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Metamath Proof Explorer

Theorem cdlemk11t

Description: Part of proof of Lemma K of Crawley p. 118. Eq. 5, line 36, p. 119. G , I stand for g, h. X represents tau. (Contributed by NM, 21-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 cdlemk11t 
|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' 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 simp11l 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> K e. HL )
13 simp11r 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> W e. H )
14 1 6 7 8 cdlemftr3 
 |-  ( ( K e. HL /\ W e. H ) -> E. b e. T ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) )
15 12 13 14 syl2anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> E. b e. T ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) )
16 nfv 
 |-  F/ b ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) )
17 nfcv 
 |-  F/_ b G
18 nfra1 
 |-  F/ b A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y )
19 nfcv 
 |-  F/_ b T
20 18 19 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 ) )
21 11 20 nfcxfr 
 |-  F/_ b X
22 17 21 nfcsbw 
 |-  F/_ b [_ G / g ]_ X
23 nfcv 
 |-  F/_ b P
24 22 23 nffv 
 |-  F/_ b ( [_ G / g ]_ X ` P )
25 nfcv 
 |-  F/_ b .<_
26 nfcv 
 |-  F/_ b I
27 26 21 nfcsbw 
 |-  F/_ b [_ I / g ]_ X
28 27 23 nffv 
 |-  F/_ b ( [_ I / g ]_ X ` P )
29 nfcv 
 |-  F/_ b .\/
30 nfcv 
 |-  F/_ b ( R ` ( I o. `' G ) )
31 28 29 30 nfov 
 |-  F/_ b ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) )
32 24 25 31 nfbr 
 |-  F/ b ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) )
33 simp11 
 |-  ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) )
34 simp12 
 |-  ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
35 simp2 
 |-  ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> b e. T )
36 simp3l 
 |-  ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> b =/= ( _I |` B ) )
37 simp3r1 
 |-  ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( R ` b ) =/= ( R ` F ) )
38 simp3r2 
 |-  ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( R ` b ) =/= ( R ` G ) )
39 36 37 38 3jca 
 |-  ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) )
40 simp13l 
 |-  ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> I e. T )
41 simp13r 
 |-  ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> I =/= ( _I |` B ) )
42 simp3r3 
 |-  ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( R ` b ) =/= ( R ` I ) )
43 40 41 42 3jca 
 |-  ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) )
44 1 2 3 4 5 6 7 8 9 10 11 cdlemk11tc 
 |-  ( ( ( ( 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 ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) ) )
45 33 34 35 39 43 44 syl113anc 
 |-  ( ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) /\ b e. T /\ ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) ) )
46 45 3exp 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( b e. T -> ( ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) -> ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) ) ) ) )
47 16 32 46 rexlimd 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( E. b e. T ( b =/= ( _I |` B ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) /\ ( R ` b ) =/= ( R ` I ) ) ) -> ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) ) ) )
48 15 47 mpd 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( [_ G / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` ( I o. `' G ) ) ) )