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Theorem cdlemg33e 33709
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
cdlemg33e  |-  ( ( ( ( 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  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 cdlemg33e
StepHypRef Expression
1 simp1 997 . . 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  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 1030 . . 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( v  e.  A  /\  v  .<_  W ) )
3 simp23l 1118 . . 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  F  e.  T
)
4 simp3 999 . . 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( P  =/= 
Q  /\  v  =/=  ( R `  F )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )
5 cdlemg12.l . . . 4  |-  .<_  =  ( le `  K )
6 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
7 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
8 cdlemg12.a . . . 4  |-  A  =  ( Atoms `  K )
9 cdlemg12.h . . . 4  |-  H  =  ( LHyp `  K
)
10 cdlemg12.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
11 cdlemg12b.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
12 cdlemg31.n . . . 4  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
135, 6, 7, 8, 9, 10, 11, 12cdlemg33c0 33701 . . 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 )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  z  .<_  ( P 
.\/  v ) ) )
141, 2, 3, 4, 13syl121anc 1235 . 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  z  .<_  ( P 
.\/  v ) ) )
15 simp11l 1108 . . . . . . . . . 10  |-  ( ( ( ( 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  K  e.  HL )
16 hlatl 32358 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  AtLat )
1715, 16syl 17 . . . . . . . . 9  |-  ( ( ( ( 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  K  e.  AtLat )
18 eqid 2402 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
1918, 8atn0 32306 . . . . . . . . 9  |-  ( ( K  e.  AtLat  /\  z  e.  A )  ->  z  =/=  ( 0. `  K
) )
2017, 19sylan 469 . . . . . . . 8  |-  ( ( ( ( ( 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  z  =/=  ( 0. `  K ) )
21 simp22l 1116 . . . . . . . . 9  |-  ( ( ( ( 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  N  =  ( 0. `  K ) )
2221adantr 463 . . . . . . . 8  |-  ( ( ( ( ( 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  N  =  ( 0. `  K ) )
2320, 22neeqtrrd 2703 . . . . . . 7  |-  ( ( ( ( ( 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  z  =/=  N )
24 simp22r 1117 . . . . . . . . 9  |-  ( ( ( ( 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  O  =  ( 0. `  K ) )
2524adantr 463 . . . . . . . 8  |-  ( ( ( ( ( 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  O  =  ( 0. `  K ) )
2620, 25neeqtrrd 2703 . . . . . . 7  |-  ( ( ( ( ( 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  z  =/=  O )
2723, 26jca 530 . . . . . 6  |-  ( ( ( ( ( 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  ( z  =/=  N  /\  z  =/= 
O ) )
2827biantrurd 506 . . . . 5  |-  ( ( ( ( ( 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  ( z  .<_  ( P  .\/  v
)  <->  ( ( z  =/=  N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v
) ) ) )
29 df-3an 976 . . . . 5  |-  ( ( z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v
) )  <->  ( (
z  =/=  N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v ) ) )
3028, 29syl6bbr 263 . . . 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  ( z  .<_  ( P  .\/  v
)  <->  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )
3130anbi2d 702 . . 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  ( ( -.  z  .<_  W  /\  z  .<_  ( P  .\/  v ) )  <->  ( -.  z  .<_  W  /\  (
z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v
) ) ) ) )
3231rexbidva 2914 . 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( E. z  e.  A  ( -.  z  .<_  W  /\  z  .<_  ( P  .\/  v
) )  <->  E. z  e.  A  ( -.  z  .<_  W  /\  (
z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v
) ) ) ) )
3314, 32mpbid 210 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  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   lecple 14914   joincjn 15895   meetcmee 15896   0.cp0 15989   Atomscatm 32261   AtLatcal 32262   HLchlt 32348   LHypclh 32981   LTrncltrn 33098   trLctrl 33156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lhyp 32985
This theorem is referenced by:  cdlemg33  33710
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