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Theorem cdlemg31b0a 31590
Description: TODO: Fix comment. (Contributed by NM, 30-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 )
cdlemg31.n  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
Assertion
Ref Expression
cdlemg31b0a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  ( N  e.  A  \/  N  =  ( 0. `  K ) ) )

Proof of Theorem cdlemg31b0a
StepHypRef Expression
1 simp1l 982 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  K  e.  HL )
2 simp21l 1075 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  P  e.  A )
3 simp23l 1079 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  v  e.  A )
4 simp22l 1077 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  Q  e.  A )
5 simp1 958 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
6 simp3l 986 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  F  e.  T )
7 eqid 2442 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
8 cdlemg12.a . . . . 5  |-  A  =  ( Atoms `  K )
9 cdlemg12.h . . . . 5  |-  H  =  ( LHyp `  K
)
10 cdlemg12.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
11 cdlemg12b.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
127, 8, 9, 10, 11trlator0 31066 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  A  \/  ( R `  F )  =  ( 0. `  K ) ) )
135, 6, 12syl2anc 644 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  (
( R `  F
)  e.  A  \/  ( R `  F )  =  ( 0. `  K ) ) )
14 simp22 992 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
15 cdlemg12.l . . . . . . . 8  |-  .<_  =  ( le `  K )
1615, 9, 10, 11trlle 31079 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
175, 6, 16syl2anc 644 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  ( R `  F )  .<_  W )
1813, 17jca 520 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  (
( ( R `  F )  e.  A  \/  ( R `  F
)  =  ( 0.
`  K ) )  /\  ( R `  F )  .<_  W ) )
19 simp23 993 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  (
v  e.  A  /\  v  .<_  W ) )
20 simp3r 987 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  v  =/=  ( R `  F
) )
2120necomd 2693 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  ( R `  F )  =/=  v )
22 cdlemg12.j . . . . . 6  |-  .\/  =  ( join `  K )
2315, 22, 7, 8, 9lhp2at0ne 30931 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  e.  A
)  /\  ( (
( ( R `  F )  e.  A  \/  ( R `  F
)  =  ( 0.
`  K ) )  /\  ( R `  F )  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( R `  F )  =/=  v )  -> 
( Q  .\/  ( R `  F )
)  =/=  ( P 
.\/  v ) )
245, 14, 2, 18, 19, 21, 23syl321anc 1207 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  ( Q  .\/  ( R `  F ) )  =/=  ( P  .\/  v
) )
2524necomd 2693 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  ( P  .\/  v )  =/=  ( Q  .\/  ( R `  F )
) )
26 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
2722, 26, 7, 82at0mat0 30420 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  v  e.  A )  /\  ( Q  e.  A  /\  ( ( R `  F )  e.  A  \/  ( R `  F
)  =  ( 0.
`  K ) )  /\  ( P  .\/  v )  =/=  ( Q  .\/  ( R `  F ) ) ) )  ->  ( (
( P  .\/  v
)  ./\  ( Q  .\/  ( R `  F
) ) )  e.  A  \/  ( ( P  .\/  v ) 
./\  ( Q  .\/  ( R `  F ) ) )  =  ( 0. `  K ) ) )
281, 2, 3, 4, 13, 25, 27syl33anc 1200 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  (
( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )  e.  A  \/  (
( P  .\/  v
)  ./\  ( Q  .\/  ( R `  F
) ) )  =  ( 0. `  K
) ) )
29 cdlemg31.n . . . 4  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
3029eleq1i 2505 . . 3  |-  ( N  e.  A  <->  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
 F ) ) )  e.  A )
3129eqeq1i 2449 . . 3  |-  ( N  =  ( 0. `  K )  <->  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
 F ) ) )  =  ( 0.
`  K ) )
3230, 31orbi12i 509 . 2  |-  ( ( N  e.  A  \/  N  =  ( 0. `  K ) )  <->  ( (
( P  .\/  v
)  ./\  ( Q  .\/  ( R `  F
) ) )  e.  A  \/  ( ( P  .\/  v ) 
./\  ( Q  .\/  ( R `  F ) ) )  =  ( 0. `  K ) ) )
3328, 32sylibr 205 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  ( N  e.  A  \/  N  =  ( 0. `  K ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   lecple 13567   joincjn 14432   meetcmee 14433   0.cp0 14497   Atomscatm 30159   HLchlt 30246   LHypclh 30879   LTrncltrn 30996   trLctrl 31053
This theorem is referenced by:  cdlemg27b  31591  cdlemg33  31606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-map 7049  df-poset 14434  df-plt 14446  df-lub 14462  df-glb 14463  df-join 14464  df-meet 14465  df-p0 14499  df-p1 14500  df-lat 14506  df-clat 14568  df-oposet 30072  df-ol 30074  df-oml 30075  df-covers 30162  df-ats 30163  df-atl 30194  df-cvlat 30218  df-hlat 30247  df-llines 30393  df-psubsp 30398  df-pmap 30399  df-padd 30691  df-lhyp 30883  df-laut 30884  df-ldil 30999  df-ltrn 31000  df-trl 31054
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