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Theorem cdlemg34 35517
Description: Use cdlemg33 to eliminate  z from cdlemg29 35510. TODO: Fix comment. (Contributed by NM, 31-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
cdlemg34  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
Distinct variable groups:    A, r    G, r    .\/ , r    .<_ , r    P, r    Q, r    W, r    F, r    H, r    K, r    N, r    v, r    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)    ./\ ( v,
r)    N( v)    O( v)    W( v)

Proof of Theorem cdlemg34
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cdlemg12.l . . 3  |-  .<_  =  ( le `  K )
2 cdlemg12.j . . 3  |-  .\/  =  ( join `  K )
3 cdlemg12.m . . 3  |-  ./\  =  ( meet `  K )
4 cdlemg12.a . . 3  |-  A  =  ( Atoms `  K )
5 cdlemg12.h . . 3  |-  H  =  ( LHyp `  K
)
6 cdlemg12.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
7 cdlemg12b.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
8 cdlemg31.n . . 3  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
9 cdlemg33.o . . 3  |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  G ) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9cdlemg33 35516 . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  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 ) ) ) )
11 simp11 1026 . . . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A  /\  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
12 simp121 1128 . . . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A  /\  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )  ->  (
v  e.  A  /\  v  .<_  W ) )
13 simp2 997 . . . . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A  /\  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )  ->  z  e.  A )
14 simp3l 1024 . . . . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A  /\  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )  ->  -.  z  .<_  W )
1513, 14jca 532 . . . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A  /\  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )  ->  (
z  e.  A  /\  -.  z  .<_  W ) )
16 simp122 1129 . . . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A  /\  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )  ->  ( F  e.  T  /\  G  e.  T )
)
17 simp3r1 1104 . . . . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A  /\  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )  ->  z  =/=  N )
18 simp3r2 1105 . . . . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A  /\  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )  ->  z  =/=  O )
1917, 18jca 532 . . . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A  /\  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )  ->  (
z  =/=  N  /\  z  =/=  O ) )
20 simp3r3 1106 . . . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A  /\  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )  ->  z  .<_  ( P  .\/  v
) )
21 simp131 1131 . . . . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A  /\  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )  ->  v  =/=  ( R `  F
) )
22 simp132 1132 . . . . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A  /\  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )  ->  v  =/=  ( R `  G
) )
2321, 22jca 532 . . . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A  /\  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )  ->  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G ) ) )
241, 2, 3, 4, 5, 6, 7, 8, 9cdlemg29 35510 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
2511, 12, 15, 16, 19, 20, 23, 24syl133anc 1251 . . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A  /\  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )  ->  (
( P  .\/  ( F `  ( G `  P ) ) ) 
./\  W )  =  ( ( Q  .\/  ( F `  ( G `
 Q ) ) )  ./\  W )
)
2625rexlimdv3a 2957 . 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  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
) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) ) )
2710, 26mpd 15 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  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
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 2815   class class class wbr 4447   ` cfv 5587  (class class class)co 6283   lecple 14561   joincjn 15430   meetcmee 15431   Atomscatm 34069   HLchlt 34156   LHypclh 34789   LTrncltrn 34906   trLctrl 34963
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-riotaBAD 33765
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-1st 6784  df-2nd 6785  df-undef 7002  df-map 7422  df-poset 15432  df-plt 15444  df-lub 15460  df-glb 15461  df-join 15462  df-meet 15463  df-p0 15525  df-p1 15526  df-lat 15532  df-clat 15594  df-oposet 33982  df-ol 33984  df-oml 33985  df-covers 34072  df-ats 34073  df-atl 34104  df-cvlat 34128  df-hlat 34157  df-llines 34303  df-lplanes 34304  df-lvols 34305  df-lines 34306  df-psubsp 34308  df-pmap 34309  df-padd 34601  df-lhyp 34793  df-laut 34794  df-ldil 34909  df-ltrn 34910  df-trl 34964
This theorem is referenced by:  cdlemg36  35519
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