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Theorem cdlemk56 31782
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 985 . 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 2804 . . . . . 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 6324 . . . . . . 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 2366 . . . . . 6  |-  X  e. 
_V
117, 10ifex 3636 . . . . 5  |-  if ( F  =  N , 
g ,  X )  e.  _V
1211rgenw 2623 . . . 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 5386 . . . 4  |-  ( A. g  e.  T  if ( F  =  N ,  g ,  X
)  e.  _V  ->  U  Fn  T )
1512, 14mp1i 11 . . 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 1030 . . . . 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 959 . . . . 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 1031 . . . . 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 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 )  ->  N  e.  T )
20 simpr 447 . . . . 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 960 . . . . 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 31775 . . . . 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 1202 . . . 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 2639 . . 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 5701 . . 3  |-  ( U : T --> T  <->  ( U  Fn  T  /\  A. f  e.  T  ( U `  f )  e.  T
) )
3215, 30, 31sylanbrc 645 . 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 985 . . 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 986 . . 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 956 . . 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 957 . . 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 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 )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3822, 1, 23, 24, 25, 2, 3, 4, 26, 27, 8, 13cdlemk55u 31777 . . 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 1195 . 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 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 )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  N  e.  T
) )
4122, 1, 23, 24, 25, 2, 3, 4, 26, 27, 8, 13cdlemk39u 31779 . . 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 1187 . 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 31573 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801   ifcif 3578   class class class wbr 4039    e. cmpt 4093    _I cid 4320   `'ccnv 4704    |` cres 4707    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   TEndoctendo 31563
This theorem is referenced by:  cdlemk56w  31784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566
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