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Theorem cdleme50lebi 30547
Description: Part of proof of Lemma D in [Crawley] p. 113. TODO: fix comment. (Contributed by NM, 9-Apr-2013.)
Hypotheses
Ref Expression
cdlemef50.b  |-  B  =  ( Base `  K
)
cdlemef50.l  |-  .<_  =  ( le `  K )
cdlemef50.j  |-  .\/  =  ( join `  K )
cdlemef50.m  |-  ./\  =  ( meet `  K )
cdlemef50.a  |-  A  =  ( Atoms `  K )
cdlemef50.h  |-  H  =  ( LHyp `  K
)
cdlemef50.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemef50.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemefs50.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdlemef50.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 ) )
Assertion
Ref Expression
cdleme50lebi  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
 Y ) ) )
Distinct variable groups:    t, s, x, y, z,  ./\    .\/ , s,
t, x, y, z    .<_ , s, t, x, y, z    A, s, t, x, y, z    B, s, t, x, y, z    D, s, x, y, z   
x, E, y, z    H, s, t, x, y, z    K, 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    X, s, t, x, y, z    Y, s, t, x, y, z
Allowed substitution hints:    D( t)    E( t, s)    F( x, y, z, t, s)

Proof of Theorem cdleme50lebi
Dummy variables  a 
b  c  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdlemef50.b . 2  |-  B  =  ( Base `  K
)
2 cdlemef50.l . 2  |-  .<_  =  ( le `  K )
3 cdlemef50.j . 2  |-  .\/  =  ( join `  K )
4 cdlemef50.m . 2  |-  ./\  =  ( meet `  K )
5 cdlemef50.a . 2  |-  A  =  ( Atoms `  K )
6 cdlemef50.h . 2  |-  H  =  ( LHyp `  K
)
7 cdlemef50.u . 2  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 cdlemef50.d . 2  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
9 cdlemefs50.e . 2  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
10 cdlemef50.f . 2  |-  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 ) )
11 eqid 2316 . 2  |-  ( ( Q  .\/  P ) 
./\  W )  =  ( ( Q  .\/  P )  ./\  W )
12 eqid 2316 . 2  |-  ( ( v  .\/  ( ( Q  .\/  P ) 
./\  W ) ) 
./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )  =  ( ( v  .\/  ( ( Q  .\/  P )  ./\  W )
)  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
13 eqid 2316 . 2  |-  ( ( Q  .\/  P ) 
./\  ( ( ( v  .\/  ( ( Q  .\/  P ) 
./\  W ) ) 
./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )  .\/  ( ( u  .\/  v )  ./\  W
) ) )  =  ( ( Q  .\/  P )  ./\  ( (
( v  .\/  (
( Q  .\/  P
)  ./\  W )
)  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )  .\/  ( ( u  .\/  v )  ./\  W
) ) )
14 eqid 2316 . 2  |-  ( 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  =  ( ( Q 
.\/  P )  ./\  ( ( ( v 
.\/  ( ( Q 
.\/  P )  ./\  W ) )  ./\  ( P  .\/  ( ( Q 
.\/  v )  ./\  W ) ) )  .\/  ( ( u  .\/  v )  ./\  W
) ) ) ) ) ,  [_ u  /  v ]_ (
( v  .\/  (
( Q  .\/  P
)  ./\  W )
)  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) ) ) 
.\/  ( a  ./\  W ) ) ) ) ,  a ) )  =  ( 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  =  ( ( Q 
.\/  P )  ./\  ( ( ( v 
.\/  ( ( Q 
.\/  P )  ./\  W ) )  ./\  ( P  .\/  ( ( Q 
.\/  v )  ./\  W ) ) )  .\/  ( ( u  .\/  v )  ./\  W
) ) ) ) ) ,  [_ u  /  v ]_ (
( v  .\/  (
( Q  .\/  P
)  ./\  W )
)  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) ) ) 
.\/  ( a  ./\  W ) ) ) ) ,  a ) )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cdlemeg49lebilem 30546 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
 Y ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   A.wral 2577   [_csb 3115   ifcif 3599   class class class wbr 4060    e. cmpt 4114   ` cfv 5292  (class class class)co 5900   iota_crio 6339   Basecbs 13195   lecple 13262   joincjn 14127   meetcmee 14128   Atomscatm 29271   HLchlt 29358   LHypclh 29991
This theorem is referenced by:  cdleme50eq  30548  cdleme50laut  30554
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-llines 29505  df-lplanes 29506  df-lvols 29507  df-lines 29508  df-psubsp 29510  df-pmap 29511  df-padd 29803  df-lhyp 29995
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