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Theorem cdlemg12d 35319
Description: TODO: FIX COMMENT (Contributed by NM, 5-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg12d  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( R `  G
)  .<_  ( ( R `
 F )  .\/  ( ( ( F `
 ( G `  P ) )  .\/  P )  ./\  ( ( F `  ( G `  Q ) )  .\/  Q ) ) ) )

Proof of Theorem cdlemg12d
StepHypRef Expression
1 simp11 1021 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp12 1022 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
3 simp13 1023 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
4 simp2l 1017 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  F  e.  T )
5 simp2r 1018 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  G  e.  T )
6 simp31 1027 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  P  =/=  Q )
7 simp33 1029 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  -.  ( R `  G
)  .<_  ( P  .\/  Q ) )
8 cdlemg12.l . . . 4  |-  .<_  =  ( le `  K )
9 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
10 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
11 cdlemg12.a . . . 4  |-  A  =  ( Atoms `  K )
12 cdlemg12.h . . . 4  |-  H  =  ( LHyp `  K
)
13 cdlemg12.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
14 cdlemg12b.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
158, 9, 10, 11, 12, 13, 14cdlemg12c 35318 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  ( G `  P ) )  ./\  ( Q  .\/  ( G `  Q
) ) )  .<_  ( ( ( ( G `  P ) 
.\/  ( F `  ( G `  P ) ) )  ./\  (
( G `  Q
)  .\/  ( F `  ( G `  Q
) ) ) ) 
.\/  ( ( ( F `  ( G `
 P ) ) 
.\/  P )  ./\  ( ( F `  ( G `  Q ) )  .\/  Q ) ) ) )
161, 2, 3, 4, 5, 6, 7, 15syl133anc 1246 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  ( G `  P ) )  ./\  ( Q  .\/  ( G `  Q
) ) )  .<_  ( ( ( ( G `  P ) 
.\/  ( F `  ( G `  P ) ) )  ./\  (
( G `  Q
)  .\/  ( F `  ( G `  Q
) ) ) ) 
.\/  ( ( ( F `  ( G `
 P ) ) 
.\/  P )  ./\  ( ( F `  ( G `  Q ) )  .\/  Q ) ) ) )
178, 9, 10, 11, 12, 13, 14trlval4 34861 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  ( R `  G ) 
.<_  ( P  .\/  Q
) ) )  -> 
( R `  G
)  =  ( ( P  .\/  ( G `
 P ) ) 
./\  ( Q  .\/  ( G `  Q ) ) ) )
181, 5, 2, 3, 6, 7, 17syl132anc 1241 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( R `  G
)  =  ( ( P  .\/  ( G `
 P ) ) 
./\  ( Q  .\/  ( G `  Q ) ) ) )
198, 11, 12, 13ltrnel 34812 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
201, 5, 2, 19syl3anc 1223 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
218, 11, 12, 13ltrnel 34812 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( ( G `  Q )  e.  A  /\  -.  ( G `  Q )  .<_  W ) )
221, 5, 3, 21syl3anc 1223 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( G `  Q )  e.  A  /\  -.  ( G `  Q )  .<_  W ) )
23 simp12l 1104 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  P  e.  A )
24 simp13l 1106 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  Q  e.  A )
2511, 12, 13ltrn11at 34820 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  ( G `  P )  =/=  ( G `  Q
) )
261, 5, 23, 24, 6, 25syl113anc 1235 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( G `  P
)  =/=  ( G `
 Q ) )
27 simp32 1028 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  -.  ( R `  F
)  .<_  ( P  .\/  Q ) )
28 simp2 992 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( F  e.  T  /\  G  e.  T
) )
298, 9, 10, 11, 12, 13, 14cdlemg10c 35312 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( ( R `  F )  .<_  ( ( G `  P ) 
.\/  ( G `  Q ) )  <->  ( R `  F )  .<_  ( P 
.\/  Q ) ) )
301, 2, 3, 28, 29syl121anc 1228 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( R `  F )  .<_  ( ( G `  P ) 
.\/  ( G `  Q ) )  <->  ( R `  F )  .<_  ( P 
.\/  Q ) ) )
3127, 30mtbird 301 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  -.  ( R `  F
)  .<_  ( ( G `
 P )  .\/  ( G `  Q ) ) )
328, 9, 10, 11, 12, 13, 14trlval4 34861 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( ( G `  P )  e.  A  /\  -.  ( G `  P ) 
.<_  W )  /\  (
( G `  Q
)  e.  A  /\  -.  ( G `  Q
)  .<_  W ) )  /\  ( ( G `
 P )  =/=  ( G `  Q
)  /\  -.  ( R `  F )  .<_  ( ( G `  P )  .\/  ( G `  Q )
) ) )  -> 
( R `  F
)  =  ( ( ( G `  P
)  .\/  ( F `  ( G `  P
) ) )  ./\  ( ( G `  Q )  .\/  ( F `  ( G `  Q ) ) ) ) )
331, 4, 20, 22, 26, 31, 32syl132anc 1241 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( R `  F
)  =  ( ( ( G `  P
)  .\/  ( F `  ( G `  P
) ) )  ./\  ( ( G `  Q )  .\/  ( F `  ( G `  Q ) ) ) ) )
3433oveq1d 6292 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( R `  F )  .\/  (
( ( F `  ( G `  P ) )  .\/  P ) 
./\  ( ( F `
 ( G `  Q ) )  .\/  Q ) ) )  =  ( ( ( ( G `  P ) 
.\/  ( F `  ( G `  P ) ) )  ./\  (
( G `  Q
)  .\/  ( F `  ( G `  Q
) ) ) ) 
.\/  ( ( ( F `  ( G `
 P ) ) 
.\/  P )  ./\  ( ( F `  ( G `  Q ) )  .\/  Q ) ) ) )
3516, 18, 343brtr4d 4472 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( P  =/=  Q  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( R `  G
)  .<_  ( ( R `
 F )  .\/  ( ( ( F `
 ( G `  P ) )  .\/  P )  ./\  ( ( F `  ( G `  Q ) )  .\/  Q ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   lecple 14553   joincjn 15422   meetcmee 15423   Atomscatm 33937   HLchlt 34024   LHypclh 34657   LTrncltrn 34774   trLctrl 34831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-riotaBAD 33633
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-undef 6994  df-map 7414  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-p1 15518  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-llines 34171  df-lplanes 34172  df-lvols 34173  df-lines 34174  df-psubsp 34176  df-pmap 34177  df-padd 34469  df-lhyp 34661  df-laut 34662  df-ldil 34777  df-ltrn 34778  df-trl 34832
This theorem is referenced by:  cdlemg12e  35320
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