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Theorem cdleme50rn 31181
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
cdleme50rn  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ran  F  =  B )
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
Allowed substitution hints:    D( t)    E( t, s)    F( x, y, z, t, s)

Proof of Theorem cdleme50rn
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 2435 . 2  |-  ( ( Q  .\/  P ) 
./\  W )  =  ( ( Q  .\/  P )  ./\  W )
12 eqid 2435 . 2  |-  ( ( v  .\/  ( ( Q  .\/  P ) 
./\  W ) ) 
./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )  =  ( ( v  .\/  ( ( Q  .\/  P )  ./\  W )
)  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
13 eqid 2435 . 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 2435 . 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, 14cdleme50rnlem 31180 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ran  F  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   [_csb 3243   ifcif 3731   class class class wbr 4204    e. cmpt 4258   ran crn 4870   ` cfv 5445  (class class class)co 6072   iota_crio 6533   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   Atomscatm 29900   HLchlt 29987   LHypclh 30620
This theorem is referenced by:  cdleme50f1o  31182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624
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