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Theorem cdlemg31b0N 34173
Description: TODO: Fix comment. (Contributed by NM, 30-May-2013.) (New usage is discouraged.)
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 )
cdlemg31.n  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
Assertion
Ref Expression
cdlemg31b0N  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
( N  e.  A  \/  N  =  ( 0. `  K ) ) )

Proof of Theorem cdlemg31b0N
StepHypRef Expression
1 simp11 1035 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  ->  K  e.  HL )
2 simp2ll 1072 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  ->  P  e.  A )
3 simp31l 1128 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
v  e.  A )
4 simp2rl 1074 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  ->  Q  e.  A )
5 simp12 1036 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  ->  W  e.  H )
61, 5jca 534 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
7 simp2l 1031 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
8 simp13 1037 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  ->  F  e.  T )
9 simp33 1043 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
( F `  P
)  =/=  P )
10 cdlemg12.l . . . . 5  |-  .<_  =  ( le `  K )
11 cdlemg12.a . . . . 5  |-  A  =  ( Atoms `  K )
12 cdlemg12.h . . . . 5  |-  H  =  ( LHyp `  K
)
13 cdlemg12.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
14 cdlemg12b.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
1510, 11, 12, 13, 14trlat 33647 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
166, 7, 8, 9, 15syl112anc 1268 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
( R `  F
)  e.  A )
17 simp2r 1032 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
1810, 12, 13, 14trlle 33662 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
196, 8, 18syl2anc 665 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
( R `  F
)  .<_  W )
2016, 19jca 534 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
( ( R `  F )  e.  A  /\  ( R `  F
)  .<_  W ) )
21 simp31 1041 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
( v  e.  A  /\  v  .<_  W ) )
22 simp32 1042 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
v  =/=  ( R `
 F ) )
2322necomd 2656 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
( R `  F
)  =/=  v )
24 cdlemg12.j . . . . . 6  |-  .\/  =  ( join `  K )
2510, 24, 11, 12lhp2atne 33511 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  e.  A
)  /\  ( (
( R `  F
)  e.  A  /\  ( R `  F ) 
.<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( R `  F )  =/=  v
)  ->  ( Q  .\/  ( R `  F
) )  =/=  ( P  .\/  v ) )
266, 17, 2, 20, 21, 23, 25syl321anc 1286 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
( Q  .\/  ( R `  F )
)  =/=  ( P 
.\/  v ) )
2726necomd 2656 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
( P  .\/  v
)  =/=  ( Q 
.\/  ( R `  F ) ) )
28 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
29 eqid 2428 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
3024, 28, 29, 112atmat0 33003 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  v  e.  A )  /\  ( Q  e.  A  /\  ( R `  F
)  e.  A  /\  ( P  .\/  v )  =/=  ( Q  .\/  ( R `  F ) ) ) )  -> 
( ( ( P 
.\/  v )  ./\  ( Q  .\/  ( R `
 F ) ) )  e.  A  \/  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )  =  ( 0. `  K ) ) )
311, 2, 3, 4, 16, 27, 30syl33anc 1279 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
( ( ( P 
.\/  v )  ./\  ( Q  .\/  ( R `
 F ) ) )  e.  A  \/  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )  =  ( 0. `  K ) ) )
32 cdlemg31.n . . . 4  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
3332eleq1i 2497 . . 3  |-  ( N  e.  A  <->  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
 F ) ) )  e.  A )
3432eqeq1i 2433 . . 3  |-  ( N  =  ( 0. `  K )  <->  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
 F ) ) )  =  ( 0.
`  K ) )
3533, 34orbi12i 523 . 2  |-  ( ( N  e.  A  \/  N  =  ( 0. `  K ) )  <->  ( (
( P  .\/  v
)  ./\  ( Q  .\/  ( R `  F
) ) )  e.  A  \/  ( ( P  .\/  v ) 
./\  ( Q  .\/  ( R `  F ) ) )  =  ( 0. `  K ) ) )
3631, 35sylibr 215 1  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  v  =/=  ( R `  F )  /\  ( F `  P )  =/=  P ) )  -> 
( N  e.  A  \/  N  =  ( 0. `  K ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   class class class wbr 4366   ` cfv 5544  (class class class)co 6249   lecple 15140   joincjn 16132   meetcmee 16133   0.cp0 16226   Atomscatm 32741   HLchlt 32828   LHypclh 33461   LTrncltrn 33578   trLctrl 33636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-iin 4245  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-1st 6751  df-2nd 6752  df-map 7429  df-preset 16116  df-poset 16134  df-plt 16147  df-lub 16163  df-glb 16164  df-join 16165  df-meet 16166  df-p0 16228  df-p1 16229  df-lat 16235  df-clat 16297  df-oposet 32654  df-ol 32656  df-oml 32657  df-covers 32744  df-ats 32745  df-atl 32776  df-cvlat 32800  df-hlat 32829  df-llines 32975  df-psubsp 32980  df-pmap 32981  df-padd 33273  df-lhyp 33465  df-laut 33466  df-ldil 33581  df-ltrn 33582  df-trl 33637
This theorem is referenced by: (None)
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