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Metamath Proof Explorer

Theorem cdlemk39s-id

Description: Substitution version of cdlemk39 with non-identity requirement on G removed. TODO: Can any commonality with cdlemk35s be exploited? (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 cdlemk39s-id 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` [_ G / g ]_ X ) .<_ ( 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 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> ( K e. HL /\ W e. H ) )
13 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> F e. T )
14 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> N e. T )
15 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` F ) = ( R ` N ) )
16 13 14 15 3jca 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) )
17 16 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) )
18 simpl3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> ( P e. A /\ -. P .<_ W ) )
19 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> G = ( _I |` B ) )
20 1 2 3 4 5 6 7 8 9 10 11 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 ) )
21 12 17 18 19 20 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> [_ G / g ]_ X = ( _I |` B ) )
22 21 fveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> ( R ` [_ G / g ]_ X ) = ( R ` ( _I |` B ) ) )
23 simpl1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> K e. HL )
24 simpl1r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> W e. H )
25 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
26 1 25 6 8 trlid0 
 |-  ( ( K e. HL /\ W e. H ) -> ( R ` ( _I |` B ) ) = ( 0. ` K ) )
27 23 24 26 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> ( R ` ( _I |` B ) ) = ( 0. ` K ) )
28 22 27 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> ( R ` [_ G / g ]_ X ) = ( 0. ` K ) )
29 hlop 
 |-  ( K e. HL -> K e. OP )
30 23 29 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> K e. OP )
31 simpl22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> G e. T )
32 1 6 7 8 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) e. B )
33 12 31 32 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> ( R ` G ) e. B )
34 1 2 25 op0le 
 |-  ( ( K e. OP /\ ( R ` G ) e. B ) -> ( 0. ` K ) .<_ ( R ` G ) )
35 30 33 34 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> ( 0. ` K ) .<_ ( R ` G ) )
36 28 35 eqbrtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G = ( _I |` B ) ) -> ( R ` [_ G / g ]_ X ) .<_ ( R ` G ) )
37 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G =/= ( _I |` B ) ) -> ( K e. HL /\ W e. H ) )
38 simpl21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G =/= ( _I |` B ) ) -> ( F e. T /\ F =/= ( _I |` B ) ) )
39 simpl22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G =/= ( _I |` B ) ) -> G e. T )
40 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G =/= ( _I |` B ) ) -> G =/= ( _I |` B ) )
41 39 40 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G =/= ( _I |` B ) ) -> ( G e. T /\ G =/= ( _I |` B ) ) )
42 simpl23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G =/= ( _I |` B ) ) -> N e. T )
43 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G =/= ( _I |` B ) ) -> ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
44 1 2 3 4 5 6 7 8 9 10 11 cdlemk39s 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` [_ G / g ]_ X ) .<_ ( R ` G ) )
45 37 38 41 42 43 44 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ G =/= ( _I |` B ) ) -> ( R ` [_ G / g ]_ X ) .<_ ( R ` G ) )
46 36 45 pm2.61dane 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ G e. T /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` [_ G / g ]_ X ) .<_ ( R ` G ) )