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Theorem cdlemg6 30883
Description: TODO: FIX COMMENT (Contributed by NM, 27-Apr-2013.)
Hypotheses
Ref Expression
cdlemg6.l  |-  .<_  =  ( le `  K )
cdlemg6.a  |-  A  =  ( Atoms `  K )
cdlemg6.h  |-  H  =  ( LHyp `  K
)
cdlemg6.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemg6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( F `  ( G `  Q )
)  =  Q )

Proof of Theorem cdlemg6
StepHypRef Expression
1 simpl1 959 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  Q  .<_  ( P ( join `  K
) ( ( ( trL `  K ) `
 W ) `  G ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simpl2l 1009 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  Q  .<_  ( P ( join `  K
) ( ( ( trL `  K ) `
 W ) `  G ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simpl2r 1010 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  Q  .<_  ( P ( join `  K
) ( ( ( trL `  K ) `
 W ) `  G ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
4 simpl31 1037 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  Q  .<_  ( P ( join `  K
) ( ( ( trL `  K ) `
 W ) `  G ) ) )  ->  F  e.  T
)
5 simpl32 1038 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  Q  .<_  ( P ( join `  K
) ( ( ( trL `  K ) `
 W ) `  G ) ) )  ->  G  e.  T
)
6 simpr 447 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  Q  .<_  ( P ( join `  K
) ( ( ( trL `  K ) `
 W ) `  G ) ) )  ->  Q  .<_  ( P ( join `  K
) ( ( ( trL `  K ) `
 W ) `  G ) ) )
7 simpl33 1039 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  Q  .<_  ( P ( join `  K
) ( ( ( trL `  K ) `
 W ) `  G ) ) )  ->  ( F `  ( G `  P ) )  =  P )
8 cdlemg6.l . . . 4  |-  .<_  =  ( le `  K )
9 cdlemg6.a . . . 4  |-  A  =  ( Atoms `  K )
10 cdlemg6.h . . . 4  |-  H  =  ( LHyp `  K
)
11 cdlemg6.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
12 eqid 2366 . . . 4  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
13 eqid 2366 . . . 4  |-  ( join `  K )  =  (
join `  K )
14 eqid 2366 . . . 4  |-  ( ( ( trL `  K
) `  W ) `  G )  =  ( ( ( trL `  K
) `  W ) `  G )
158, 9, 10, 11, 12, 13, 14cdlemg6e 30882 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  Q  .<_  ( P ( join `  K
) ( ( ( trL `  K ) `
 W ) `  G ) )  /\  ( F `  ( G `
 P ) )  =  P ) )  ->  ( F `  ( G `  Q ) )  =  Q )
161, 2, 3, 4, 5, 6, 7, 15syl133anc 1206 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  Q  .<_  ( P ( join `  K
) ( ( ( trL `  K ) `
 W ) `  G ) ) )  ->  ( F `  ( G `  Q ) )  =  Q )
17 simpl1 959 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  -.  Q  .<_  ( P ( join `  K ) ( ( ( trL `  K
) `  W ) `  G ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
18 simpl2l 1009 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  -.  Q  .<_  ( P ( join `  K ) ( ( ( trL `  K
) `  W ) `  G ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
19 simpl2r 1010 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  -.  Q  .<_  ( P ( join `  K ) ( ( ( trL `  K
) `  W ) `  G ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
20 simpl31 1037 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  -.  Q  .<_  ( P ( join `  K ) ( ( ( trL `  K
) `  W ) `  G ) ) )  ->  F  e.  T
)
21 simpl32 1038 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  -.  Q  .<_  ( P ( join `  K ) ( ( ( trL `  K
) `  W ) `  G ) ) )  ->  G  e.  T
)
22 simpr 447 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  -.  Q  .<_  ( P ( join `  K ) ( ( ( trL `  K
) `  W ) `  G ) ) )  ->  -.  Q  .<_  ( P ( join `  K
) ( ( ( trL `  K ) `
 W ) `  G ) ) )
23 simpl33 1039 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  -.  Q  .<_  ( P ( join `  K ) ( ( ( trL `  K
) `  W ) `  G ) ) )  ->  ( F `  ( G `  P ) )  =  P )
248, 9, 10, 11, 12, 13, 14cdlemg4 30877 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P ( join `  K ) ( ( ( trL `  K
) `  W ) `  G ) )  /\  ( F `  ( G `
 P ) )  =  P ) )  ->  ( F `  ( G `  Q ) )  =  Q )
2517, 18, 19, 20, 21, 22, 23, 24syl133anc 1206 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P )
)  =  P ) )  /\  -.  Q  .<_  ( P ( join `  K ) ( ( ( trL `  K
) `  W ) `  G ) ) )  ->  ( F `  ( G `  Q ) )  =  Q )
2616, 25pm2.61dan 766 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  ( G `  P ) )  =  P ) )  -> 
( F `  ( G `  Q )
)  =  Q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   lecple 13423   joincjn 14288   Atomscatm 29524   HLchlt 29611   LHypclh 30244   LTrncltrn 30361   trLctrl 30418
This theorem is referenced by:  cdlemg7aN  30885  cdlemg8a  30887  cdlemg8c  30889  cdlemg11a  30897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-map 6917  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-llines 29758  df-lplanes 29759  df-lvols 29760  df-lines 29761  df-psubsp 29763  df-pmap 29764  df-padd 30056  df-lhyp 30248  df-laut 30249  df-ldil 30364  df-ltrn 30365  df-trl 30419
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