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Metamath Proof Explorer

Theorem cdlemkid

Description: The value of the tau function (in Lemma K of Crawley p. 118) on the identity relation. (Contributed by NM, 25-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 cdlemkid 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> [_ G / g ]_ X = ( _I |` B ) )

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 7 fvexi 
 |-  T e. _V
13 nfv 
 |-  F/ b ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) )
14 nfcv 
 |-  F/_ b G
15 nfra1 
 |-  F/ b A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y )
16 nfcv 
 |-  F/_ b T
17 15 16 nfriota 
 |-  F/_ b ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
18 11 17 nfcxfr 
 |-  F/_ b X
19 14 18 nfcsbw 
 |-  F/_ b [_ G / g ]_ X
20 19 nfeq1 
 |-  F/ b [_ G / g ]_ X = ( _I |` B )
21 20 a1i 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> F/ b [_ G / g ]_ X = ( _I |` B ) )
22 1 2 3 4 5 6 7 8 9 10 11 cdlemkid4 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> [_ G / g ]_ X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( _I |` B ) ) ) )
23 eqeq1 
 |-  ( ( _I |` B ) = [_ G / g ]_ X -> ( ( _I |` B ) = ( _I |` B ) <-> [_ G / g ]_ X = ( _I |` B ) ) )
24 23 adantl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( _I |` B ) = [_ G / g ]_ X ) -> ( ( _I |` B ) = ( _I |` B ) <-> [_ G / g ]_ X = ( _I |` B ) ) )
25 eqidd 
 |-  ( ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) -> ( _I |` B ) = ( _I |` B ) )
26 25 a1i 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> ( ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) -> ( _I |` B ) = ( _I |` B ) ) )
27 1 2 3 4 5 6 7 8 9 10 11 cdlemkid5 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> [_ G / g ]_ X e. T )
28 1 6 7 8 cdlemftr2 
 |-  ( ( K e. HL /\ W e. H ) -> E. b e. T ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) )
29 28 3ad2ant1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> E. b e. T ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) )
30 13 21 22 24 26 27 29 riotasv3d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ T e. _V ) -> [_ G / g ]_ X = ( _I |` B ) )
31 12 30 mpan2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> [_ G / g ]_ X = ( _I |` B ) )