Metamath Proof Explorer

Theorem cdleml7

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 cdleml7 
|- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( ( U o. s ) ` h ) = ( ( _I |` T ) ` 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 1 2 3 4 5 6 7 8 9 10 11 12 13 cdleml6 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( U e. E /\ ( U ` ( s ` h ) ) = h ) )
15 14 simprd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( U ` ( s ` h ) ) = h )
16 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( K e. HL /\ W e. H ) )
17 14 simpld 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> U e. E )
18 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> s e. E )
19 simp2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> h e. T )
20 4 5 12 tendocoval 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ s e. E ) /\ h e. T ) -> ( ( U o. s ) ` h ) = ( U ` ( s ` h ) ) )
21 16 17 18 19 20 syl121anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( ( U o. s ) ` h ) = ( U ` ( s ` h ) ) )
22 fvresi 
 |-  ( h e. T -> ( ( _I |` T ) ` h ) = h )
23 22 3ad2ant2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( ( _I |` T ) ` h ) = h )
24 15 21 23 3eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( ( U o. s ) ` h ) = ( ( _I |` T ) ` h ) )