Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme51finvtrN Unicode version

Theorem cdleme51finvtrN 30016
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
Dummy variables  a  b  c  u  v are mutually distinct and distinct from all other variables.
Allowed substitution groups:    D( t)    T( x, y, z, t, s)    E( t, s)    F( x, y, z, t, s)

Proof of Theorem cdleme51finvtrN
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 2286 . . 3  |-  ( ( Q  .\/  P ) 
./\  W )  =  ( ( Q  .\/  P )  ./\  W )
12 eqid 2286 . . 3  |-  ( ( v  .\/  ( ( Q  .\/  P ) 
./\  W ) ) 
./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )  =  ( ( v  .\/  ( ( Q  .\/  P )  ./\  W )
)  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
13 eqid 2286 . . 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 2286 . . 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 30014 . 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 30015 . . 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 2360 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 5    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1625    e. wcel 1687    =/= wne 2449   A.wral 2546   [_csb 3084   ifcif 3568   class class class wbr 4026    e. cmpt 4080   `'ccnv 4689   ` cfv 5223  (class class class)co 5821   iota_crio 6292   Basecbs 13144   lecple 13211   joincjn 14074   meetcmee 14075   Atomscatm 28722   HLchlt 28809   LHypclh 29442   LTrncltrn 29559
This theorem is referenced by:  cdlemg1finvtrlemN  30033
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3831  df-iun 3910  df-iin 3911  df-br 4027  df-opab 4081  df-mpt 4082  df-id 4310  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-1st 6085  df-2nd 6086  df-iota 6254  df-undef 6293  df-riota 6301  df-map 6771  df-poset 14076  df-plt 14088  df-lub 14104  df-glb 14105  df-join 14106  df-meet 14107  df-p0 14141  df-p1 14142  df-lat 14148  df-clat 14210  df-oposet 28635  df-ol 28637  df-oml 28638  df-covers 28725  df-ats 28726  df-atl 28757  df-cvlat 28781  df-hlat 28810  df-llines 28956  df-lplanes 28957  df-lvols 28958  df-lines 28959  df-psubsp 28961  df-pmap 28962  df-padd 29254  df-lhyp 29446  df-laut 29447  df-ldil 29562  df-ltrn 29563
  Copyright terms: Public domain W3C validator