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Theorem cdleme51finvtrN 31040
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: fix comment. (Contributed by NM, 14-Apr-2013.) (New usage is discouraged.)
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 ) )
cdleme50ltrn.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdleme51finvtrN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  `' F  e.  T )
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)    T( x, y, z, t, s)    E( t, s)    F( x, y, z, t, s)

Proof of Theorem cdleme51finvtrN
Dummy variables  a 
b  c  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdlemef50.b . . 3  |-  B  =  ( Base `  K
)
2 cdlemef50.l . . 3  |-  .<_  =  ( le `  K )
3 cdlemef50.j . . 3  |-  .\/  =  ( join `  K )
4 cdlemef50.m . . 3  |-  ./\  =  ( meet `  K )
5 cdlemef50.a . . 3  |-  A  =  ( Atoms `  K )
6 cdlemef50.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdlemef50.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 cdlemef50.d . . 3  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
9 cdlemefs50.e . . 3  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
10 cdlemef50.f . . 3  |-  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 2404 . . 3  |-  ( ( Q  .\/  P ) 
./\  W )  =  ( ( Q  .\/  P )  ./\  W )
12 eqid 2404 . . 3  |-  ( ( v  .\/  ( ( Q  .\/  P ) 
./\  W ) ) 
./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )  =  ( ( v  .\/  ( ( Q  .\/  P )  ./\  W )
)  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
13 eqid 2404 . . 3  |-  ( ( 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 2404 . . 3  |-  ( 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, 14cdleme51finvN 31038 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  `' F  =  ( 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 ) ) )
16 cdleme50ltrn.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
171, 2, 3, 4, 5, 6, 11, 12, 13, 14, 16cdleme50ltrn 31039 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( 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 ) )  e.  T )
18173com23 1159 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( 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 ) )  e.  T )
1915, 18eqeltrd 2478 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  `' F  e.  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   [_csb 3211   ifcif 3699   class class class wbr 4172    e. cmpt 4226   `'ccnv 4836   ` cfv 5413  (class class class)co 6040   iota_crio 6501   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583
This theorem is referenced by:  cdlemg1finvtrlemN  31057
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  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 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587
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