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Metamath Proof Explorer

Theorem cdlemk39u

Description: Part of proof of Lemma K of Crawley p. 118. Line 31, p. 119. Trace-preserving property of the value of tau, represented by ( UG ) . (Contributed by NM, 31-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 ) )
cdlemk5.u 
|- U = ( g e. T |-> if ( F = N , g , X ) )
Assertion cdlemk39u 
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` ( U ` G ) ) .<_ ( R ` G ) )

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 cdlemk5.u 
 |-  U = ( g e. T |-> if ( F = N , g , X ) )
13 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F = N ) -> F = N )
14 simpl2r 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F = N ) -> G e. T )
15 11 12 cdlemk40t 
 |-  ( ( F = N /\ G e. T ) -> ( U ` G ) = G )
16 13 14 15 syl2anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F = N ) -> ( U ` G ) = G )
17 16 fveq2d 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F = N ) -> ( R ` ( U ` G ) ) = ( R ` G ) )
18 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> K e. HL )
19 18 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> K e. Lat )
20 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( K e. HL /\ W e. H ) )
21 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> G e. T )
22 1 6 7 8 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) e. B )
23 20 21 22 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` G ) e. B )
24 1 2 latref 
 |-  ( ( K e. Lat /\ ( R ` G ) e. B ) -> ( R ` G ) .<_ ( R ` G ) )
25 19 23 24 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` G ) .<_ ( R ` G ) )
26 25 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F = N ) -> ( R ` G ) .<_ ( R ` G ) )
27 17 26 eqbrtrd 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F = N ) -> ( R ` ( U ` G ) ) .<_ ( R ` G ) )
28 simpl1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= N ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) )
29 simpl2l 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= N ) -> ( R ` F ) = ( R ` N ) )
30 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= N ) -> F =/= N )
31 simpl2r 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= N ) -> G e. T )
32 simpl3 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= N ) -> ( P e. A /\ -. P .<_ W ) )
33 1 2 3 4 5 6 7 8 9 10 11 12 cdlemk39u1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= N /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` ( U ` G ) ) .<_ ( R ` G ) )
34 28 29 30 31 32 33 syl131anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ F =/= N ) -> ( R ` ( U ` G ) ) .<_ ( R ` G ) )
35 27 34 pm2.61dane 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` ( U ` G ) ) .<_ ( R ` G ) )