17 |
16
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) /\ ( s e. E /\ s =/= .0. ) ) /\ U = .0. ) -> ( U o. s ) = ( _I |` T ) ) |
18 |
|
coeq1 |
|- ( U = .0. -> ( U o. s ) = ( .0. o. s ) ) |
19 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) /\ ( s e. E /\ s =/= .0. ) ) -> ( K e. HL /\ W e. H ) ) |
20 |
|
simp3l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) /\ ( s e. E /\ s =/= .0. ) ) -> s e. E ) |
21 |
1 4 5 12 13
|
tendo0mul |
|- ( ( ( K e. HL /\ W e. H ) /\ s e. E ) -> ( .0. o. s ) = .0. ) |
22 |
19 20 21
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) /\ ( s e. E /\ s =/= .0. ) ) -> ( .0. o. s ) = .0. ) |
23 |
18 22
|
sylan9eqr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) /\ ( s e. E /\ s =/= .0. ) ) /\ U = .0. ) -> ( U o. s ) = .0. ) |
24 |
17 23
|
eqtr3d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) /\ ( s e. E /\ s =/= .0. ) ) /\ U = .0. ) -> ( _I |` T ) = .0. ) |
25 |
24
|
ex |
|- ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) /\ ( s e. E /\ s =/= .0. ) ) -> ( U = .0. -> ( _I |` T ) = .0. ) ) |
26 |
25
|
necon3d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) /\ ( s e. E /\ s =/= .0. ) ) -> ( ( _I |` T ) =/= .0. -> U =/= .0. ) ) |
27 |
15 26
|
mpd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( h e. T /\ h =/= ( _I |` B ) ) /\ ( s e. E /\ s =/= .0. ) ) -> U =/= .0. ) |