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Theorem cdlemeg46fvaw 34095
Description: Show that  ( F `
 R ) is an atom not under  W when  R is an atom not under  W. (Contributed by NM, 1-Apr-2013.)
Hypotheses
Ref Expression
cdlemef46g.b  |-  B  =  ( Base `  K
)
cdlemef46g.l  |-  .<_  =  ( le `  K )
cdlemef46g.j  |-  .\/  =  ( join `  K )
cdlemef46g.m  |-  ./\  =  ( meet `  K )
cdlemef46g.a  |-  A  =  ( Atoms `  K )
cdlemef46g.h  |-  H  =  ( LHyp `  K
)
cdlemef46g.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemef46g.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemefs46g.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdlemef46g.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) ) ,  x ) )
cdlemef46.v  |-  V  =  ( ( Q  .\/  P )  ./\  W )
cdlemef46.n  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
cdlemefs46.o  |-  O  =  ( ( Q  .\/  P )  ./\  ( N  .\/  ( ( u  .\/  v )  ./\  W
) ) )
cdlemef46.g  |-  G  =  ( a  e.  B  |->  if ( ( Q  =/=  P  /\  -.  a  .<_  W ) ,  ( iota_ c  e.  B  A. u  e.  A  ( ( -.  u  .<_  W  /\  ( u 
.\/  ( a  ./\  W ) )  =  a )  ->  c  =  ( if ( u  .<_  ( Q  .\/  P ) ,  ( iota_ b  e.  B  A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q 
.\/  P ) )  ->  b  =  O ) ) ,  [_ u  /  v ]_ N
)  .\/  ( a  ./\  W ) ) ) ) ,  a ) )
Assertion
Ref Expression
cdlemeg46fvaw  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q
)  ->  ( ( G `  R )  e.  A  /\  -.  ( G `  R )  .<_  W ) )
Distinct variable groups:    t, s, x, y, z, A    B, s, t, x, y, z    D, s, x, y, z   
x, E, y, z    H, s, t, x, y, z    .\/ , s, t, x, y, z    K, s, t, x, y, z    .<_ , s, t, x, y, z    ./\ , s, t, x, y, z    P, s, t, x, y, z    Q, s, t, x, y, z    R, s, t, x, y, z    U, s, t, x, y, z    W, s, t, x, y, z    a, b, c, u, v, A    B, a, b, c, u, v   
v, D    G, s,
t, x, y, z    H, a, b, c, u, v    .\/ , a, b, c, u, v    K, a, b, c, u, v    .<_ , a, b, c, u, v    ./\ , a, b, c, u, v    N, a, b, c    O, a, b, c    P, a, b, c, u, v    Q, a, b, c, u, v    R, a, b, c, u, v    V, a, b, c    W, a, b, c, u, v   
x, u, y, z, N    x, O, y, z    v, t    u, V    x, v, y, z, V
Allowed substitution hints:    D( u, t, a, b, c)    U( v, u, a, b, c)    E( v, u, t, s, a, b, c)    F( x, y, z, v, u, t, s, a, b, c)    G( v, u, a, b, c)    N( v, t, s)    O( v, u, t, s)    V( t, s)

Proof of Theorem cdlemeg46fvaw
StepHypRef Expression
1 simp11 1039 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q
)  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp13 1041 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q
)  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
3 simp12 1040 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q
)  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 simp2 1010 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q
)  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
5 cdlemef46g.b . . 3  |-  B  =  ( Base `  K
)
6 cdlemef46g.l . . 3  |-  .<_  =  ( le `  K )
7 cdlemef46g.j . . 3  |-  .\/  =  ( join `  K )
8 cdlemef46g.m . . 3  |-  ./\  =  ( meet `  K )
9 cdlemef46g.a . . 3  |-  A  =  ( Atoms `  K )
10 cdlemef46g.h . . 3  |-  H  =  ( LHyp `  K
)
11 cdlemef46.v . . 3  |-  V  =  ( ( Q  .\/  P )  ./\  W )
12 cdlemef46.n . . 3  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
13 cdlemefs46.o . . 3  |-  O  =  ( ( Q  .\/  P )  ./\  ( N  .\/  ( ( u  .\/  v )  ./\  W
) ) )
14 cdlemef46.g . . 3  |-  G  =  ( a  e.  B  |->  if ( ( Q  =/=  P  /\  -.  a  .<_  W ) ,  ( iota_ c  e.  B  A. u  e.  A  ( ( -.  u  .<_  W  /\  ( u 
.\/  ( a  ./\  W ) )  =  a )  ->  c  =  ( if ( u  .<_  ( Q  .\/  P ) ,  ( iota_ b  e.  B  A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q 
.\/  P ) )  ->  b  =  O ) ) ,  [_ u  /  v ]_ N
)  .\/  ( a  ./\  W ) ) ) ) ,  a ) )
155, 6, 7, 8, 9, 10, 11, 12, 13, 14cdleme46fvaw 34080 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( G `  R )  e.  A  /\  -.  ( G `  R )  .<_  W ) )
161, 2, 3, 4, 15syl31anc 1272 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q
)  ->  ( ( G `  R )  e.  A  /\  -.  ( G `  R )  .<_  W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   A.wral 2739   [_csb 3365   ifcif 3883   class class class wbr 4405    |-> cmpt 4464   ` cfv 5585   iota_crio 6256  (class class class)co 6295   Basecbs 15133   lecple 15209   joincjn 16201   meetcmee 16202   Atomscatm 32841   HLchlt 32928   LHypclh 33561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-riotaBAD 32537
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-undef 7025  df-preset 16185  df-poset 16203  df-plt 16216  df-lub 16232  df-glb 16233  df-join 16234  df-meet 16235  df-p0 16297  df-p1 16298  df-lat 16304  df-clat 16366  df-oposet 32754  df-ol 32756  df-oml 32757  df-covers 32844  df-ats 32845  df-atl 32876  df-cvlat 32900  df-hlat 32929  df-llines 33075  df-lplanes 33076  df-lvols 33077  df-lines 33078  df-psubsp 33080  df-pmap 33081  df-padd 33373  df-lhyp 33565
This theorem is referenced by:  cdlemeg46fjgN  34100  cdlemeg46fjv  34102  cdlemeg46frv  34104  cdlemeg46vrg  34106  cdlemeg46rgv  34107  cdlemeg46req  34108
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