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Theorem cdlemg7aN 34109
Description: TODO: FIX COMMENT (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
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
Ref Expression
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 of Theorem cdlemg7aN
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 simp1l 1012 . . 3  |-  ( ( ( 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 )
2 simp1r 1013 . . 3  |-  ( ( ( 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 )
3 simp2r 1015 . . 3  |-  ( ( ( 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 ) )
4 cdlemg7.b . . . 4  |-  B  =  ( Base `  K
)
5 cdlemg7.l . . . 4  |-  .<_  =  ( le `  K )
6 eqid 2438 . . . 4  |-  ( join `  K )  =  (
join `  K )
7 eqid 2438 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
8 cdlemg7.a . . . 4  |-  A  =  ( Atoms `  K )
9 cdlemg7.h . . . 4  |-  H  =  ( LHyp `  K
)
104, 5, 6, 7, 8, 9lhpmcvr2 33508 . . 3  |-  ( ( ( 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 ) )
111, 2, 3, 10syl21anc 1217 . 2  |-  ( ( ( 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 ) )
12 simp11 1018 . . . . 5  |-  ( ( ( ( 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 ) )
13 simp2 989 . . . . . 6  |-  ( ( ( ( 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
)
14 simp3l 1016 . . . . . 6  |-  ( ( ( ( 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 )
1513, 14jca 532 . . . . 5  |-  ( ( ( ( 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 ) )
16 simp12r 1102 . . . . 5  |-  ( ( ( ( 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 ) )
17 simp131 1123 . . . . 5  |-  ( ( ( ( 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
)
18 simp132 1124 . . . . 5  |-  ( ( ( ( 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
)
19 simp3r 1017 . . . . 5  |-  ( ( ( ( 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 )
20 cdlemg7.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
214, 5, 6, 7, 8, 9, 20cdlemg7fvN 34108 . . . . 5  |-  ( ( ( 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 ) ) )
2212, 15, 16, 17, 18, 19, 21syl123anc 1235 . . . 4  |-  ( ( ( ( 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 1101 . . . . . 6  |-  ( ( ( ( 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 1125 . . . . . 6  |-  ( ( ( ( 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 )
255, 8, 9, 20cdlemg6 34107 . . . . . 6  |-  ( ( ( 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 )
2612, 23, 15, 17, 18, 24, 25syl123anc 1235 . . . . 5  |-  ( ( ( ( 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 )
2726oveq1d 6101 . . . 4  |-  ( ( ( ( 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 ) ) )
2822, 27, 193eqtrd 2474 . . 3  |-  ( ( ( ( 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 )
2928rexlimdv3a 2838 . 2  |-  ( ( ( 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 ) )
3011, 29mpd 15 1  |-  ( ( ( 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 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2711   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   lecple 14237   joincjn 15106   meetcmee 15107   Atomscatm 32748   HLchlt 32835   LHypclh 33468   LTrncltrn 33585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-riotaBAD 32444
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-undef 6784  df-map 7208  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-llines 32982  df-lplanes 32983  df-lvols 32984  df-lines 32985  df-psubsp 32987  df-pmap 32988  df-padd 33280  df-lhyp 33472  df-laut 33473  df-ldil 33588  df-ltrn 33589  df-trl 33643
This theorem is referenced by:  cdlemg7N  34110
  Copyright terms: Public domain W3C validator