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Theorem cdlemg33d 35861
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 ) ) )
cdlemg33.o  |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  G ) ) )
Assertion
Ref Expression
cdlemg33d  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  =  ( 0. `  K )  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )
Distinct variable groups:    A, r    G, r    .\/ , r    .<_ , r    P, r    Q, r    W, r    F, r    z, A    z, F, r    H, r, z   
z,  .\/    K, r, z   
z,  .<_    N, r, z    z, P    z, Q    z, R    z, T    z, W    z,
v, r    z, G    z, O, r
Allowed substitution hints:    A( v)    P( v)    Q( v)    R( v, r)    T( v, r)    F( v)    G( v)    H( v)    .\/ ( v)    K( v)    .<_ ( v)    ./\ ( z,
v, r)    N( v)    O( v)    W( v)

Proof of Theorem cdlemg33d
StepHypRef Expression
1 simp1 996 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  =  ( 0. `  K )  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
2 simp21 1029 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  =  ( 0. `  K )  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( v  e.  A  /\  v  .<_  W ) )
3 simp22r 1116 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  =  ( 0. `  K )  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  O  e.  A
)
4 simp22l 1115 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  =  ( 0. `  K )  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  N  =  ( 0. `  K ) )
53, 4jca 532 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  =  ( 0. `  K )  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( O  e.  A  /\  N  =  ( 0. `  K
) ) )
6 simp23r 1118 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  =  ( 0. `  K )  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  G  e.  T
)
7 simp23l 1117 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  =  ( 0. `  K )  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  F  e.  T
)
86, 7jca 532 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  =  ( 0. `  K )  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( G  e.  T  /\  F  e.  T ) )
9 simp3 998 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  =  ( 0. `  K )  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( P  =/= 
Q  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )
10 cdlemg12.l . . . 4  |-  .<_  =  ( le `  K )
11 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
12 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
13 cdlemg12.a . . . 4  |-  A  =  ( Atoms `  K )
14 cdlemg12.h . . . 4  |-  H  =  ( LHyp `  K
)
15 cdlemg12.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
16 cdlemg12b.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
17 cdlemg33.o . . . 4  |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  G ) ) )
18 cdlemg31.n . . . 4  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
1910, 11, 12, 13, 14, 15, 16, 17, 18cdlemg33c 35860 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( O  e.  A  /\  N  =  ( 0. `  K ) )  /\  ( G  e.  T  /\  F  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
O  /\  z  =/=  N  /\  z  .<_  ( P 
.\/  v ) ) ) )
201, 2, 5, 8, 9, 19syl131anc 1241 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  =  ( 0. `  K )  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
O  /\  z  =/=  N  /\  z  .<_  ( P 
.\/  v ) ) ) )
21 3ancoma 980 . . . 4  |-  ( ( z  =/=  O  /\  z  =/=  N  /\  z  .<_  ( P  .\/  v
) )  <->  ( z  =/=  N  /\  z  =/= 
O  /\  z  .<_  ( P  .\/  v ) ) )
2221anbi2i 694 . . 3  |-  ( ( -.  z  .<_  W  /\  ( z  =/=  O  /\  z  =/=  N  /\  z  .<_  ( P 
.\/  v ) ) )  <->  ( -.  z  .<_  W  /\  ( z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v ) ) ) )
2322rexbii 2969 . 2  |-  ( E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/=  O  /\  z  =/=  N  /\  z  .<_  ( P 
.\/  v ) ) )  <->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )
2420, 23sylib 196 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  =  ( 0. `  K )  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   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-lplanes 34651  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:  cdlemg33  35863
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