Metamath Proof Explorer

Theorem cdlemk39s

Description: Substitution version of cdlemk39 . TODO: Can any commonality with cdlemk35s be exploited? (Contributed by NM, 23-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 
|- ( ( ( 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 ) )

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 simp22l 
 |-  ( ( ( 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 ) ) ) -> G e. T )
13 1 2 3 4 5 6 7 8 9 10 11 cdlemk39 
 |-  ( ( ( 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 ` X ) .<_ ( R ` g ) )
14 13 sbcth 
 |-  ( G e. T -> [. G / g ]. ( ( ( 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 ` X ) .<_ ( R ` g ) ) )
15 sbcimg 
 |-  ( G e. T -> ( [. G / g ]. ( ( ( 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 ` X ) .<_ ( R ` g ) ) <-> ( [. G / g ]. ( ( 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 ) ) ) -> [. G / g ]. ( R ` X ) .<_ ( R ` g ) ) ) )
16 14 15 mpbid 
 |-  ( G e. T -> ( [. G / g ]. ( ( 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 ) ) ) -> [. G / g ]. ( R ` X ) .<_ ( R ` g ) ) )
17 eleq1 
 |-  ( g = G -> ( g e. T <-> G e. T ) )
18 neeq1 
 |-  ( g = G -> ( g =/= ( _I |` B ) <-> G =/= ( _I |` B ) ) )
19 17 18 anbi12d 
 |-  ( g = G -> ( ( g e. T /\ g =/= ( _I |` B ) ) <-> ( G e. T /\ G =/= ( _I |` B ) ) ) )
20 19 3anbi2d 
 |-  ( g = G -> ( ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( g e. T /\ g =/= ( _I |` B ) ) /\ N e. T ) <-> ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) ) )
21 20 3anbi2d 
 |-  ( g = G -> ( ( ( 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 ) ) ) <-> ( ( 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 ) ) ) ) )
22 21 sbcieg 
 |-  ( G e. T -> ( [. G / g ]. ( ( 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 ) ) ) <-> ( ( 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 ) ) ) ) )
23 sbcbr12g 
 |-  ( G e. T -> ( [. G / g ]. ( R ` X ) .<_ ( R ` g ) <-> [_ G / g ]_ ( R ` X ) .<_ [_ G / g ]_ ( R ` g ) ) )
24 csbfv2g 
 |-  ( G e. T -> [_ G / g ]_ ( R ` X ) = ( R ` [_ G / g ]_ X ) )
25 csbfv 
 |-  [_ G / g ]_ ( R ` g ) = ( R ` G )
26 25 a1i 
 |-  ( G e. T -> [_ G / g ]_ ( R ` g ) = ( R ` G ) )
27 24 26 breq12d 
 |-  ( G e. T -> ( [_ G / g ]_ ( R ` X ) .<_ [_ G / g ]_ ( R ` g ) <-> ( R ` [_ G / g ]_ X ) .<_ ( R ` G ) ) )
28 23 27 bitrd 
 |-  ( G e. T -> ( [. G / g ]. ( R ` X ) .<_ ( R ` g ) <-> ( R ` [_ G / g ]_ X ) .<_ ( R ` G ) ) )
29 16 22 28 3imtr3d 
 |-  ( G e. T -> ( ( ( 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 ) ) )
30 12 29 mpcom 
 |-  ( ( ( 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 ) )