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Theorem cdlemg31b0a 35847
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 1020 . . 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 1113 . . 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 1117 . . 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 1115 . . 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 996 . . . 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 1024 . . . 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 2467 . . . . 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 35323 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  A  \/  ( R `  F )  =  ( 0. `  K ) ) )
135, 6, 12syl2anc 661 . . 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 1030 . . . . 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 35336 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
175, 6, 16syl2anc 661 . . . . . 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 532 . . . . 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 1031 . . . . 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 1025 . . . . . 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 2738 . . . . 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 35188 . . . . 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 1250 . . . 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 2738 . . 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 34677 . . 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 1243 . 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 2544 . . 3  |-  ( N  e.  A  <->  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
 F ) ) )  e.  A )
3129eqeq1i 2474 . . 3  |-  ( N  =  ( 0. `  K )  <->  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `
 F ) ) )  =  ( 0.
`  K ) )
3230, 31orbi12i 521 . 2  |-  ( ( N  e.  A  \/  N  =  ( 0. `  K ) )  <->  ( (
( P  .\/  v
)  ./\  ( Q  .\/  ( R `  F
) ) )  e.  A  \/  ( ( P  .\/  v ) 
./\  ( Q  .\/  ( R `  F ) ) )  =  ( 0. `  K ) ) )
3328, 32sylibr 212 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   lecple 14579   joincjn 15448   meetcmee 15449   0.cp0 15541   Atomscatm 34416   HLchlt 34503   LHypclh 35136   LTrncltrn 35253   trLctrl 35310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-map 7434  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34329  df-ol 34331  df-oml 34332  df-covers 34419  df-ats 34420  df-atl 34451  df-cvlat 34475  df-hlat 34504  df-llines 34650  df-psubsp 34655  df-pmap 34656  df-padd 34948  df-lhyp 35140  df-laut 35141  df-ldil 35256  df-ltrn 35257  df-trl 35311
This theorem is referenced by:  cdlemg27b  35848  cdlemg33  35863
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