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Theorem cdlemg7aN 30741
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 981 . . 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 982 . . 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 984 . . 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 2389 . . . 4  |-  ( join `  K )  =  (
join `  K )
7 eqid 2389 . . . 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 30140 . . 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 1183 . 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 987 . . . . 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 958 . . . . . 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 985 . . . . . 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 519 . . . . 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 1071 . . . . 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 1092 . . . . 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 1093 . . . . 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 986 . . . . 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 30740 . . . . 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 1201 . . . 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 1070 . . . . . 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 1094 . . . . . 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 30739 . . . . . 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 1201 . . . . 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 6037 . . . 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 2425 . . 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 2777 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   E.wrex 2652   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   Basecbs 13398   lecple 13465   joincjn 14330   meetcmee 14331   Atomscatm 29380   HLchlt 29467   LHypclh 30100   LTrncltrn 30217
This theorem is referenced by:  cdlemg7N  30742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-map 6958  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-llines 29614  df-lplanes 29615  df-lvols 29616  df-lines 29617  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104  df-laut 30105  df-ldil 30220  df-ltrn 30221  df-trl 30275
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