Metamath Proof Explorer

Theorem cdlemkid4

Description: Lemma for cdlemkid . (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 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 ) ) ) )

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 simp3r 
 |-  ( ( ( 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 = ( _I |` B ) )
13 1 6 7 idltrn 
 |-  ( ( K e. HL /\ W e. H ) -> ( _I |` B ) e. T )
14 13 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 ) ) ) -> ( _I |` B ) e. T )
15 12 14 eqeltrd 
 |-  ( ( ( 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 e. T )
16 11 csbeq2i 
 |-  [_ G / g ]_ X = [_ G / g ]_ ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
17 csbriota 
 |-  [_ G / g ]_ ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) ) = ( iota_ z e. T [. G / g ]. A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
18 16 17 eqtri 
 |-  [_ G / g ]_ X = ( iota_ z e. T [. G / g ]. A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
19 18 a1i 
 |-  ( G e. T -> [_ G / g ]_ X = ( iota_ z e. T [. G / g ]. A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) ) )
20 sbcralg 
 |-  ( G e. T -> ( [. G / g ]. A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) <-> A. b e. T [. G / g ]. ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) ) )
21 sbcimg 
 |-  ( G e. T -> ( [. G / g ]. ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) <-> ( [. G / g ]. ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> [. G / g ]. ( z ` P ) = Y ) ) )
22 sbc3an 
 |-  ( [. G / g ]. ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) <-> ( [. G / g ]. b =/= ( _I |` B ) /\ [. G / g ]. ( R ` b ) =/= ( R ` F ) /\ [. G / g ]. ( R ` b ) =/= ( R ` g ) ) )
23 sbcg 
 |-  ( G e. T -> ( [. G / g ]. b =/= ( _I |` B ) <-> b =/= ( _I |` B ) ) )
24 sbcg 
 |-  ( G e. T -> ( [. G / g ]. ( R ` b ) =/= ( R ` F ) <-> ( R ` b ) =/= ( R ` F ) ) )
25 sbcne12 
 |-  ( [. G / g ]. ( R ` b ) =/= ( R ` g ) <-> [_ G / g ]_ ( R ` b ) =/= [_ G / g ]_ ( R ` g ) )
26 csbconstg 
 |-  ( G e. T -> [_ G / g ]_ ( R ` b ) = ( R ` b ) )
27 csbfv 
 |-  [_ G / g ]_ ( R ` g ) = ( R ` G )
28 27 a1i 
 |-  ( G e. T -> [_ G / g ]_ ( R ` g ) = ( R ` G ) )
29 26 28 neeq12d 
 |-  ( G e. T -> ( [_ G / g ]_ ( R ` b ) =/= [_ G / g ]_ ( R ` g ) <-> ( R ` b ) =/= ( R ` G ) ) )
30 25 29 bitrid 
 |-  ( G e. T -> ( [. G / g ]. ( R ` b ) =/= ( R ` g ) <-> ( R ` b ) =/= ( R ` G ) ) )
31 23 24 30 3anbi123d 
 |-  ( G e. T -> ( ( [. G / g ]. b =/= ( _I |` B ) /\ [. G / g ]. ( R ` b ) =/= ( R ` F ) /\ [. G / g ]. ( R ` b ) =/= ( R ` g ) ) <-> ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) )
32 22 31 bitrid 
 |-  ( G e. T -> ( [. G / g ]. ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) <-> ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) )
33 sbceq2g 
 |-  ( G e. T -> ( [. G / g ]. ( z ` P ) = Y <-> ( z ` P ) = [_ G / g ]_ Y ) )
34 32 33 imbi12d 
 |-  ( G e. T -> ( ( [. G / g ]. ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> [. G / g ]. ( z ` P ) = Y ) <-> ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) )
35 21 34 bitrd 
 |-  ( G e. T -> ( [. G / g ]. ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) <-> ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) )
36 35 ralbidv 
 |-  ( G e. T -> ( A. b e. T [. G / g ]. ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) <-> A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) )
37 20 36 bitrd 
 |-  ( G e. T -> ( [. G / g ]. A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) <-> A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) )
38 37 riotabidv 
 |-  ( G e. T -> ( iota_ z e. T [. G / g ]. A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) ) = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) )
39 19 38 eqtrd 
 |-  ( G e. T -> [_ G / g ]_ X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) )
40 15 39 syl 
 |-  ( ( ( 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 ` P ) = [_ G / g ]_ Y ) ) )
41 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( K e. HL /\ W e. H ) )
42 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) )
43 simpl3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
44 simpl3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> G = ( _I |` B ) )
45 simprlr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> b e. T )
46 simprr1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> b =/= ( _I |` B ) )
47 45 46 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( b e. T /\ b =/= ( _I |` B ) ) )
48 1 2 3 4 5 6 7 8 9 10 cdlemkid2 
 |-  ( ( ( 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 ) ) ) ) -> [_ G / g ]_ Y = P )
49 41 42 43 44 47 48 syl113anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> [_ G / g ]_ Y = P )
50 49 eqeq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( z ` P ) = [_ G / g ]_ Y <-> ( z ` P ) = P ) )
51 simprll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> z e. T )
52 1 2 5 6 7 ltrnideq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ z e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( z = ( _I |` B ) <-> ( z ` P ) = P ) )
53 41 51 43 52 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( z = ( _I |` B ) <-> ( z ` P ) = P ) )
54 50 53 bitr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( z ` P ) = [_ G / g ]_ Y <-> z = ( _I |` B ) ) )
55 54 exp44 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> ( z e. T -> ( b e. T -> ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( ( z ` P ) = [_ G / g ]_ Y <-> z = ( _I |` B ) ) ) ) ) )
56 55 imp41 
 |-  ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ z e. T ) /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) -> ( ( z ` P ) = [_ G / g ]_ Y <-> z = ( _I |` B ) ) )
57 56 pm5.74da 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ z e. T ) /\ b e. T ) -> ( ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) <-> ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( _I |` B ) ) ) )
58 57 ralbidva 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ z e. T ) -> ( A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) <-> A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( _I |` B ) ) ) )
59 58 riotabidva 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( _I |` B ) ) ) )
60 40 59 eqtrd 
 |-  ( ( ( 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 ) ) ) )