Metamath Proof Explorer

Theorem cdlemk55b

Description: Lemma for cdlemk55 . (Contributed by NM, 26-Jul-2013)

Ref Expression
Hypotheses cdlemk5.b 
|- B = ( Base ` K )
cdlemk5.l 
|- .<_ = ( le ` K )
cdlemk5.j 
|- .\/ = ( join ` K )
cdlemk5.m 
|- ./\ = ( meet ` K )
cdlemk5.a 
|- A = ( Atoms ` K )
cdlemk5.h 
|- H = ( LHyp ` K )
cdlemk5.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk5.r 
|- R = ( ( trL ` K ) ` W )
cdlemk5.z 
|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
cdlemk5.y 
|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
cdlemk5.x 
|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
Assertion cdlemk55b 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) = ( R ` I ) ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )

Proof

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 cdlemk5.x 
 |-  X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
12 simp1ll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) = ( R ` I ) ) ) -> K e. HL )
13 simp1lr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) = ( R ` I ) ) ) -> W e. H )
14 1 6 7 8 cdlemftr2 
 |-  ( ( K e. HL /\ W e. H ) -> E. j e. T ( j =/= ( _I |` B ) /\ ( R ` j ) =/= ( R ` G ) /\ ( R ` j ) =/= ( R ` ( G o. I ) ) ) )
15 12 13 14 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) = ( R ` I ) ) ) -> E. j e. T ( j =/= ( _I |` B ) /\ ( R ` j ) =/= ( R ` G ) /\ ( R ` j ) =/= ( R ` ( G o. I ) ) ) )
16 simp11 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) = ( R ` I ) ) ) /\ j e. T /\ ( j =/= ( _I |` B ) /\ ( R ` j ) =/= ( R ` G ) /\ ( R ` j ) =/= ( R ` ( G o. I ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) )
17 simp12 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) = ( R ` I ) ) ) /\ j e. T /\ ( j =/= ( _I |` B ) /\ ( R ` j ) =/= ( R ` G ) /\ ( R ` j ) =/= ( R ` ( G o. I ) ) ) ) -> ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) )
18 simp13 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) = ( R ` I ) ) ) /\ j e. T /\ ( j =/= ( _I |` B ) /\ ( R ` j ) =/= ( R ` G ) /\ ( R ` j ) =/= ( R ` ( G o. I ) ) ) ) -> ( I e. T /\ ( R ` G ) = ( R ` I ) ) )
19 simp2 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) = ( R ` I ) ) ) /\ j e. T /\ ( j =/= ( _I |` B ) /\ ( R ` j ) =/= ( R ` G ) /\ ( R ` j ) =/= ( R ` ( G o. I ) ) ) ) -> j e. T )
20 simp3 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) = ( R ` I ) ) ) /\ j e. T /\ ( j =/= ( _I |` B ) /\ ( R ` j ) =/= ( R ` G ) /\ ( R ` j ) =/= ( R ` ( G o. I ) ) ) ) -> ( j =/= ( _I |` B ) /\ ( R ` j ) =/= ( R ` G ) /\ ( R ` j ) =/= ( R ` ( G o. I ) ) ) )
21 1 2 3 4 5 6 7 8 9 10 11 cdlemk55a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( I e. T /\ ( R ` G ) = ( R ` I ) ) /\ j e. T /\ ( j =/= ( _I |` B ) /\ ( R ` j ) =/= ( R ` G ) /\ ( R ` j ) =/= ( R ` ( G o. I ) ) ) ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )
22 16 17 18 19 20 21 syl113anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) = ( R ` I ) ) ) /\ j e. T /\ ( j =/= ( _I |` B ) /\ ( R ` j ) =/= ( R ` G ) /\ ( R ` j ) =/= ( R ` ( G o. I ) ) ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )
23 22 rexlimdv3a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) = ( R ` I ) ) ) -> ( E. j e. T ( j =/= ( _I |` B ) /\ ( R ` j ) =/= ( R ` G ) /\ ( R ` j ) =/= ( R ` ( G o. I ) ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) ) )
24 15 23 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( I e. T /\ ( R ` G ) = ( R ` I ) ) ) -> [_ ( G o. I ) / g ]_ X = ( [_ G / g ]_ X o. [_ I / g ]_ X ) )