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Theorem cdlemg31b0N 30942
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 986 . . 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 1023 . . 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 1079 . . 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 1025 . . 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 987 . . . . 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 518 . . . 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 982 . . . 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 988 . . . 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 994 . . . 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 30417 . . . 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 1187 . . 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 983 . . . . 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 30432 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
196, 8, 18syl2anc 642 . . . . . 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 518 . . . . 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 992 . . . . 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 993 . . . . . 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 2612 . . . . 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 30282 . . . . 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 1205 . . . 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 2612 . . 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 2366 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
3024, 28, 29, 112atmat0 29774 . . 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 1198 . 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 2429 . . 3  |-  ( N  e.  A  <->  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
 F ) ) )  e.  A )
3432eqeq1i 2373 . . 3  |-  ( N  =  ( 0. `  K )  <->  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
 F ) ) )  =  ( 0.
`  K ) )
3533, 34orbi12i 507 . 2  |-  ( ( N  e.  A  \/  N  =  ( 0. `  K ) )  <->  ( (
( P  .\/  v
)  ./\  ( Q  .\/  ( R `  F
) ) )  e.  A  \/  ( ( P  .\/  v ) 
./\  ( Q  .\/  ( R `  F ) ) )  =  ( 0. `  K ) ) )
3631, 35sylibr 203 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 set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   lecple 13423   joincjn 14288   meetcmee 14289   0.cp0 14353   Atomscatm 29512   HLchlt 29599   LHypclh 30232   LTrncltrn 30349   trLctrl 30406
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-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-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 29425  df-ol 29427  df-oml 29428  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-llines 29746  df-psubsp 29751  df-pmap 29752  df-padd 30044  df-lhyp 30236  df-laut 30237  df-ldil 30352  df-ltrn 30353  df-trl 30407
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