Metamath Proof Explorer

Theorem cdlemg7aN

Description: TODO: FIX COMMENT. (Contributed by NM, 28-Apr-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemg7.b 
|- B = ( Base ` K )
cdlemg7.l 
|- .<_ = ( le ` K )
cdlemg7.a 
|- A = ( Atoms ` K )
cdlemg7.h 
|- H = ( LHyp ` K )
cdlemg7.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion cdlemg7aN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( F ` ( G ` X ) ) = X )

Proof

Step Hyp Ref Expression
1 cdlemg7.b 
 |-  B = ( Base ` K )
2 cdlemg7.l 
 |-  .<_ = ( le ` K )
3 cdlemg7.a 
 |-  A = ( Atoms ` K )
4 cdlemg7.h 
 |-  H = ( LHyp ` K )
5 cdlemg7.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> K e. HL )
7 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> W e. H )
8 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( X e. B /\ -. X .<_ W ) )
9 eqid 
 |-  ( join ` K ) = ( join ` K )
10 eqid 
 |-  ( meet ` K ) = ( meet ` K )
11 1 2 9 10 3 4 lhpmcvr2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> E. r e. A ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) )
12 6 7 8 11 syl21anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> E. r e. A ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) )
13 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( K e. HL /\ W e. H ) )
14 simp2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> r e. A )
15 simp3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> -. r .<_ W )
16 14 15 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( r e. A /\ -. r .<_ W ) )
17 simp12r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( X e. B /\ -. X .<_ W ) )
18 simp131 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> F e. T )
19 simp132 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> G e. T )
20 simp3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X )
21 1 2 9 10 3 4 5 cdlemg7fvN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( r e. A /\ -. r .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( F ` ( G ` X ) ) = ( ( F ` ( G ` r ) ) ( join ` K ) ( X ( meet ` K ) W ) ) )
22 13 16 17 18 19 20 21 syl123anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( F ` ( G ` X ) ) = ( ( F ` ( G ` r ) ) ( join ` K ) ( X ( meet ` K ) W ) ) )
23 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( P e. A /\ -. P .<_ W ) )
24 simp133 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( F ` ( G ` P ) ) = P )
25 2 3 4 5 cdlemg6 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( r e. A /\ -. r .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( F ` ( G ` r ) ) = r )
26 13 23 16 18 19 24 25 syl123anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( F ` ( G ` r ) ) = r )
27 26 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( ( F ` ( G ` r ) ) ( join ` K ) ( X ( meet ` K ) W ) ) = ( r ( join ` K ) ( X ( meet ` K ) W ) ) )
28 22 27 20 3eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) /\ r e. A /\ ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) ) -> ( F ` ( G ` X ) ) = X )
29 28 rexlimdv3a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( E. r e. A ( -. r .<_ W /\ ( r ( join ` K ) ( X ( meet ` K ) W ) ) = X ) -> ( F ` ( G ` X ) ) = X ) )
30 12 29 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ -. X .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( F ` ( G ` X ) ) = X )