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Theorem cdleme48gfv 31031
Description: TODO: fix comment. (Contributed by NM, 9-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
cdleme48gfv  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  ->  ( G `  ( F `  X )
)  =  X )
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    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    V, a, b, c    W, a, b, c, u, v, x, y, z   
u, N, x, y, z    x, O, y, z    v, t    u, V    x, v, y, z, V    D, a, b, c    E, a, b, c    F, a, b, c, u, v   
t, N    U, a,
b, c, v    t, V    s, a, t, b, c, x, y, z, u, v    X, a, c, s, t, u, v, x, z
Allowed substitution hints:    D( u, t)    U( u)    E( v, u, t, s)    F( x, y, z, t, s)    G( v, u, a, b, c)    N( v, s)    O( v, u, t, s)    V( s)    X( y, b)

Proof of Theorem cdleme48gfv
StepHypRef Expression
1 simpll 731 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
2 simprl 733 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  P  =/=  Q )
3 simplr 732 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  X  e.  B )
4 simprr 734 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  -.  X  .<_  W )
53, 4jca 519 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( X  e.  B  /\  -.  X  .<_  W ) )
6 cdlemef46g.b . . . 4  |-  B  =  ( Base `  K
)
7 cdlemef46g.l . . . 4  |-  .<_  =  ( le `  K )
8 cdlemef46g.j . . . 4  |-  .\/  =  ( join `  K )
9 cdlemef46g.m . . . 4  |-  ./\  =  ( meet `  K )
10 cdlemef46g.a . . . 4  |-  A  =  ( Atoms `  K )
11 cdlemef46g.h . . . 4  |-  H  =  ( LHyp `  K
)
12 cdlemef46g.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
13 cdlemef46g.d . . . 4  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
14 cdlemefs46g.e . . . 4  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
15 cdlemef46g.f . . . 4  |-  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 ) )
16 cdlemef46.v . . . 4  |-  V  =  ( ( Q  .\/  P )  ./\  W )
17 cdlemef46.n . . . 4  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
18 cdlemefs46.o . . . 4  |-  O  =  ( ( Q  .\/  P )  ./\  ( N  .\/  ( ( u  .\/  v )  ./\  W
) ) )
19 cdlemef46.g . . . 4  |-  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 ) )
206, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19cdleme48gfv1 31030 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( G `  ( F `  X ) )  =  X )
211, 2, 5, 20syl12anc 1182 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( G `  ( F `  X
) )  =  X )
2215cdleme31fv2 30887 . . . . . 6  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( F `  X )  =  X )
2322adantll 695 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  -> 
( F `  X
)  =  X )
24 simplr 732 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  ->  X  e.  B )
2523, 24eqeltrd 2486 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  -> 
( F `  X
)  e.  B )
26 simpr 448 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  ->  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )
27 necom 2656 . . . . . . 7  |-  ( Q  =/=  P  <->  P  =/=  Q )
2827a1i 11 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  -> 
( Q  =/=  P  <->  P  =/=  Q ) )
2923breq1d 4190 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  -> 
( ( F `  X )  .<_  W  <->  X  .<_  W ) )
3029notbid 286 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  -> 
( -.  ( F `
 X )  .<_  W 
<->  -.  X  .<_  W ) )
3128, 30anbi12d 692 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  -> 
( ( Q  =/= 
P  /\  -.  ( F `  X )  .<_  W )  <->  ( P  =/=  Q  /\  -.  X  .<_  W ) ) )
3226, 31mtbird 293 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  ->  -.  ( Q  =/=  P  /\  -.  ( F `  X )  .<_  W ) )
3319cdleme31fv2 30887 . . . 4  |-  ( ( ( F `  X
)  e.  B  /\  -.  ( Q  =/=  P  /\  -.  ( F `  X )  .<_  W ) )  ->  ( G `  ( F `  X
) )  =  ( F `  X ) )
3425, 32, 33syl2anc 643 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  -> 
( G `  ( F `  X )
)  =  ( F `
 X ) )
3534, 23eqtrd 2444 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  -> 
( G `  ( F `  X )
)  =  X )
3621, 35pm2.61dan 767 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  ->  ( G `  ( F `  X )
)  =  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   [_csb 3219   ifcif 3707   class class class wbr 4180    e. cmpt 4234   ` cfv 5421  (class class class)co 6048   iota_crio 6509   Basecbs 13432   lecple 13499   joincjn 14364   meetcmee 14365   Atomscatm 29758   HLchlt 29845   LHypclh 30478
This theorem is referenced by:  cdleme48fgv  31032  cdlemeg49lebilem  31033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992  df-lplanes 29993  df-lvols 29994  df-lines 29995  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482
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