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Theorem cdlemk56 30410
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
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 990 . 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 2766 . . . . . 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 6276 . . . . . . 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 2328 . . . . . 6  |-  X  e. 
_V
117, 10ifex 3597 . . . . 5  |-  if ( F  =  N , 
g ,  X )  e.  _V
1211rgenw 2585 . . . 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 5308 . . . 4  |-  ( A. g  e.  T  if ( F  =  N ,  g ,  X
)  e.  _V  ->  U  Fn  T )
1512, 14mp1i 13 . . 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 1035 . . . . 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 964 . . . . 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 1036 . . . . 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 1037 . . . . 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 449 . . . . 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 965 . . . . 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 30403 . . . . 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 1207 . . . 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 2601 . . 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 5619 . . 3  |-  ( U : T --> T  <->  ( U  Fn  T  /\  A. f  e.  T  ( U `  f )  e.  T
) )
3215, 30, 31sylanbrc 648 . 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 990 . . 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 991 . . 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 961 . . 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 962 . . 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 992 . . 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 30405 . . 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 1200 . 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 963 . . 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 30407 . . 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 1192 . 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 30201 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   _Vcvv 2763   ifcif 3539   class class class wbr 3997    e. cmpt 4051    _I cid 4276   `'ccnv 4660    |` cres 4663    o. ccom 4665    Fn wfn 4668   -->wf 4669   ` cfv 4673  (class class class)co 5792   iota_crio 6263   Basecbs 13111   lecple 13178   joincjn 14041   meetcmee 14042   Atomscatm 28703   HLchlt 28790   LHypclh 29423   LTrncltrn 29540   trLctrl 29597   TEndoctendo 30191
This theorem is referenced by:  cdlemk56w  30412
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-map 6742  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-llines 28937  df-lplanes 28938  df-lvols 28939  df-lines 28940  df-psubsp 28942  df-pmap 28943  df-padd 29235  df-lhyp 29427  df-laut 29428  df-ldil 29543  df-ltrn 29544  df-trl 29598  df-tendo 30194
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