Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemk56 Unicode version

Theorem cdlemk56 31085
Description: Part of Lemma K of [Crawley] p. 118. Line 11, p. 120, "tau is in Delta" i.e.  U is a trace-preserving endormorphism. (Contributed by NM, 31-Jul-2013.)
Hypotheses
Ref Expression
cdlemk5.b  |-  B  =  ( Base `  K
)
cdlemk5.l  |-  .<_  =  ( le `  K )
cdlemk5.j  |-  .\/  =  ( join `  K )
cdlemk5.m  |-  ./\  =  ( meet `  K )
cdlemk5.a  |-  A  =  ( Atoms `  K )
cdlemk5.h  |-  H  =  ( LHyp `  K
)
cdlemk5.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk5.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk5.z  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
cdlemk5.y  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
cdlemk5.x  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
cdlemk5.u  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
cdlemk5.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
cdlemk56  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  U  e.  E )
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b, z,  ./\    .<_ , b   
z, g,  .<_    .\/ , b,
z    A, b, g, z    B, b, z    F, b, g, z    H, b, g, z    K, b, g, z    N, b, g, z    P, b, z    R, b, z    T, b, z    W, b, g, z    z, Y
Allowed substitution hints:    U( z, g, b)    E( z, g, b)    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk56
Dummy variables  f  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdlemk5.l . 2  |-  .<_  =  ( le `  K )
2 cdlemk5.h . 2  |-  H  =  ( LHyp `  K
)
3 cdlemk5.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
4 cdlemk5.r . 2  |-  R  =  ( ( trL `  K
) `  W )
5 cdlemk5.e . 2  |-  E  =  ( ( TEndo `  K
) `  W )
6 simp11 987 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 vex 2902 . . . . . 6  |-  g  e. 
_V
8 cdlemk5.x . . . . . . 7  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
9 riotaex 6489 . . . . . . 7  |-  ( iota_ z  e.  T A. b  e.  T  ( (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  g )
)  ->  ( z `  P )  =  Y ) )  e.  _V
108, 9eqeltri 2457 . . . . . 6  |-  X  e. 
_V
117, 10ifex 3740 . . . . 5  |-  if ( F  =  N , 
g ,  X )  e.  _V
1211rgenw 2716 . . . 4  |-  A. g  e.  T  if ( F  =  N , 
g ,  X )  e.  _V
13 cdlemk5.u . . . . 5  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
1413fnmpt 5511 . . . 4  |-  ( A. g  e.  T  if ( F  =  N ,  g ,  X
)  e.  _V  ->  U  Fn  T )
1512, 14mp1i 12 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  U  Fn  T )
16 simpl11 1032 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
17 simpl2 961 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T )  ->  ( R `  F )  =  ( R `  N ) )
18 simpl12 1033 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T )  ->  F  e.  T )
19 simpl13 1034 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T )  ->  N  e.  T )
20 simpr 448 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T )  ->  f  e.  T )
21 simpl3 962 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
22 cdlemk5.b . . . . . 6  |-  B  =  ( Base `  K
)
23 cdlemk5.j . . . . . 6  |-  .\/  =  ( join `  K )
24 cdlemk5.m . . . . . 6  |-  ./\  =  ( meet `  K )
25 cdlemk5.a . . . . . 6  |-  A  =  ( Atoms `  K )
26 cdlemk5.z . . . . . 6  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
27 cdlemk5.y . . . . . 6  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
2822, 1, 23, 24, 25, 2, 3, 4, 26, 27, 8, 13cdlemk35u 31078 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  f  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( U `  f
)  e.  T )
2916, 17, 18, 19, 20, 21, 28syl231anc 1204 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T )  ->  ( U `  f )  e.  T )
3029ralrimiva 2732 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  A. f  e.  T  ( U `  f )  e.  T
)
31 ffnfv 5833 . . 3  |-  ( U : T --> T  <->  ( U  Fn  T  /\  A. f  e.  T  ( U `  f )  e.  T
) )
3215, 30, 31sylanbrc 646 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  U : T
--> T )
33 simp11 987 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T  /\  h  e.  T )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T
) )
34 simp12 988 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T  /\  h  e.  T )  ->  ( R `  F )  =  ( R `  N ) )
35 simp2 958 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T  /\  h  e.  T )  ->  f  e.  T )
36 simp3 959 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T  /\  h  e.  T )  ->  h  e.  T )
37 simp13 989 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T  /\  h  e.  T )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3822, 1, 23, 24, 25, 2, 3, 4, 26, 27, 8, 13cdlemk55u 31080 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  f  e.  T  /\  h  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( U `  ( f  o.  h
) )  =  ( ( U `  f
)  o.  ( U `
 h ) ) )
3933, 34, 35, 36, 37, 38syl131anc 1197 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T  /\  h  e.  T )  ->  ( U `  ( f  o.  h ) )  =  ( ( U `  f )  o.  ( U `  h )
) )
40 simpl1 960 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T
) )
4122, 1, 23, 24, 25, 2, 3, 4, 26, 27, 8, 13cdlemk39u 31082 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  (
( R `  F
)  =  ( R `
 N )  /\  f  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  ( U `  f
) )  .<_  ( R `
 f ) )
4240, 17, 20, 21, 41syl121anc 1189 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  f  e.  T )  ->  ( R `  ( U `  f ) )  .<_  ( R `  f ) )
431, 2, 3, 4, 5, 6, 32, 39, 42istendod 30876 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T )  /\  ( R `  F )  =  ( R `  N )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  U  e.  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   _Vcvv 2899   ifcif 3682   class class class wbr 4153    e. cmpt 4207    _I cid 4434   `'ccnv 4817    |` cres 4820    o. ccom 4822    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   iota_crio 6478   Basecbs 13396   lecple 13463   joincjn 14328   meetcmee 14329   Atomscatm 29378   HLchlt 29465   LHypclh 30098   LTrncltrn 30215   trLctrl 30272   TEndoctendo 30866
This theorem is referenced by:  cdlemk56w  31087
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-map 6956  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-lplanes 29613  df-lvols 29614  df-lines 29615  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-lhyp 30102  df-laut 30103  df-ldil 30218  df-ltrn 30219  df-trl 30273  df-tendo 30869
  Copyright terms: Public domain W3C validator