Metamath Proof Explorer


Theorem cdleml6

Description: Part of proof of Lemma L of Crawley p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013)

Ref Expression
Hypotheses cdleml6.b
|- B = ( Base ` K )
cdleml6.j
|- .\/ = ( join ` K )
cdleml6.m
|- ./\ = ( meet ` K )
cdleml6.h
|- H = ( LHyp ` K )
cdleml6.t
|- T = ( ( LTrn ` K ) ` W )
cdleml6.r
|- R = ( ( trL ` K ) ` W )
cdleml6.p
|- Q = ( ( oc ` K ) ` W )
cdleml6.z
|- Z = ( ( Q .\/ ( R ` b ) ) ./\ ( ( h ` Q ) .\/ ( R ` ( b o. `' ( s ` h ) ) ) ) )
cdleml6.y
|- Y = ( ( Q .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
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 ) )
cdleml6.u
|- U = ( g e. T |-> if ( ( s ` h ) = h , g , X ) )
cdleml6.e
|- E = ( ( TEndo ` K ) ` W )
cdleml6.o
|- .0. = ( f e. T |-> ( _I |` B ) )
Assertion cdleml6
|- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( U e. E /\ ( U ` ( s ` h ) ) = h ) )

Proof

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 ) )