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Theorem cdlemg7aN 35822
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 1020 . . 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 1021 . . 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 1023 . . 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 2467 . . . 4  |-  ( join `  K )  =  (
join `  K )
7 eqid 2467 . . . 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 35221 . . 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 1227 . 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 1026 . . . . 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 997 . . . . . 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 1024 . . . . . 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 1110 . . . . 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 1131 . . . . 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 1132 . . . . 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 1025 . . . . 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 35821 . . . . 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 1245 . . . 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 1109 . . . . . 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 1133 . . . . . 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 35820 . . . . . 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 1245 . . . . 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 6310 . . . 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 2512 . . 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 2961 . 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 973    = wceq 1379    e. wcel 1767   E.wrex 2818   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   meetcmee 15449   Atomscatm 34461   HLchlt 34548   LHypclh 35181   LTrncltrn 35298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-riotaBAD 34157
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-undef 7014  df-map 7434  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696  df-lvols 34697  df-lines 34698  df-psubsp 34700  df-pmap 34701  df-padd 34993  df-lhyp 35185  df-laut 35186  df-ldil 35301  df-ltrn 35302  df-trl 35356
This theorem is referenced by:  cdlemg7N  35823
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