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Metamath Proof Explorer

Theorem cdlemkfid3N

Description: TODO: is this useful or should it be deleted? (Contributed by NM, 29-Jul-2013) (New usage is discouraged.)

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 ) ) ) )
Assertion cdlemkfid3N 
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> [_ G / g ]_ Y = ( G ` P ) )

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 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G e. T )
12 10 cdlemk41 
 |-  ( G e. T -> [_ G / g ]_ Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) )
13 11 12 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> [_ G / g ]_ Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) )
14 simp1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F = N ) )
15 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T )
16 simp21r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F =/= ( _I |` B ) )
17 simp23l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> b e. T )
18 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` b ) =/= ( R ` F ) )
19 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
20 1 2 3 4 5 6 7 8 9 cdlemkfid2N 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ b e. T ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> Z = ( b ` P ) )
21 14 15 16 17 18 19 20 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> Z = ( b ` P ) )
22 21 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( Z .\/ ( R ` ( G o. `' b ) ) ) = ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) )
23 22 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) )
24 simp1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
25 simp23r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> b =/= ( _I |` B ) )
26 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` b ) =/= ( R ` G ) )
27 26 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` G ) =/= ( R ` b ) )
28 1 2 3 4 5 6 7 8 cdlemkfid1N 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( b e. T /\ b =/= ( _I |` B ) /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` b ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) = ( G ` P ) )
29 24 17 25 11 27 19 28 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( b ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) = ( G ` P ) )
30 13 23 29 3eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ ( b e. T /\ b =/= ( _I |` B ) ) ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> [_ G / g ]_ Y = ( G ` P ) )